Mirror symmetry for log Calabi-Yau surfaces I

Mark Gross; Paul Hacking; Sean Keel

doi:10.1007/s10240-015-0073-1

Mirror symmetry for log Calabi-Yau surfaces I

Gross, Mark; Hacking, Paul; Keel, Sean 2015-03-25 00:00:00 by MARK GROSS, PAUL HACKING, and SEAN KEEL ABSTRACT We give a canonical synthetic construction of the mirror family to pairs (Y, D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule deﬁned in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y, D). CONTENTS Introduction ........................................................ 65 0.1. The main theorems ................................................ 65 0.2. The symplectic heuristic ............................................. 68 0.3. Outline of the proof ................................................ 70 0.4. Further directions ................................................. 73 1. Basics .......................................................... 75 1.1. Looijenga pairs . ................................................. 75 1.2. Tropical Looijenga pairs . ............................................ 77 1.3. The Mumford degeneration and Givental’s construction ............................ 81 2. Modiﬁed Mumford degenerations ........................................... 84 2.1. The uncorrected degeneration .......................................... 85 2.2. Scattering diagrams on B . ............................................ 90 2.3. Broken lines .................................................... 93 2.4. The algebra structure ............................................... 101 3. The canonical scattering diagram ........................................... 102 3.1. Deﬁnition ..................................................... 102 3.2. Consistency: overview of the proof ........................................ 107 3.3. Consistency: reduction to the Gross-Siebert locus ................................ 109 3.4. Step V: the proof of Theorem 3.25 and the connection with [GPS09] .................... 124 4. Smoothness: around the Gross-Siebert locus ..................................... 127 5. The relative torus .................................................... 133 6. Extending the family over boundary strata ...................................... 134 6.1. Theorem 0.2 in the case that (Y, D) has a toric model ............................. 135 6.2. Proof of Theorems 0.1 and 0.2 in general .................................... 139 6.3. The case that (Y, D) is positive .......................................... 141 7. Looijenga’s conjecture ................................................. 145 7.1. Duality of cusp singularities ............................................ 145 7.2. Cusp family .................................................... 147 7.3. Thickening of the cusp family .......................................... 152 7.4. Smoothness .................................................... 163 Acknowledgements ..................................................... 166 References ......................................................... 166 Introduction 0.1. The main theorems. — Throughout the paper (Y, D) with D = D +···+ D 1 n will denote a smooth rational projective surface over an algebraically closed ﬁeld k of characteristic zero, with D∈|−K | a singular nodal curve. The divisor D is necessarily either an irreducible rational nodal curve, or a cycle of n ≥ 2 smooth rational curves. We DOI 10.1007/s10240-015-0073-1 66 MARK GROSS, PAUL HACKING, AND SEAN KEEL call (Y, D) a Looijenga pair for, as far as we know, their rich geometry was ﬁrst investigated in [L81]. We cyclically order the components of D and take indices modulo n.Byas- sumption there is a holomorphic symplectic 2-form , unique up to scaling, on Y \ D, with simple poles along D, and thus U := Y \ D is a log Calabi-Yau surface. Our main result is a canonical synthetic construction of the mirror family to such a pair. The construction gives an embedded smoothing of the n-vertex V ⊂ A ,deﬁned as, for n ≥ 3, the n-cycle of coordinate planes in A : 2 2 2 n V := A ∪ A ∪···∪ A ⊂ A . x ,x x ,x x ,x x ,...,x 1 2 2 3 n 1 1 n (See (1.7)and (1.8) for the deﬁnition of V and V .) This family is in general parameter- 1 2 ized roughly by the formal completion of the afﬁne toric variety Spec k[NE(Y)] along the union of toric boundary strata corresponding to contractions f : Y → Y. Here NE(Y) de- notes the monoid NE(Y) ∩ A (Y, Z) where NE(Y) ⊂ A (Y, R) is the cone generated R 1 R 1 by effective curve classes. This is just an approximate statement of our result, as NE(Y) is not in general ﬁnitely generated. More precisely, ﬁx (Y, D),D = D + ··· + D as above. Let B (Z) be the set 1 n 0 of pairs (E, n) where E is a prime divisor on some blowup of Y along which has a pole and n is a positive integer. Set B(Z) := B (Z)∪{0}. Later we will describe this set as the set of integer points in a natural integral afﬁne manifold, the dual intersection complex, or tropicalization, of the pair (Y, D).Let v ∈ B(Z) be the pair (D , 1). Choose i i σ ⊂ A (Y, R) a strictly convex rational polyhedral cone containing NE(Y) ,let P := P 1 R σ ∩ A (Y, Z) be the associated monoid, and set R := k[P] to be the associated k-algebra. P 1 For each monomial ideal I ⊂ R, consider the free R := R/I-module (0.1)A := R · ϑ . I I q q∈B(Z) D n Let m ⊂ R denote the maximal monomial ideal. Let T := G be the torus with charac- ter group χ(T ) having basis e indexed by the components D ⊂ D. There is a homo- D i D gp D morphism T → Spec k[P ] induced by C → (C · D )e ,so T acts on Spec R . i D I Theorem 0.1. —Let I ⊂ R be a monomial ideal with I = m. In Sections 2 and 3,we construct a ﬁnitely generated R -algebra structure on A , determined by relative Gromov-Witten invariants I I of (Y, D) counting rational curves meeting D in a single point. In Section 5,weconstruct a T action on Spec A . This induces a ﬂat T -equivariant map f : X := Spec A → Spec R I I I with closed ﬁbre V . By taking the limit over all such I, this yields a formal ﬂat family f : X → S := Spf R, m m where R is the completion of R with respect to the ideal m. The generic ﬁbre of f is smooth in the sense of Deﬁnition 4.2,so f is a formal smoothing of V . n MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 67 We use the notation ϑ for generators of our algebra, as the construction ﬁts into a more general family of constructions which includes, as a special case, theta functions on Abelian varieties. The history of such functions is as follows. Tyurin conjectured the existence of canonical theta functions (i.e., a basis of global sections) for polarized K3 surfaces, see [Ty99]. In 2007, discussions of the ﬁrst author with Abouzaid and Siebert involving a tropicalization of the Fukaya category gave a stronger hint as to the existence of theta functions on arbitrary degenerations of Calabi-Yau manifolds in the context of the Gross-Siebert program. In particular, these discussions led to what is now understood to be a variant of the notion of broken line. The latter notion was introduced in [G09]. These were initially used to construct canonical perturbations of the Landau-Ginzburg potential for P . Broken lines were then used for constructing mirror Landau-Ginzburg potentials for varieties with effective anti- canonical divisor in the setting of the Gross-Siebert program by Carl, Pumperla and Siebert in [CPS]. The authors show that the mirrors to such varieties as constructed in [GS07] carry a canonical Landau-Ginzburg potential obtained by using broken lines to lift monomial functions on the central ﬁbre of a toric degeneration to the toric degenera- tion. Simultaneously, we used these same lifts to allow an extension of the construction developed by Gross and Siebert to prove the above main theorem. The main innovations we have introduced here are that we use theta functions to provide partial compactiﬁ- cations of certain canonically constructed deformations, and that these canonically con- structed deformations, along with the theta functions, can be constructed relying only on the Gromov-Witten theory of (Y, D). The key point is that it is easy to build deformations of the punctured n-vertex V := V \{0}, but it is difﬁcult to extend these to deformations of V . This is effectively done by using theta functions to embed a suitably chosen defor- mation of V in afﬁne space, where the closure may then be taken. This extension would be impossible without the existence of theta functions. This result can be viewed as log analogs of Tyurin’s conjecture. In work in progress we apply similar ideas to obtain Tyurin’s conjecture in the K3 case as well, and construct canonical bases for cluster algebras, to cite two other generalizations. These are large topics and will be expanded on elsewhere. See also [GSTheta] for more motivation from mirror symmetry, and upcoming papers [GHKS]and [K3]. Continuing with (Y, D), P and R as above, our second main theorem is: Theorem 0.2. — There is a unique smallest radical monomial ideal J ⊂ R with the following properties: (1) For every monomial ideal I with J ⊂ I there is a ﬁnitely generated R -algebra structure on N N A compatible with the R -algebra structure on A of Theorem 0.1 for all N > 0. I I+m I+m (2) If the intersection matrix (D · D ) is not negative semi-deﬁnite then J = 0. In general, the i j zero locus V(J) ⊂ Spec R contains the union of the closed toric strata corresponding to faces F of σ such that there exists an i such that [D] ∈ F. P i 68 MARK GROSS, PAUL HACKING, AND SEAN KEEL (3) Let R denote the J-adic completion of R and S := Spf R the associated formal scheme. The algebras A determine a canonical T -equivariant formal ﬂat family of afﬁne surfaces f : X → S J J max(n,3) with ﬁbre V over 0.The ϑ determine a canonical embedding X ⊂ A × S . n q J J Remark 0.3. — When NE(Y) ⊂ P ⊂ P ⊂ A (Y),then J ⊂ J and the formal family X for P comes from the family for P by base-change. In this sense the family is indepen- dent of the choice of P. Remark 0.4. — Note that in the case that the intersection matrix (D · D ) is not i j negative semi-deﬁnite (which includes the case that D supports an ample divisor), The- orem 0.2 tells us that our construction gives a family over Spec R, so in particular the construction is algebraic. In this paper, we will not address the question as to in what sense our construction can be proved to be a mirror family. We expect, however, that our families constructed by the above theorems are mirror to U = Y\ D in the sense of homological mirror symmetry in the case k = C. Further justiﬁcation for our construction yielding the mirror family comes from the heuristic description of the construction in terms of symplectic geometry as discussed below. The third main result of this paper is an application of our general construction, following from a more detailed analysis of the case where the matrix (D · D ) is negative i j deﬁnite: Theorem 0.5 (Looijenga’s conjecture). — A 2-dimensional cusp singularity is smoothable if and only if the exceptional cycle of the dual cusp occurs as an anti-canonical cycle on a smooth projective rational surface. This was conjectured by Looijenga in [L81], where he also proved the forward implication. Partial results were obtained in [FM83]and [FP84]. 0.2. The symplectic heuristic. — Much of what we do in this paper, following the philosophy of the Gross–Siebert program, is to tropicalize the SYZ picture [SYZ96]. Thus it is helpful to review informally this picture in the context of mirrors to Looijenga pairs (Y, D). The SYZ picture will be a heuristic philosophical guide, and hence we make no effort to be rigorous. Here we follow the exposition from [A07] concerning SYZ on the complement of an anti-canonical divisor, itself a generalization of ideas of Cho and Oh for interpreting the Landau-Ginzburg mirror of a toric variety in terms of counting Maslov index two holomorphic disks [CO06]. For the most part we follow Auroux’s notation, except that we use Y instead of his X, and our X is his M. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 69 We ﬁx a Kähler form ω on Y, and a nowhere vanishing holomorphic 2-form on U := Y\ D. Now suppose we have a ﬁbration f : U → B by special Lagrangian 2-tori (i.e., a ﬁbre L of f satisﬁes Im | = ω| = 0). Then the SYZ mirror X of (U,ω) is the L L dual torus ﬁbration f : X → B. This can be thought of as a moduli space of pairs (L,∇) consisting of a special Lagrangian ﬁbre L of f equipped with a unitary connection ∇ modulo gauge equivalence, or equivalently a holonomy map hol : H (L, Z) → U(1) ⊂ ∇ 1 C . The complex structure on X is subtle, speciﬁed by so-called instanton corrections. In this picture we can deﬁne local holomorphic functions on X associated to a basis of H (Y, L, Z) (in a neighbourhood of a ﬁbre of f corresponding to a non-singular ﬁbre Lof f ) as follows. For A ∈ H (Y, L, Z) deﬁne A ∗ (0.2) z := exp −2π ω hol (∂ A) : X → C . By choosing a splitting of H (Y, L, Z) H (L, Z) we can pick out local coordinates on 2 1 X which deﬁne a complex structure. See [A07], Lemma 2.7. Note that as the ﬁbre L varies, the relative homology group H (Y, L, Z) forms a local system over B ⊂ B, where 2 0 B is the subset of points with non-singular ﬁbres. This local system has monodromy, and as a consequence, the functions z are only well-deﬁned locally. However, there are also well-deﬁned global functions ϑ ,...,ϑ on X. These are 1 n deﬁned locally in neighbourhoods of ﬁbres of f corresponding to ﬁbres of f not bounding holomorphic disks contained in U, via a (rough) expression (0.3) ϑ = n z , i β β∈H (Y,L,Z) where n is a count of so-called Maslov index two disks with boundary on L representing the class β and intersecting D transversally in one point lying in D . (We note that in our setting the Maslov index μ of a holomorphic disk f : → Y with boundary lying on a special Lagrangian torus L ⊂ Yis given by μ = 2deg f D. See [A07], Lemma 3.1.) In the case that D is ample, there are, for generic L, only ﬁnitely many such disks; it is not known how to treat the general case in this symplectic setting. For ϑ to make sense the moduli space of Maslov index 2 disks with boundary on L must deform smoothly with the Lagrangian L. This fails for Lagrangians that bound holomorphic disks contained in U (Maslov index zero disks). This is a real codimension one condition on L, and thus deﬁnes canonical walls in the afﬁne manifold B. When we cross the wall the ϑ are discontinuous. But the discontinuity is corrected by a holomor- phic change of variable in the local coordinates z , according to [A07], Proposition 3.9: [∂β]·[∂α] β β α (0.4) z → z · h z where here α ∈ H (Y, L , Z) represents the class of the Maslov index zero disk with 2 0 boundary on L a Lagrangian ﬁbre over a point on the wall, and h(q) is a generating func- tion counting such holomorphic disks. Thus we can deﬁne a new complex manifold, with 70 MARK GROSS, PAUL HACKING, AND SEAN KEEL the same local coordinates, by composing the obvious gluing induced by identiﬁcations of ﬁbres of the local system on B with ﬁbres H (Y, L, Z) with the automorphism (0.4). 0 2 These regluings are the instanton corrections, and the modiﬁed manifold X should be the mirror. By construction it comes with canonical global holomorphic functions ϑ . In particular, the sum W = ϑ is a well-deﬁned global function, the Landau–Ginzburg potential. 0.3. Outline of the proof. — We now outline how we realise the symplectic SYZ heuristic in terms of algebraic geometry. There are three principal issues to consider: • What information about a putative SYZ ﬁbration can be seen inside algebraic geometry? • What is the analogue of a Maslov index two disk in algebraic geometry? • How do we obtain the mirror by gluing together varieties? The philosophy for dealing with the ﬁrst and third issues was developed by Gross and Siebert in [GS07]. For the ﬁrst item, while we cannot build an SYZ ﬁbration f : U → B in general, we can roughly describe B as a combinatorial object. Given the Looijenga pair (Y, D), we build a space B homeomorphic to R along with a decomposition of B into cones. We construct (B, ) as the dual intersection complex of (Y, D).For each double point of D, we take a copy of the ﬁrst quadrant in R , with the axes labelled by the two irreducible components of D (assuming D is not irreducible) passing through the double point. We then identify edges of these cones if they are labelled with the same irreducible component of D. We thus get a topological space abstractly homeomorphic to R subdivided into cones. This is (B, ).InSection 1.2, we show how we can put an additional structure on B, namely the structure of an afﬁne manifold with singularities. Indeed, we can give B := B\{0} a system of coordinate charts whose transition maps are integral afﬁne linear transformations. The afﬁne structure does not extend across the origin unless (Y, D) is in fact a toric pair, in which case we recover the fan deﬁning Y. The manifold B can be viewed as the base of the SYZ ﬁbration “seen from a great distance.” In general the base of an SYZ ﬁbration has the structure of an afﬁne manifold with singularities. Singular ﬁbres of the ﬁbration occur over the singular points. One would expect f : U → B to have a number of singular ﬁbres in general, hence B will have a number of singular points. So the above construction moves all these singular points to the origin. Next, let us consider the third item. Fixing (Y, D) with D = D +··· + D ,let 1 n P ⊂ A (Y, Z) be a ﬁnitely generated monoid containing the classes of all effective curves on Y, obtained by choosing a strictly convex rational polyhedral cone σ ⊂ A (Y, R) P 1 containing the Mori cone. Let m be the maximal monomial ideal in the ring k[P],Ia monomial ideal with radical m,and let R = k[P]/I. We will describe the basic pieces we will glue together to describe a scheme over S := Spec R whose special ﬁbre is V := V \{0}. Assume that the components D are I I n i n MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 71 numbered in cyclic order, with indices taken modulo n.Wecan deﬁneanopencover of V by taking sets, for 1 ≤ i ≤ n, U = V(X X ) ⊂ A × (G ) . i i−1 i+1 m X X ,X i i−1 i+1 Note as subsets of V , they are disjoint except for U := U ∩ U = (G ) . i,i+1 i i+1 m X ,X i i+1 In V they are glued in the obvious way, i.e., via the canonical inclusions U ={X = 0}⊂ U , U ={X = 0}⊂ U . i,i+1 i+1 i i,i+1 i i+1 A deformation of V over S is obtained by gluing thickenings of the U I i −D [D ] i 2 (0.5)U := V X X − z X ⊂ S × A × (G ) i,I i−1 i+1 I m X i X ,X i i−1 i+1 [D ] where z ∈ k[P] is the corresponding monomial. The overlaps are relative tori, U = S × G , and the gluings are the obvious ones. The details are given in Sec- i,i+1,I I tion 2.1. This gluing gives a ﬂat family X → S , which can be viewed as being analogous to the naive complex structure on the mirror described as the moduli of smooth special Lagrangian ﬁbres with U(1) connection. There is no reason in general to believe that X → S can be extended to a ﬂat deformation X → S of V . The reason is that such an X should be an afﬁne scheme, I I n I and hence have many functions, while X as constructed tends to have few functions. The only case where X extends to give a deformation of V is when (Y, D) is a toric pair. In this case, we recover an inﬁnitesimal version of Givental’s mirror family, which then easily extends to Givental’s mirror construction. We review this case in Section 1.3. To rectify this problem, we need to translate the instanton corrections of the sym- plectic heuristic. We do this using the notion of scattering diagram, here a variant of similar notions introduced in [KS06]and [GS07]. For us, a scattering diagram D will be a collection of pairs (d, f ) where d is a ray emanating from the origin of B with rational slope, and f is a kind of function attached to the ray. Any scattering diagram will dictate how to modify both the deﬁnition of the open sets U and the gluings of U with U . The precise details of this modiﬁcation i,I i,I i+1,I are given in Section 2.2. Brieﬂy, the rays deﬁne automorphisms of the open sets U i,i+1,I analogous to (0.4), and are used to modify the gluing. While any scattering diagram can be used to obtain a modiﬁed ﬂat deformation X , we need to choose D correctly to have a chance of extending this deformation I,D to V . The symplectic heuristic can be used to motivate the choice of the canonical scattering diagram. The functions f chosen are generating functions for certain Gromov-Witten invariants, intuitively counting ﬁnite maps A → U. Heuristically, each holomorphic disk contributing can be approximated by a proper rational curve meeting D in a single point. 72 MARK GROSS, PAUL HACKING, AND SEAN KEEL Thus the canonical scattering diagram encodes the chamber structure seen in the symplectic heuristic. But there still remains the question of extending X to a ﬂat de- I,D formation of V . To do so, we need to construct enough functions on X . This is where I,D the concept of theta function comes in. The symplectic heuristic suggests that there should be a canonical choice of holomorphic functions on X arising from a count of Maslov I,D index two holomorphic disks. Rather than trying to ﬁnd an algebro-geometric analogue of a Maslov index two holomorphic disk, one instead deﬁnes the counts using tropical ge- ometry. In particular, we use the notion of broken line, introduced in [G09] and developed further by [CPS] simultaneously with this work, to provide the count. A broken line is es- sentially a tropical analogue of a Maslov index two disk. They are piecewise linear paths which only bend when they cross rays of the scattering diagram D, in ways prescribed by the functions attached to the rays. For any point p ∈ B with integral coordinates, we can use a count of broken lines to deﬁne a function on U for any i. This procedure is described in Section 2.3. Since i,I this procedure is dependent on the scattering diagram D, we can then ask whether these functions on the various U glue. We say D is consistent if they always glue. If these i,I functions do glue, then we call the resulting global function on X a theta function, writing I,D it as ϑ . The bulk of the argument in this paper occurs in Section 3, where we prove that the canonical scattering diagram described above is in fact consistent. This argument is rather involved, so we leave it to Section 3.2 to give an overview of the full argument for consistency. Crucial to the argument is a reduction to methods of [CPS] using the main results of [GPS09]. Once consistency is proved, this gives global functions ϑ on X for each p ∈ I,D B with integral coordinates. Let v denote the ﬁrst integral point along the ray of corresponding to the divisor D , and write ϑ := ϑ . Then we can use the functions i i v ϑ ,...,ϑ to embed (in the case that n ≥ 3) X in A × S . Taking the closure of the 1 n I I,D image gives the desired deformation X → S of V . I I n This construction essentially proves the ﬁrst main theorem, Theorem 0.1.The statement about the scheme-theoretic singular locus of f is dealt with in Section 4.There we again make a connection with the techniques of [GS07]. The crucial point is to show the singularity 0 ∈ V is formally smoothed, and for this, we need to work in a family where we have a local model for the behaviour near 0, much as Gross and Siebert have in [GS07]. More work is required for Theorem 0.2. We need to show that the construction above, which really only produces a family over the completion of Spec k[P] at the zero- dimensional torus orbit of this scheme, extends across completions along larger strata. Since the coordinate rings of the families constructed above are generated by theta func- tions, we proceed by studying the products of theta functions. In general, one expects the product of two theta functions to be a formal series of theta functions. However, in many cases one can control the terms sufﬁciently in these products to obtain the desired exten- MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 73 sions. This relies on a tropical interpretation of the product of theta functions, given in Section 2.4, as well as the existence of a torus action on our families, given in Section 5. This torus action only exists because of the canonical nature of our scattering diagrams. Complete details for the arguments are given in the last section, Section 6. Turning to Theorem 0.5, the main point is that Looijenga’s conjecture is really a form of mirror symmetry. We start with a pair (Y, D) such that the intersection matrix (D · D ) is negative deﬁnite. Thus D can be contracted analytically to give a cusp sin- i j gularity p ∈ Y . (By deﬁnition, a cusp singularity is a surface singularity whose minimal resolution is a cycle of rational curves.) For the sake of exposition, assume this contraction is algebraic, so that there is a divisor L on Y which is the pull-back of an ample divisor gp on Y . We choose the monoid P so L ∩ Pis a face P of P, with P generated by the bdy bdy classes [D ],...,[D ]. The main goal is to extend our construction to a formal neigh- 1 n bourhood of Spec k[P ]⊂ Spec k[P]. The problem is that Theorem 0.2 explicitly does bdy not apply in this case. The main difﬁculty is that the charts (0.5)overlap toomuchwhen [D ] all the z are invertible (in fact the ﬁbres over such points in Spec k[P ] coincide under bdy the natural gluing maps). There is no way to glue these charts compatibly. However, this can be done after shrinking these charts to analytic open subsets and working over an analytic open neighborhood of the zero-dimensional stratum of Spec k[P ].Herewe bdy work of course with k = C only. In doing so, we ﬁnd over a general point of Spec k[P ] the dual cusp singular- bdy ity to p ∈ Y . Thus we see that our mirror symmetry construction naturally produces the dual cusp. We then would like to extend the family constructed over thickenings of Spec k[P ]. We use the same techniques as those used to prove Theorem 0.1.However, bdy the construction of theta functions is considerably more delicate. In general, theta func- tions are described as a sum of monomials associated to broken lines. In the situation of Theorem 0.1, these sums are always ﬁnite. However, in the current situation, they are al- ways inﬁnite. Thus there are serious convergence issues, and this makes the proof rather technical. A delicate analysis of the combinatorics of broken lines is necessary to prove convergence. Once convergence is shown, we then argue that the formal family produced actu- ally gives a smoothing of the cusp singularity. This follows from the fact proved in Theo- rem 0.1 that we already have a smoothing of the n-vertex in a formal neighbourhood of the zero-dimensional stratum, but again the argument is slightly delicate. All details are giveninSection 7. 0.4. Further directions. — Here we will brieﬂy indicate the results of further study of our mirror construction, to be given in sequal paper [GHKII], as well as connections with other recent work. There are three broad classes of behaviour for our construction, depending on the properties of the intersection matrix (D · D ): the matrix can be negative deﬁnite, i j negative semi-deﬁnite but not negative deﬁnite, or not negative semi-deﬁnite. The ﬁrst 74 MARK GROSS, PAUL HACKING, AND SEAN KEEL case is analyzed here in detail to prove Theorem 0.5. We will discuss the third case in the sequel paper. We call the case that the intersection matrix is not negative semi-deﬁnite the positive case. It holds if and only if U is the minimal resolution of an afﬁne surface, see Lemma 6.9. In this case, the cone NE(Y) is rational polyhedral, so we may take P = NE(Y). Further- more, the ideal J of Theorem 0.2 equals 0. Thus our construction deﬁnes an algebraic family over Spec k[NE(Y)], with smooth generic ﬁbre. We will show in Part II that the restriction of this family to the structure torus X → T := Pic(Y) ⊗ G = Spec k A (Y) ⊂ Spec k NE(Y) Y m 1 is close to a universal family of deformations of U = Y \ D. More precisely, we will show independently of the positivity of the intersection matrix that our formal family has a simple and canonical (ﬁbrewise) compactiﬁcation to aformalfamily (Z, D) of Looijenga pairs (with X = Z \ D), equivariant for the action D D of T ⊂ T , the subtorus generated by the components of D. The theta functions are T eigenfunctions, see Section 5. In the positive case this extends naturally over all of Spec k[NE(Y)], and its restric- tion (Z, D) → T comes with a trivialization of the boundary D = D × T realizing it Y ∗ Y as the universal family of Looijenga pairs (Z, D ) deformation equivalent to (Y, D) to- gether with an isomorphism D → D constructed in [GHK12]. In particular, choosing Z ∗ such an isomorphism D → D for our original pair (Y, D) canonically identiﬁes it with a ﬁbre of the family (Z, D)/T . More importantly, the restrictions of the theta functions ϑ to U ⊂ X endow the afﬁne surface U = Y \ D with canonical functions. We give a modular interpretation of the quotient of Z \ D → T by T as the universal deforma- tion of U (this shows in particular the quotient depends only on U, e.g., is independent of the choice of compactiﬁcation U ⊂ Y), and give a unique geometric characterisation of the theta function basis of H (U, O ). The fact that (Y, D) appears as a ﬁbre is perhaps a bit surprising as, after all, we set out to construct the mirror and have obtained the original surface back. Note however that dual Lagrangian torus ﬁbrations in dimension 2 are topologically equivalent by Poincaré duality, so this is consistent with the SYZ formulation of mirror symmetry. To illustrate, in Example 6.12 we explicitly compute the theta functions in the case (Y, D) is the del Pezzo of degree 5 together with a cycle of 5 (−1)-curves. In Ex- ample 6.13, we give the expression in the case of a triangle of lines on a cubic surface, deferring in this case the proof until Part II. In each of these cases there is a characterisa- tion of the ϑ in terms of classical geometry. In a different direction, in [GHKK], along with M. Kontsevich, we extend many of the methods introduced in this paper to prove a number of signiﬁcant conjectures about cluster varieties. In particular, the technology of theta functions leads to a proof of positivity of the Laurent phenomenon, and a proof of the Fock-Goncharov dual basis MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 75 conjecture for a broad class of cluster varieties. The latter can be viewed as a generaliza- tion of the construction of theta functions on Y \ D in the positive case, described above. In fact, in the case of cluster varieties associated to a skew-symmetric matrix of rank two, the Fock-Goncharov X variety ﬁbres over a torus with ﬁbres being interiors of Looijenga pairs. This is described in detail in [GHK13]. In this case, the general construction of theta functions in [GHKK] coincides with the ones constructed here. Let us end with some mild speculation in all dimensions suggested by the above discussion. By a Looijenga pair we mean a dlt pair (Y, D) (e.g., a simple normal crossings pair) with K + D trivial and (Y, D) having a zero-dimensional log canonical center. (In the simple normal crossing case, this means there is an intersection point of dim Y different components of D.) By a log Calabi-Yau with maximal boundary we mean a variety U which can be realized as the interior Y \ D of a Looijenga pair. See Section 1 of [GHK13] for background on these notions. We expect that many of the results in this paper will extend to Looijenga pairs of all dimensions. This generalization will require the further development of the technology of logarithmic Gromow-Witten invariants, [GS11],[AC11]. We should obtain in complete generality a mirror family X → Spf k[P] for suitable monoids P. Furthermore, one would expect in the case that U = Y \ Dis afﬁne that this family extends to X → Spec k[P]. Using the two-dimensional case as a guide, the general ﬁbre of X → Spec k[P] should itself be the interior of a Looijenga ¯ ¯ pair (X, E),with X \ E afﬁne by construction. Thus we can then repeat the process to obtain a family X → Spec k[P ], and it would be expected, as taking mirror twice should return to where we started, that X → Spec k[P ] contains a ﬁbre isomorphic to U. The family X carries our canonically deﬁned theta functions, indexed by tropical points of the mirror. This leads us to propose: Conjecture 0.6. —Let U be an afﬁne log Calabi-Yau variety with maximal boundary. Then H (U, O ) has a canonical basis of theta functions indexed by tropical points of the mirror. The structure constants for multiplication of theta functions can be described combinatorially in terms of broken lines. Versions of this conjecture have been proven for cluster varieties in many cases in [GHKK]. 1. Basics 1.1. Looijenga pairs. Deﬁnition 1.1. —A Looijenga pair (Y, D) is a smooth rational projective surface Y together with a reduced nodal curve D∈|−K | with at least one singular point. Note that for a Looijenga pair, p (D) = 1 by adjunction. Since H (Y, O ) = 0 a Y by rationality of Y, D is connected. Applying adjunction to each irreducible component 76 MARK GROSS, PAUL HACKING, AND SEAN KEEL of D, one sees easily that D is either an irreducible genus one curve with a single node, or a cycle of smooth rational curves. We will always write D = D +···+ D , with a cyclic 1 n ordering of the irreducible components, and take the indices modulo n. We will need a few basic facts about Looijenga pairs, which we collect here. Deﬁnition 1.2. —Let (Y, D) be a Looijenga pair. ˜ ˜ (1) A toric blow-up of (Y, D) is a birational morphism π : Y → Y such that if D is the −1 ˜ ˜ ˜ reduced scheme structure on π (D), then (Y, D) is a Looijenga pair. In particular, Y is smooth. ¯ ¯ (2) A toric model of (Y, D) is a birational morphism (Y, D) → (Y, D) to a smooth toric ¯ ¯ ¯ surface Y with its toric boundary D such that D → D is an isomorphism. Note that if π : Y → Y is the blow-up of a node of D, then π is a toric blow-up. ˜ ˜ Proposition 1.3. —Given (Y, D) there exists a toric blowup (Y, D) which has a toric model ˜ ˜ ¯ ¯ (Y, D) → (Y, D). Proof. — First observe: (1) Let p : Y → Y be the blowdown of a (−1)-curve not contained in D, and D := p (D) ⊂ Y . If the proposition holds for (Y , D ) then it holds for (Y, D). (2) Let Y → Y be the blowup at a node of D, and D ⊂ Y the reduced inverse image of D. The proposition holds for (Y , D ) if and only if it holds for (Y, D). By using (1) and (2) repeatedly we may assume Y is minimal, and thus is either a ruled surface or is P . In the latter case, by blowing up a node of D we reduce to the ruled case. So we have q : Y → P a ruling. We next consider the number of components of D which are ﬁbres of q. There cannot be more than two such components, for otherwise D cannot be a cycle. If there are precisely two such components, then D necessarily has precisely four components, and it is then easy to check that D is the toric boundary of Y, for a suitable choice of toric structure on Y. In this case the proposition obviously holds. Otherwise let D ⊂ D be the union of components not contained in ﬁbres. If D has a node, then we can blowup the node, blowdown the strict transform of the ﬁbre through the node, increasing the number of components of D contained in ﬁbres. After carrying out this procedure for each node of D ,weare then in oneoftwo cases. Case I. D has two components contained in ﬁbres, and then we are done. Case II. D consists of a ﬁbre f and a non-singular irreducible two-section D of q. Note that since D + f ∼−K and Y is isomorphic to the Hirzebruch surface F for Y e some e, we can write Pic Y = ZC ⊕ Zf ,with C =−e and −K = 2C + (e + 2)f .Thus 0 Y 0 D ∼ 2C + (e + 1)f and C · D =−e + 1. Since C is not contained in D , e = 0or1. 0 0 0 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 77 If e = 0, then there is a second ruling q : Y → P ,with D and f sections of this ruling. In this case, we follow the same procedure as above of blowing up nodes for this new ruling, arriving in Case I. If e = 1, then C is disjoint from D . Blowing down C ,weobtain P , and can then 0 0 blowup one of the nodes of the image of D ∪ f . Using this new ruled surface, we can again blowup a node and ﬁnd ourselves back in Case I. 1.2. Tropical Looijenga pairs. — We explain how to tropicalize a Looijenga pair, ﬁrst recalling the following basic deﬁnition. Fix a lattice M Z . In what follows, we will always use the notation M = M⊗ R,N = Hom (M, Z) and N = N⊗ R.Wedenote R Z Z R Z by Aff(M) the group of afﬁne linear transformations of the lattice M. Recall the following deﬁnitions from [GS06]. Deﬁnition 1.4. —An integral afﬁne manifold B is a (real) manifold B with an atlas of −1 charts {ψ : U → M } such that ψ ◦ ψ ∈ Aff(M) for all i, j. i i R i An integral afﬁne manifold with singularities B is a (real) manifold B with an open subset B ⊂ B which carries the structure of an integral afﬁne manifold, and such that := B \ B ,the 0 0 singular locus of B, is a locally ﬁnite union of locally closed submanifolds of codimension at least two. If B is an integral afﬁne manifold with singularities, there is a local system on B consisting B 0 of ﬂat integral vector ﬁelds: if y ,..., y are local integral afﬁne coordinates, then is locally given 1 n B by linear combinations of the vector ﬁelds ∂/∂ y ,...,∂/∂ y .If B is clear from context, we drop the 1 n subscript B. Similarly, is the dual local system, locally generated by dy ,..., dy . B 1 n We will be primarily interested in dim B = 2 in this paper, in which case will consist, in all our examples, of a ﬁnite number of points. All integral afﬁne manifolds we encounter will in fact be linear, in the sense that the coordinate transformations are in fact linear rather than just afﬁne linear. We associate to a Looijenga pair (Y, D) apair (B, ), where B is homeomorphic to R and has the structure of integral afﬁne manifold with one singularity at the origin, and is a decomposition of B into cones. We call (B, ) the tropicalization of (Y, D),and the fan of (Y, D). The idea is that we pretend that (Y, D) is toric and we try to build the associated fan. More precisely, the construction is as follows. For each node p := D ∩ D of D we take a rank two lattice M with basis i,i+1 i i+1 i,i+1 v ,v ,and thecone σ ⊂ M ⊗ R generated by v and v .Wethenglue σ to i i+1 i,i+1 i,i+1 Z i i+1 i,i+1 σ along the rays ρ := R v to obtain a piecewise-linear manifold B homeomorphic i−1,i i ≥0 i to R and a decomposition ={σ | 1 ≤ i ≤ n}∪{ρ | 1 ≤ i ≤ n}∪{0}. i,i+1 i We deﬁne an integral afﬁne structure on B\{0} by deﬁning charts ψ : U → M (where i i R M = Z ). Here (1.1)U = Int(σ ∪ σ ) i i−1,i i,i+1 78 MARK GROSS, PAUL HACKING, AND SEAN KEEL and ψ is deﬁned on the closure of U by i i ψ (v ) = (1, 0), ψ (v ) = (0, 1), ψ (v ) = −1,−D , i i−1 i i i i+1 with ψ linear on σ and σ . The reason for choosing these particular vectors is i i−1,i i,i+1 that they form the one-dimensional rays of a fan deﬁning a toric variety such that the divisor D corresponding to the ray generated by v has self-intersection D . i i We note this construction makes sense even when n = 1, i.e., the anti-canonical divisor D is an irreducible nodal curve. In this case there is one cone σ , and opposite 1,1 sides of the cone are identiﬁed. (Moreover, the integral afﬁne charts are deﬁned using 2 2 the integer D − 2 instead of D . This is the degree of the normal bundle of the map 1 1 from the normalization of D to Y.) However, this case will often complicate arguments in this paper, so we will usually replace Y with a surface obtained by blowing up the node of D, and replace D with the reduced inverse image of D under the blowup. This does not change the underlying integral afﬁne manifold with singularities, but reﬁnes the decomposition , exactly as in the toric case: ˜ ˜ Deﬁnition 1.5. —Given (B, ),a reﬁnement is a pair (B, ),where is a decomposition of B into rational polyhedral cones reﬁning ,eachconeof integral afﬁne isomorphic to the ﬁrst quadrant in R . Lemma 1.6. — There is a one-to-one correspondence between toric blow-ups of (Y, D) and ˜ ˜ ˜ ˜ reﬁnements of (B, ).Furthermore,if (Y, D) is a non-singular toric blow-up of (Y, D),and (B, ) ˜ ˜ ˜ is the afﬁne manifold with singularities constructed from (Y, D), then B and B are isomorphic as integral afﬁne manifolds with singularities in such a way that is the corresponding reﬁnement of . Proof.—Let π : Y → Y be a toric blow-up. It follows from the condition that −1 −1 π (D) is an anti-canonical divisor that π : Y \ π (Sing(D)) → Y \ Sing(D) is an red isomorphism. Indeed, if this restriction of π has an exceptional divisor, it must have discrepancy a(E, Y, D) =−1. But by [KM98], Cor. 2.31, (3), the smallest discrepancy occurring is 0. Thus necessarily π is a blow-up along a subscheme supported on Sing(D).Let x ∈ Sing(D) be a double point of D, corresponding to a cone σ ∈ .Note σ can be viewed as a rational polyhedral cone deﬁning a non-singular toric variety X A .Then étale locally near x, the pair (Y, D) is isomorphic to the pair (X ,∂ X ).One canthen σ σ check that in this local model, the only possible blow-ups satisfying the deﬁnition of toric blow-ups come from subdivisions of the cone σ , i.e., toric blow-ups of X .Indeed,the exceptional divisors of toric blowups are the only divisors with discrepancy −1. This gives the desired correspondence. The second statement is then easily checked. Example 1.7. — It is easy to see that if Y is a non-singular toric surface and D = ∂ Y is the toric boundary of D, then in fact the afﬁne structure on B extends across the MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 79 origin, identifying (B, ) with (M , ),where is the fan for Y. Indeed, if ρ ∈ R Y Y j Y corresponds to the divisor D and ρ = R v with v ∈ M primitive, then it is a standard j j ≥0 j j fact that v + (D ) v + v = 0. i−1 i i i+1 Since Y is non-singular, there is always a linear identiﬁcation ϕ : M → Z taking v to i i−1 (1, 0), v to (0, 1),and thus v must map to (−1,−D ).Soon U ,achartfor theafﬁne i i+1 i −1 structure on B is ψ = ϕ ◦ ψ : U → M .The maps ψ glue to give an integral afﬁne i i R i i i isomorphism B → M . In fact, the converse is also true: Lemma 1.8. — If the afﬁne structure on B = B\{0} extends across the origin, then Y is toric and D = ∂ Y. Proof. — We ﬁrst note that by Lemma 1.6, we can replace (Y, D) with a non- singular toric blow-up without affecting the afﬁne manifold B. By Proposition 1.3,we ¯ ¯ ¯ can thus assume the existence of a toric model π : (Y, D) → (Y, D).If D is the image of 2 2 D under this map, then D ≥ D . i i ¯ ¯ We ﬁrst claim that (Y, D) is isomorphic to (Y, D) if and only if equality holds for every i. Indeed, if equality holds for a given i,then π can’t contract any curves which intersect D . On the other hand, π can’t contract any curves contained in Y \ D since then D would not be an anti-canonical cycle. Now assume that (Y, D) is not toric, so that π is not an isomorphism. Let (M , ) ¯ ¯ be the fan for the toric pair (Y, D),withrays ρ¯ ,..., ρ¯ corresponding to ρ ,...,ρ .In 1 n 1 n general, B \ ρ has a coordinate chart ψ : B \ ρ → M , constructed by gluing together 1 1 R coordinate charts for U ,..., U . This can be done so that σ is mapped to the cone of 2 n 1,2 generated by ρ¯ and ρ¯ . It is now enough to show the following: 1 2 Claim. — For suitable choice of ρ , in fact ψ is injective and ψ(B \ ρ ) is strictly contained in 1 1 M \¯ ρ . R 1 To show this, ﬁrst let us analyze the effects of one blow-up on these charts. Let (Y, D) → (Y , D ) be given by a blow-up of a single point p ∈ D for some i,where ¯ ¯ (Y , D ) is obtained from (Y, D) by a sequence of blow-ups with centers at smooth points of the boundary. Let (B , ) be the tropicalization of (Y , D ). Let us examine the dif- ference between the charts ψ : B \ ρ → M and ψ : B \ ρ → M deﬁned as above. 1 R R If i = 1, then B \ ρ and B \ ρ are afﬁne isomorphic and ψ , ψ agree. Otherwise, let σ = σ ⊂ B, with σ ⊂ B deﬁned similarly. Then σ and σ are afﬁne 1,i j−1,j 1,i j=2 1,i 1,i isomorphic and ψ| = ψ | . On the other hand, ψ| = T ◦ ψ | where σ \ρ σ \ρ B\σ i B \σ 1 1 1,i 1,i 1,i 1,i T : M → M is the shear T (m) = m +m, n v ,where v is a primitive generator of i R R i i i i ψ (ρ ) and n ∈ N is primitive, annihilates v , and is positive on ψ (σ ). i i i i,i+1 80 MARK GROSS, PAUL HACKING, AND SEAN KEEL FIG.1.— (B, ) for Example 1.9 2 2 Now note that D > D for at least one i, and by choosing ρ appropriately, we can i i assume that this is the case for some i = 1. Furthermore, we can also assume that if there 2 2 ¯ ¯ is a ρ¯ ∈ with ρ¯ =−ρ¯ , then there is an i with D > D with i = 1, j . Applying the j j 1 i i above description of the change of the coordinate charts under one blow-up repeatedly then shows the claim. Now if the afﬁne structure on B extended across the origin, then ψ would extend to an isomorphism ψ : B → M , which contradicts the claim. Example 1.9. — Let Y be a del Pezzo surface of degree 5. Thus Y is isomorphic to the blowup of P in 4 points in general position. The surface Y contains exactly 10 (−1)-curves. It is easy to ﬁnd an anti-canonical cycle D of length 5 among these 10 curves. In this case, consider B \ ρ .Eachchart ψ : U → M can be composed with an 1 i i R integral linear function on M in such a way that the charts ψ ,ψ ,ψ and ψ glue to R 2 3 4 5 give a chart ψ : B \ ρ → M . This can be done, for example, with 1 R ψ(v ) = (1, 0), ψ (v ) = (0, 1), ψ (v ) = (−1, 1), 1 2 3 ψ(v ) = (−1, 0), ψ (v ) = (0,−1). 4 5 We can then take a chart ψ : U ∪ U → M which agrees with ψ on σ , and hence 5 1 R 5,1 takes the values ψ (v ) = (0,−1), ψ (v ) = (1,−1), ψ (v ) = (1, 0), 5 1 2 see Figure 1. Thus B, as an afﬁne manifold, can be constructed by cutting M along the posi- tive real axis, and then identifying the two copies of the cone σ via an integral linear 1,2 transformation. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 81 Example 1.10. — Suppose given a Looijenga pair (Y, D) with D ≤−2for all i and D is negative deﬁnite (which is equivalent to D ≤−3for some i). Then we have an ¯ ¯ analytic contraction p : Y → Ywith Y having a single cusp singularity. This case will lead to our proof of Looijenga’s conjecture. We can describe (B, ) as follows. Let M = Z and take v ,v to be a basis for M, and deﬁne v for i ∈ Z by the relation 0 1 i (1.2) v + D v + v = 0. i−1 i i+1 i mod n We deﬁne an inﬁnite fan in M whose two-dimensional cones are the cones generated by v and v , i ∈ Z. It is easy to check that these cones do indeed form a fan and that i i+1 the support of the fan | | is a strictly convex cone. If we deﬁne T ∈ SL(M) by T(v ) = v 0 n and T(v ) = v ,then T(v ) = v for each i. Necessarily T takes | | to itself, so the 1 n+1 i i+n boundary rays of the closure of | | are real eigenspaces for T. Hence T is hyperbolic, i.e., Tr T > 2. We now obtain (B, ) by dividing out | | by the action of T. 1.3. The Mumford degeneration and Givental’s construction. — The toric case of Theo- rem 0.1 yields Givental’s construction for mirrors of toric varieties in the surface case, and can also be seen as a special case of a construction due to Mumford [Mum]. Mumford’s construction in general produces degenerations of arbitrary toric varieties; the construc- tion as we review it here only gives degenerations of the algebraic torus. This should be regarded as a warmup for our general construction. gp A toric monoid P is a (commutative) monoid whose Grothendieck group P is a gp gp ﬁnitely generated free Abelian group and P = P ∩ σ ,where σ ⊆ P ⊗ R is a convex P P Z rational polyhedral cone. Let M = Z be a lattice, for some arbitrary rank n.Fix afan in M = M ⊗ R, whose support, | |, is convex. In what follows, we view B=| | as an R Z afﬁne manifold with boundary. We denote by the set of maximal cones in . max We now generalize the usual notion of a convex piecewise linear function on a fan. If one is interested in R-valued convex functions, then one can take P = N, σ = R . P ≥0 Then the following deﬁnition yields the notion of a piecewise linear R-valued function with integral slopes, and convexity here means upper convexity, i.e., the function is the supremum of a collection of linear functions. gp Deﬁnition 1.11. —A -piecewise linear function ϕ :| |→ P is a continuous function gp gp such that for each σ ∈ , ϕ| is given by an element ϕ ∈ Hom (M, P ) = N ⊗ P . max σ σ Z Z For each codimension one cone ρ ∈ contained in two maximal cones σ ,σ ∈ ,wecan + − max write ϕ − ϕ = n ⊗ κ σ σ ρ ρ,ϕ + − gp where n ∈ N is the unique primitive element annihilating ρ and positive on σ ,and κ ∈ P .We ρ + ρ,ϕ call κ the bending parameter. Note (as the notation suggests) it depends only on the codimension one ρ,ϕ cone ρ (not on the ordering of σ ,σ ). + − 82 MARK GROSS, PAUL HACKING, AND SEAN KEEL gp We say a -piecewise linear function ϕ :| |→ P is P-convex (strictly P-convex) if × × for every codimension one cone ρ ∈ , κ ∈ P(κ ∈ P \ P ,where P is the group of invertible ρ,ϕ ρ,ϕ elements of P). Example 1.12. —Takeacomplete fan in M . This deﬁnes a toric variety Y = gp Y , which we assume is non-singular. We let P ⊂ P be given by the cone of effective curves, NE(Y) ⊂ A (Y, Z). Each codimension one cone ρ ∈ corresponds to a one-dimensional toric stratum D ⊂ ∂ Y, hence a class [D ]∈ NE(Y) = P. If ω ∈ (1), the set of rays of , we also write D ρ ω for the corresponding toric divisor. (1) Lemma 1.13. —Deﬁne s : T := Z → M to send the basis element t , ω ∈ (1) to the ﬁrst lattice point m on ω. Then A (Y, Z) β → (D · β)t 1 ω ω ω∈ (1) identiﬁes A (Y, Z) with Ker(s), giving rise to an exact sequence (1.3)0 → A (Y, Z) → T → M → 0. Then there is a unique -piecewise linear section ϕ˜ : M → T satisfying ϕ( ˜ m ) = t .Let π : T → ω ω gp A (Y, Z) be any splitting, and set ϕ := π ◦˜ ϕ. Then ϕ : M → A (Y, Z) = P is -piecewise 1 1 linear and strictly P-convex, with (1.4) κ =[D ] ρ,ϕ ρ for each codimension one cone ρ ∈ . Up to a linear function, ϕ is the unique -piecewise linear map with these bending parameters. Proof. — The exact sequence is standard. Since is a complete non-singular fan, it is clear that there exists such a unique ϕ˜ . To calculate the kink along a codimension one ρ ∈ , suppose ρ is generated by basis vectors e ,..., e and ρ is contained in 1 n−1 n−1 two maximal cones, generated by e ,..., e and e ,..., e , e := −e + a e .Let 1 n 1 n−1 n i i n i=1 t ,..., t , t be the generators of T mapping to e ,..., e , e respectively. Then the kink 1 n 1 n n n n−1 is κ = t + t − a t . On the other hand, if D ,..., D , D are the divisors corre- ρ,ϕ˜ n i i 1 n n i=1 n sponding to the rays generated by e ,..., e , e respectively, then D · D = D · D = 1 1 n n ρ ρ n n and using the rational function z ,D is linearly equivalent to −a D plus a sum of toric i i divisors disjoint from D .Thus D · D =−a and we see that κ is the image of [D ] ρ i ρ i ρ,ϕ˜ ρ under the inclusion A (Y, Z) → T .Thus κ = π(κ )=[D ] as required. 1 ρ,π◦˜ ϕ ρ,ϕ˜ ρ MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 83 gp Given a -piecewise linear and P-convex function ϕ :| |→ P we can deﬁne a gp monoid P ⊂ M × P by (1.5)P := m,ϕ(m) + p | m∈| |, p ∈ P . This is the set of integral points lying above the graph of ϕ, in the sense given by the gp partial order on P deﬁned by p ≥ p if p − p ∈ P. The convexity of ϕ is equivalent 1 2 1 2 to P being closed under addition. Furthermore, we have a natural inclusion P → P ϕ ϕ given by p → (0, p), which gives us a morphism f : Spec k[P ]→ Spec k[P]. This morphism is ﬂat as k[P ] is freely generated as a k[P]-module by all elements of (m,ϕ(m)) the form z , m ∈| |. It is easy to see that a general ﬁbre of f is isomorphic to the gp algebraic torus Spec k[M]: in fact, if we consider the big torus orbit U = Spec k[P ]⊂ −1 Spec k[P], f (U) = U × Spec k[M]. We now describe the ﬁbres over other toric strata of Spec k[P].Let x ∈ Spec k[P] be a point in the torus orbit corresponding to a face Q ⊂ P. Then by replacing P with the localized monoid P − Q obtained by inverting all elements of Q, we may assume that x is contained in the smallest toric stratum of Spec k[P]. Consider the composed map gp gp × ϕ¯ :| |−→P → P /P . Note ϕ¯ is also piecewise linear. Let be the fan (of convex but not necessarily strictly convex cones) whose maximal cones are the maximal domains of linearity of ϕ¯.Then −1 f (x) can be written as −1 f (x) = Spec k[ ]. Here, k[ ]= kz m∈M∩| | with multiplication given by m+m z if m, m lieinacommonconeof , m m (1.6) z · z = 0 otherwise. −1 In particular, the irreducible components of f (x) are the toric varieties Spec k[σ ∩ M] for σ ∈ . max In the particular case that rank M = 2and deﬁnes a non-singular complete sur- face with n toric divisors, suppose ϕ is strictly convex. If x is a point of the smallest toric 84 MARK GROSS, PAUL HACKING, AND SEAN KEEL −1 n stratum of Spec k[P],then f (x) is just V ⊂ A , the reduced cyclic union of coordi- nate A ’s: 2 2 2 n V = A ∪ A ∪···∪ A ⊂ A . x ,x x ,x x ,x x ,...,x 1 2 2 3 n 1 1 n We call V the vertex, or more speciﬁcally, the n-vertex. We will need in the sequel the degenerate case of the n-vertex for n = 2. This is a union of two afﬁne planes and can be described as the double cover 2 2 2 2 2 (1.7) V = Spec k[x , x , y]/ y − x x = A ∪ A . 2 1 2 1 2 x ,x x ,x 1 2 2 1 Of course, this does not appear as a central ﬁbre of a Mumford degeneration. Analo- gously, one can deﬁne 2 3 (1.8) V = Spec k[x, y, z]/ xyz − x − z , the afﬁne cone over a nodal curve embedded in weighted projective space WP (3, 1, 2). Example 1.14. — In Example 1.12, with the choice of ϕ given by Lemma 1.13,the family Spec k[P ]→ Spec k NE(Y) in fact gives the family of mirror manifolds to the toric variety Y, as constructed by Given- tal [Giv]. In fact, the mirror of a toric variety also includes the data of a Landau-Ginzburg potential, which is a regular function. If Y is Fano, the potential is (m ,ϕ(m )) ρ ρ W = z where we sum over all rays ρ ∈ ,and m ∈ M denotes the primitive generator of ρ . If Y is not Fano, the potential receives corrections which can be viewed as coming from degenerate holomorphic disks on Y with irreducible components mapping into D. 2. Modiﬁed Mumford degenerations In this section, we ﬁx (Y, D) a Looijenga pair, and let (B, ) be the tropicalisation of (Y, D) deﬁned in Section 1.2.The fan contains rays ρ ,...,ρ corresponding to 1 n divisors D ,..., D , ordered cyclically. As usual, we write the two-dimensional cones of 1 n as σ being the cone with edges ρ and ρ , with indices taken modulo n. i,i+1 i i+1 We explain how to generalize Mumford’s degeneration, to give a canonical formal deformation of V = V \{0} associated to (Y, D) if n ≥ 3. Locally on B the picture is n 0 n MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 85 toric and we have Mumford’s degenerations described in Section 1.3.AsMumford’s con- struction is functorial, the deformations built locally patch together canonically: this is a minor variation on the ideas of [GS07]. In particular, Sections 2.1 and 2.2 are variations of ideas in [GS07]. However, it differs crucially in several respects which prevent us from just referring to [GS07]. First, we work with piecewise linear functions with values in a gp vector space P rather than just R. This allows us to construct higher dimensional for- mal families, namely over the completion of Spec k[P] at the zero-dimensional stratum. Second, by avoiding a description of local models in codimension at least two, we avoid some of the technical complexities of [GS07]. Here are the details. 2.1. The uncorrected degeneration. — We ﬁx some notation. For any locally constant sheaf F on B , and any simply connected subset τ ⊂ B we write F for the stalk of this 0 0 τ local system at any point of τ (as any two such stalks are canonically identiﬁed by parallel transport). In particular, we apply this for the sheaf of integral constant vector ﬁelds, as well as for the sheaf := ⊗ R. R Z −1 For each cone τ ∈ with dim τ = 1 or 2, we write τ for the localized fan of convex (but not strictly convex) cones in described as follows. If dim τ = 2, then τ,R −1 −1 τ just consists of the single cone .If dim τ = 1, then τ consists of three τ,R cones: the tangent line to τ and the two half-spaces bounded by the tangent line to τ . gp Let P ⊆ P be a toric monoid as in Section 1.3. gp Deﬁnition 2.1. —A (P -valued) -piecewise linear multivalued function on B is a gp collection ϕ ={ϕ } with ϕ a -piecewise linear function on U with values in P . i i i gp −1 Note this is equivalent to giving a ρ -piecewise linear function ϕ : → P for each i R,ρ i i R ray ρ ∈ . Two such functions ϕ, ϕ are said to be equivalent if ϕ − ϕ is linear for each i. Note i i gp the equivalence class of ϕ is determined by the collection of bending parameters κ ∈ P .Wesay the ρ,ϕ function is convex (strictly convex) if κ ∈ P (κ ∈ P \ P )for each ρ . ρ,ϕ ρ,ϕ We drop the modiﬁers and P when they are clear from context. Construction 2.2. — The collection {ϕ } determines a local system P on B as fol- i 0 lows. First, we can construct an afﬁne manifold P which comes along with the structure gp of P -principal bundle π : P → B and a piecewise linear section ϕ : B → P as follows: 0 0 0 0 gp gp gp we glue U × P to U × P along (U ∩ U ) × P by i i+1 i i+1 R R R (x, p) → x, p + ϕ (x) − ϕ (x) . i+1 i By construction we have local sections x → (x,ϕ (x)) which patch to give a piecewise gp linear section ϕ. One checks immediately the isomorphism class (of the P -principal bundle together with the section) depends only on the equivalence class of {ϕ }.The bundle P → B can be viewed as a tropical analogue of a sum of line bundles, and {ϕ } 0 0 i 86 MARK GROSS, PAUL HACKING, AND SEAN KEEL yield a section of this vector bundle. Convexity is analogous to holomorphicity of the section. We then deﬁne −1 P := π ϕ ∗ P P 0 0 on B .Wehaveanexact sequence gp (2.1)0 → P → P−→ → 0 of local systems on B ,where r is the derivative of π . gp gp Note over U , the description of P as U × P gives a splitting P| | × P . i 0 i U U R i i Example 2.3. — Our standard example, fundamental to this paper, will be as fol- lows. Suppose P is a monoid which comes with a homomorphism η : NE(Y) → Pof monoids. Choose ϕ by specifying ϕ on U by the formula i i κ = η [D ] . ρ ,ϕ i i i Such a ϕ is well-deﬁned up to linear functions, and always exists. This is always convex, and is strictly convex provided η([D ]) is not invertible for any i. Now suppose given a piecewise linear multivalued P-convex function ϕ on B. We explain how Mumford’s construction determines a canonical formal deformation of V , restricting to the case n ≥ 3 for ease of exposition. −1 For each τ ∈ with dimτ> 0, ϕ determines a canonically deﬁned τ - piecewise linear section ϕ : → P of the projection P → .If U ∩ τ = ∅, τ R,τ R,τ R,τ R,τ i −1 we use the representative ϕ on U and extend it linearly on each cone in the fan τ i i −1 to obtain a P-convex piecewise linear function on τ , which we also write as ϕ .Then the section ϕ is deﬁned as in Construction 2.2 by x → (x,ϕ (x)), using the splitting τ i gp P = × P . We note a different choice of representative of ϕ leads to a different R,τ R,τ i choice of splitting and the same section ϕ , so this section is well-deﬁned. Now deﬁne the toric monoid P ⊂ P by ϕ τ (2.2)P := q ∈ P | q = p + ϕ (m) for some p ∈ P, m ∈ . ϕ τ τ τ By the deﬁnition of convexity of ϕ, we have canonical inclusions (2.3)P ⊂ P ⊂ P ϕ ϕ ρ ρ σ whenever ρ ⊂ σ ∈ .If ρ ∈ is a ray with ρ ⊂ σ ∈ we have the equality ± max (2.4)P ∩ P = P . ϕ ϕ ϕ σ σ ρ + − MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 87 Deﬁnition 2.4 (Monomial ideals). — A (monoid) ideal of a monoid P is a subset I ⊂ P such that p ∈ I, q ∈ P implies p + q ∈ I. An ideal determines a monomial ideal in the monoid ring k[P], generated by monomials z for p ∈ I. We also denote this ideal by I, hopefully with no confusion. As a consequence, we shall sometimes write certain ideal operations either additively or multiplicatively, i.e., for J ⊂ P, kJ={p +···+ p | p ∈ J, 1 ≤ i ≤ k}, 1 k i and the corresponding monomial ideal is J . Let m = P \ P . This is the unique maximal ideal of P, deﬁning a monomial ideal m ⊂ k[P]. Note k[P]/m k[P ]. We say an ideal I ⊂ P is m-primary if m = I := {p ∈ P| there exists a positive integer k such that kp ∈ I}, in which case the same holds for the associated monomial ideal I ⊂ k[P]. Recall from Section 1.3 that we are only considering toric monoids P, i.e., monoids which are the intersection of rational polyhedral cones σ with lattices. Such monoids are always ﬁnitely generated, so that k[P] is Noetherian. If σ is strictly convex, then m is the maximal ideal corresponding to the unique torus ﬁxed point of Spec k[P]. Fix an ideal I ⊂ P, and recalling that we write R = k[P], set R := k[P]/I. We deﬁne for τ ∈ ,dimτ> 0, the ring R := k[P ]⊗ R , τ,I ϕ R I noting that P acts naturally on P by addition. So Spec R is a base-change of the ϕ τ,I −1 Mumford degeneration induced by ϕ on the localized fan τ . One observes Proposition 2.5. —Let v denote the primitive generator of the tangent ray to ρ , for each i. Then i i ρ,ϕ viewing z ∈ k[P] as determining an element in R ,wehave R [X , X , X ] I i−1 i+1 (2.5) = R ρ ,I −D κ i ρ ,ϕ (X X − z X ) i−1 i+1 via the map ϕ (v ) ρ j X → z , j ∈{i − 1, i, i + 1}. Furthermore, there are natural maps ψ : R → R ,ψ : R → R ρ ,− ρ ,I σ ,I ρ ,+ ρ ,I σ ,I i i i−1,i i i i,i+1 88 MARK GROSS, PAUL HACKING, AND SEAN KEEL induced by the inclusions P ⊆ P which induce isomorphisms ϕ ϕ ρ σ ∼ ∼ (R ) R ,(R ) R . = = ρ ,I X σ ,I ρ ,I X σ ,I i i−1 i−1,i i i+1 i,i+1 Proof. — We need to check that the ideal on the left-hand side is mapped to zero, as the rest is obvious. Note by construction of B, v + D v + v = 0 aselementsof , i−1 i i+1 ρ ρ i so one sees in fact that ϕ (v ) + ϕ (v ) = κ − D ϕ (v ). The result then follows ρ i−1 ρ i+1 ρ ,ϕ ρ i i i i ρ i easily. Remark 2.6. — Since Spec R → Spec R is a base-change of the Mumford de- ρ ,I I generation, we can in fact say what a ﬁbre of this morphism is over a closed point x in the smallest toric stratum of Spec k[P], i.e., a point in Spec R . This depends on whether −1 κ ∈ P is invertible or not. If it is not invertible, then the ﬁbre is Spec k[ρ ] ρ ,ϕ i i ±1 Spec k[X , X , X ]/(X X ).If κ is invertible, then the ﬁbre is Spec k[Z ].In i−1 i+1 i−1 i+1 ρ ,ϕ i i this latter case, if ρ ⊂ σ ,infactthe map R → R induced by the inclusion P ⊆ P i ρ ,I σ,I ϕ ϕ i ρ σ is an isomorphism. Somewhat more generally, if J ⊂ P is a radical ideal with κ ∈ J, then in fact ρ ,ϕ −1 R = R [ρ ]. ρ ,J J i i For τ ∈ ,dim τ ≥ 1, set U := Spec R . τ,I τ,I The maps ψ induce open immersions U → U and U → U . Denoting ρ ,± σ ,I ρ ,I σ ρ ,I i i−1,i i i,i+1 i the image of each of these immersions as U and U respectively, we note ρ ,σ ,I ρ ,σ ,I i i−1,i i i,i+1 that ∼ ∼ ρ ,ϕ (2.6)U ∩ U = Spec(R ) = (G ) × Spec(R ) ρ ,σ ,I ρ ,σ ,I ρ ,I X X m I z i i−1,i i i,i+1 i i−1 i+1 κρ ,ϕ Note that if κ ∈ I then the localization (k[P]/I) is zero, and the intersection is ρ ,ϕ z empty. We can now deﬁne our analogue of the Mumford degeneration. Construction 2.7. — Suppose that the number of irreducible components n of D satisﬁes n ≥ 3, that ϕ is a PL multivalued function, and I ⊂ Pan ideal such that κ ∈ I ρ,ϕ for all rays ρ ∈ . Then there are canonical identiﬁcations of open subsets ∼ ∼ U ⊃ U U U ⊂ U = = ρ ,I ρ ,σ ,I σ ,I ρ ,σ ,I ρ ,I i i i,i+1 i,i+1 i+1 i,i+1 i+1 which generate an equivalence relation on U , and the quotient by this equivalence ρ ,I relation deﬁnes a scheme X over Spec R . One checks easily that the canonical isomorphisms of U ⊆ U and U ⊆ U ρ ,σ ,I ρ ,I ρ ,σ ,I ρ ,I i i,i+1 i i+1 i,i+1 i+1 satisfy the requirements for gluing data for schemes along open subsets, see e.g., [H77], Ex. II 2.12. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 89 Remark 2.8. —X only depends on the equivalence class of ϕ, since the monoids P are canonically deﬁned, independently of the choice of representative for ϕ. We ﬁrst analyze this construction in the purely toric case: Lemma 2.9. —For (Y, D) toric and ϕ a single-valued convex function on B = M , X is an open subscheme of the Mumford degeneration Spec k[P ]/Ik[P ],and ϕ ϕ 0 o H X , O = k[P ]/Ik[P ]. X ϕ ϕ Proof. — Note that for τ ∈ , τ = {0}, the monoid P is isomorphic to the local- ization of P along the face {(m,ϕ(m))| m ∈ τ ∩ M}.Thus Spec k[P ] is an open subset of ϕ ϕ Spec k[P ] and Spec k[P ]⊗ k[P]/I is an open subset of Spec k[P ]/Ik[P ]. Further- ϕ ϕ k[P] ϕ ϕ more, the gluing procedure constructing X is clearly compatible with these inclusions, so X is an open subscheme of Spec k[P ]/Ik[P ]. Next, looking at the ﬁbre over a closed ϕ ϕ point, one sees easily that the underlying topological space of these ﬁbres is obtained just by removing the zero-dimensional torus orbit from the corresponding ﬁbre of the Mumford degeneration. The closed ﬁbres of the Mumford degeneration are S by [A02], 2.3.19. Thus by Lemma 2.10, the result follows. Lemma 2.10. —Let π : X → S be a ﬂat family of surfaces such that the ﬁbre X satisﬁes Serre’s condition S for each s ∈ S.Let i: X ⊂ X be the inclusion of an open subset such that the complement has ﬁnite ﬁbres. Then i O = O . Similarly, if F is a coherent sheaf on S then ∗ X X ∗ ∗ i (O ⊗ π F ) = O ⊗ π F . ∗ X Proof. — For the ﬁrst statement see, e.g., [H04], Lemma A.3, (the assumption that the ﬁbres are semi log canonical is not used). The second statement follows from the ﬁrst by dévissage. Deﬁnition 2.11. —Let B (Z) denote the set of points of B with integral coordinates in an 0 0 integral afﬁne chart. We also write B(Z) = B (Z)∪{0}. Given the description of Remark 2.6, the following lemma is obvious. Lemma 2.12. — Suppose n ≥ 3 and we are given a convex multivalued piecewise linear function ϕ and a radical monomial ideal J ⊂ P such that κ ∈ J for all rays ρ ∈ .Thenif x ∈ Spec R ρ,ϕ J is a closed point, the ﬁbre of X → Spec R over x is (Spec k[ ])\{0}.Here, k[ ] denotes the k- algebra with a k-basis {z | m ∈ B(Z)} with multiplication given exactly as in (1.6), and 0 is the closed m o point whose ideal is generated by {z | m = 0}. In particular, the ﬁbre is isomorphic to V .Furthermore, with R [ ]:= R ⊗ k[ ], J J k X Spec R [ ] \ (Spec R )×{0}. J J J 90 MARK GROSS, PAUL HACKING, AND SEAN KEEL 2.2. Scattering diagrams on B.— Next we translate into algebraic geometry the in- stanton corrections. To construct our mirror family we will use the canonical scattering can diagram D deﬁned in Section 3.1, (which is the translation of the instanton corrections associated to Maslov index zero disks), but as the regluing process works for any scattering diagram (and we will make use of this greater generality in [K3]), we carry it out for an arbitrary scattering diagram. We continue with the notation of the previous sections, with (Y, D), (B, ),Pan arbitrary toric monoid, and ϕ given. We also ﬁx a monomial ideal J ⊂ Psuch thatJ = J. Denote by R the completion of k[P] with respect to the ideal J, and for any τ ∈ , τ = 0, denote by k[P ] the completion of the ring k[P ] with respect to the ideal Jk[P ]. ϕ ϕ τ τ We will now deﬁne a scattering diagram, which encodes a modiﬁcation of the con- struction of X . Unlike the previous subsection, where we assumed n ≥ 3 for ease of exposition throughout, in this subsection we can allow any number of irreducible com- ponents of D except where noted. Deﬁnition 2.13. —A scattering diagram for the data (B, ), P,ϕ, and J is a set D = (d, f ) where (1) d ⊂ B is a ray in B with endpoint the origin with rational slope. d may coincide with a ray of , or lie in the interior of a two-dimensional cone of . (2) Let τ ∈ be the smallest cone containing d.Then f is a formal sum d d f = 1 + c z ∈ k[P ] d p ϕ for c ∈ k and p running over elements of P such that r(p) = 0 and r(p) is tangent p ϕ to d. Here r is deﬁned by (2.1). We further require that d satisfy one of the following two properties: (a) For those p with c = 0,r(p), viewed as a tangent vector at an interior point of d, points towards the origin, in which case we say that d is an outgoing ray. (b) For those p with c = 0,r(p) points away from the origin, in which case we say that d is an incoming ray. (3) If dim τ = 2 or if dim τ = 1 and κ ∈ J,then f ≡ 1mod J. d d τ ,ϕ d (4) For any ideal I ⊂ P with I = J, there are only a ﬁnite number of (d, f ) ∈ D such that ≡ 1mod Ik[P ]. d ϕ d MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 91 Construction 2.14. — We now explain how a scattering diagram D is used to modify the construction of X , as given in Construction 2.7. Suppose we are given a scattering diagram D for the data (B, ),P, ϕ and J, and an ideal I with I = J. We assume that κ ∈ Jfor all rays ρ ∈ and that n ≥ 3 as in Construction 2.7. ρ,ϕ We will use the scattering diagram D to modify both the deﬁnition of the rings R as well as the gluings of the schemes deﬁned by these rings. First, we modify the ρ ,I deﬁnition of R , setting ρ ,I ±1 R [X , X , X ] I i−1 i+1 (2.7)R := ρ ,I i 2 −D κ i ρ ,ϕ (X X − z X f ) i−1 i+1 ρ i i ±1 Here f is an element of R [X ] deﬁned by ρ I i i f = f mod Ik[P ], ρ d ϕ i ρ (d,f )∈D d=ρ ϕ (v ) ρ i identifying X with z as in Proposition 2.5. Note this is a generalization of the old deﬁnition of R ,whichweobtainif f = 1. Thus we continue to use the same notation. ρ ,I ρ i i Retaining the deﬁnition R = k[P ]⊗ R for dim σ = 2 from the previous sub- σ,I ϕ R I section,wenotethatthere aremaps ψ : R → R ,ψ : R → R , ρ ,− ρ ,I σ ,I ρ ,+ ρ ,I σ ,I i i i−1,i i i i,i+1 given by ϕ (v ) ϕ (v ) ϕ (v ) ρ i ρ i−1 ρ i+1 i i i ψ (X ) = z ,ψ (X ) = z ,ψ (X ) = f z , ρ ,− i ρ ,− i−1 ρ ,− i+1 ρ i i i i (2.8) ϕ (v ) ϕ (v ) ϕ (v ) ρ ρ ρ i i−1 i+1 i i i ψ (X ) = z ,ψ (X ) = f z ,ψ (X ) = z . ρ ,+ i ρ ,+ i−1 ρ ρ ,+ i+1 i i i i Furthermore, ψ induce isomorphisms ρ ,± ψ : (R ) → R ,ψ : (R ) → R . ρ ,+ ρ ,I X σ ,I ρ ,− ρ ,I X σ ,I i i i+1 i,i+1 i i i−1 i−1,i Set for τ ∈ \{0} U := Spec R . τ,I τ,I One checks easily that the natural map U → Spec R is ﬂat. The maps ψ induces ρ,I I ρ ,± canonical embeddings U , U → U , and we denote their image by U σ ,I σ ,I ρ ,I ρ ,σ ,I i−1,i i,i+1 i i i−1,i and U respectively. Note that (2.6) continues to hold. ρ ,σ ,I i i,i+1 Next, consider (d, f ) ∈ D with τ = σ ∈ .Let γ be a path in B which crosses d d max 0 d transversally at time t . Then deﬁne θ : R → R γ,d σ,I σ,I 92 MARK GROSS, PAUL HACKING, AND SEAN KEEL FIG.2.—Thepath γ .The solid lines indicate the fan, the dotted lines are additional rays in D.The solid lines may also support rays in D by n ,r(p) p p d θ z = z f γ,d where n ∈ is primitive and satisﬁes, with m a non-zero tangent vector of d, n , m= 0, n ,γ (t ) < 0. d d 0 If γ is not differentiable at t , which might occur for broken lines, see Deﬁnition 2.16, this inequality is interpreted to mean that n is positive at γ(t − ) and negative at γ(t + ) d 0 0 for > 0 small. Note that f is invertible in R since f ≡ 1mod Jk[P ],so f − 1is d σ,I d ϕ d nilpotent in R . σ,I Let D ⊂ D be the ﬁnite set of rays (d, f ) with f ≡ 1mod Ik[P ].For apath γ I d d ϕ wholly contained in the interior of σ ∈ and crossing elements of D transversally, max I we deﬁne θ := θ ◦···◦ θ , γ,D γ,d γ,d n 1 where γ crosses precisely the elements (d , f ), ...,(d , f ) of D ,inthe givenorder. 1 d n d I 1 n Note that if two rays d , d in fact coincide as subsets of B, then θ and θ com- i i+1 γ,d γ,d i i+1 mute, so the ordering is not important for overlapping rays. To construct X , we modify the gluings of the sets U along the open subsets ρ,I I,D U .For each i, we have canonical identiﬁcations of open subsets ρ,σ,I ∼ ∼ U ⊃ U U U ⊂ U = = ρ ,I ρ ,σ ,I σ ,I ρ ,σ ,I ρ ,I i i i,i+1 i,i+1 i+1 i,i+1 i+1 We can modify this identiﬁcation via any automorphism of U . We do this by choosing i,i+1 apath γ :[0, 1]→ B whose image is contained in the interior of σ ,with γ (0) a point i i,i+1 i in σ close to ρ and γ (1) ∈ σ close to ρ , chosen so that γ crosses every ray i,i+1 i i i,i+1 i+1 i (d, f ) of D with τ = σ exactly once, see Figure 2. d I d i,i+1 We then obtain an automorphism θ : R → R , γ ,D σ ,I σ ,I i i,i+1 i,i+1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 93 hence, after taking Spec, an isomorphism θ : U → U . γ ,D ρ ,σ ,I ρ ,σ ,I i i+1 i,i+1 i i,i+1 We now deﬁne X by dividing out U by the equivalence relation given by identi- ρ ,I I,D i i fying x ∈ U ⊆ U with θ (x) ∈ U ⊆ U . ρ ,σ ,I ρ ,I γ ,D ρ ,σ ,I ρ ,I i+1 i,i+1 i+1 i i i,i+1 i 2.3. Broken lines. — We continue to ﬁx a rational surface with anti-canonical cycle (Y, D) as usual, D having an arbitrary number of irreducible components, giving (B, ), as well as a monoid P, a multivalued P-convex function ϕ on B, J ⊂ P an ideal with J = J, and a scattering diagram D for this data. Broken lines were introduced in [G09] and their theory was further developed in [CPS]. Deﬁnition 2.15. —Let B be an integral afﬁne manifold. An integral afﬁne map γ : (t , t ) → 1 2 B from an open interval (t , t ) is a continuous map such that for any integral afﬁne coordinate chart 1 2 n −1 n ψ : U → R of B, ψ ◦ γ : γ (U) → R is integral afﬁne, i.e., is given by t → tv + bfor some n n v ∈ Z and b ∈ R . Note that for an integral afﬁne map, γ (t) ∈ . B,γ (t) Deﬁnition 2.16. —A broken line γ in (B, ) for q ∈ B (Z) with endpoint Q ∈ B is 0 0 a proper continuous piecewise integral afﬁne map γ : (−∞, 0]→ B with only a ﬁnite number of domains of linearity, together with, for each L ⊂ (−∞, 0] a maximal connected domain of linearity q × −1 of γ , a choice of monomial m = c z where c ∈ k and q ∈ (L,γ (P )| ), satisfying the L L L L L following properties. (1) For the unique unbounded domain of linearity L, γ| goes off to inﬁnity in a cone σ ∈ L max as t →−∞,and q ∈ σ . Furthermore, using the identiﬁcation of the stalk P for x ∈ σ ϕ (q) with P ,m = z . σ L (2) For each L and t ∈ L, −r(q ) = γ (t),where r isdeﬁnedin(2.1). Also γ(0) = Q ∈ B . L 0 (3) Let t ∈ (−∞, 0) be apointatwhich γ is not linear, passing from domain of linearity L to L .If γ(t) ∈ τ ∈ , then P = P , so that we can view q ∈ P and r(q ) ∈ . γ(t) τ L τ L τ Let d ,..., d ∈ D be the rays of D that contain γ(t), with attached functions f . Then 1 p d we require that γ passes from one side of these rays to the other at time t, so that θ is γ,d deﬁned. Let n = n be the primitive element of used to deﬁne θ . Expand d γ,d j τ j n,r(q ) (2.9) f j=1 as a formal power series in k[P ]. Then there is a term cz in this sum with m = m · cz . L L 94 MARK GROSS, PAUL HACKING, AND SEAN KEEL Remark 2.17. — Using the notation of item (3) above, by item (2) of the deﬁnition, (2.10) n, r(q ) > 0. This is vital to interpret (2.9). Indeed, if τ is a ray, f need not be invertible in k[P ],so d ϕ i τ (2.10) tells us that (2.9) makes sense in this ring. Example 2.18. — We give a ﬁrst example of broken lines, in the case where B is as given in Example 1.10 and D=∅, so that there is no possibility of bending. Nevertheless, there is quite non-trivial behaviour. For an example including bending, see Example 3.7 after the introduction of the canonical scattering diagram. Given q ∈ B (Z),Q ∈ B general, we can choose lifts q˜, Q to the universal cover 0 0 ˜ ˜ ˜ ˜ B =| |\{0} of B .Let π : B → B be the covering map. Fixing the lift Q, for any lift 0 0 0 0 q˜ we obtain a broken line γ : (−∞, 0]→ B given by γ(t) = π(Q − tq˜). As this has one ϕ (q) domain of linearity L, we decorate L with the monomial z ,where q ∈ σ ∈ .Note there are an inﬁnite number of such broken lines, one for each lift of q. Dealing with this non-ﬁniteness is a key part of the proof of Looijenga’s conjecture in Section 7. The next lemma and corollary are crucial for interpreting the monomials m : Lemma 2.19. —Let σ ,σ ∈ be the two maximal cones containing the ray ρ ∈ .If − + max −1 q ∈ P with −r(q) ∈ Int(ρ σ ) ⊂ ⊗ R, then ϕ + ρ Z q ∈ P = P ∩ P . ϕ ϕ ϕ ρ σ σ − + Proof. — By the deﬁnitions there exist p,κ ∈ Pand n ∈ annihilating the ρ,ϕ ρ tangent space to ρ and positive on σ such that q = ϕ r(q) + p ϕ −r(q) = ϕ −r(q) + n ,−r(q) κ . σ σ ρ ρ,ϕ + − Since n ,−r(q) > 0, q = ϕ r(q) + p + n ,−r(q) κ ∈ P . σ ρ ρ,ϕ ϕ + σ An immediate consequence of this lemma is Corollary 2.20. (1) Let γ :[t , t ]→ B be integral afﬁne. Suppose that γ(t ) ∈ τ , γ(t ) ∈ τ . Suppose 1 2 0 1 1 2 2 −1 also we are given a section q ∈ (γ P ) such that −r(q) = γ (t) for each t. If q(t ) ∈ P ⊂ P = P , 1 ϕ τ γ(t ) τ 1 1 1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 95 then q(t ) ∈ P ⊂ P = P . 2 ϕ τ γ(t ) τ 2 2 (2) If γ is a broken line, t ∈ L a maximal domain of linearity with γ(t) ∈ τ , then q ∈ P ⊂ P = P . L ϕ τ γ(t) Proof. — The ﬁrst item follows immediately from the lemma. The second item fol- lows from the fact that if t 0 lies in the unbounded domain of linearity with γ(t) ∈ σ , ϕ (q) then m = z ∈ P by construction. Then this holds for all t by item (1) and Deﬁni- L ϕ tion 2.16,(3). The convexity of ϕ puts further restrictions on the monomial decorations of a broken line. Deﬁnition 2.21. —Let J ⊂ P be a proper monoid ideal. For p ∈ J there exists a maximal k ≥ 1 such that p = p +···+ p with p ∈ J.Wedeﬁne ord (p) to be this maximum, and deﬁne 1 k i J ord (p) = 0 if p ∈ P \ J. For x ∈ τ,q ∈ P ,deﬁne ord (q) := ord (q − ϕ (r(q))). This measures how high q is ϕ J,x J τ above the graph of ϕ .If γ is a broken line and t ∈ L a maximal domain of linearity, deﬁne ord (t) = ord (q ), J,γ J,γ (t) L using γ(t) ∈ τ and q ∈ P ⊂ P . L ϕ γ(t) Lemma 2.22. —Let γ be a broken line. Then if t < t , ord (t) ≤ ord t , J,γ J,γ with strict inequality if either t and t lie in different domains of linearity or for some t with t < t < t , γ(t ) lies in a ray ρ ∈ with bending parameter κ ∈ J. ρ,ϕ Proof. — This is immediate from the deﬁnitions and the proof of Lemma 2.19. Deﬁnition 2.23. —For I an ideal in P with I = J,let Supp (D) := d where the union is over all (d, f ) ∈ D such that f ≡ 1mod Ik[P ]. By Deﬁnition 2.13, (4), this d d ϕ is a ﬁnite union. 96 MARK GROSS, PAUL HACKING, AND SEAN KEEL Deﬁnition 2.24. —Let I be an ideal of P with I = J,and let Q ∈ B \ Supp (D), Q ∈ τ ∈ .For q ∈ B (Z),deﬁne (2.11)Lift (q) := Mono(γ ) ∈ k[P ]/I · k[P ], Q ϕ ϕ τ τ where the sum is over all broken lines γ for q with endpoint Q,and Mono(γ ) denotes the monomial attached to the last domain of linearity of γ . The word “Lift” is used to indicate that this is, as we shall q o o show, a lifting of a monomial z on X to X . The fact that Lift (q) lies in the stated ring follows J,D I,D from: Lemma 2.25. —Let Q ∈ σ ∈ ,q ∈ B (Z).Let I be an ideal with I = J. Assume max 0 that κ ∈ J for at least one ray ρ ∈ . Then the following hold: ρ,ϕ (1) The collection of γ in Deﬁnition 2.24 with Mono(γ ) ∈ I · k[P ] is ﬁnite. (2) If one boundary ray of the connected component of B \ Supp (D) containing Q is a ray ρ ∈ , then Mono(γ ) ∈ k[P ], and the collection of γ with Mono(γ ) ∈ I · k[P ] is ﬁnite. Proof. — Note there is some k such that J ⊂ I because k[P] is Noetherian. If γ is a broken line with Mono(γ ) ∈ I · k[P ],then γ crosses the rays of in a cyclic order. Indeed, this follows from condition (3) of Deﬁnition 2.16, as a broken line must cross from one side to the other of each ray of D it intersects. From this, the hypotheses on the κ ρ,ϕ imply that in any set of at least n consecutive rays of that it crosses, there is at least one ray ρ with κ ∈ J. By Lemma 2.22,ord increases every time γ crosses such a ray, ρ,ϕ J,γ and also every time γ bends at a ray d not contained in a ray of .Onceord ≥ k, J,γ Mono(γ ) ∈ I · k[P ]. Hence there is an absolute bound on the number of rays of that γ can cross, and the number of times γ can bend. When γ crosses a ray, there are a ﬁnite number of terms in (2.9) modulo I · k[P ], as the exponent is always positive, see Remark 2.17. Thus there are a ﬁnite number of possible choices of bend, and hence only a ﬁnite number of possible choices for the exponent of Mono(γ ) modulo I · k[P ] once the initial monomial of γ is ﬁxed. Given any prescribed sequence of bends and initial direction, one sees that there is only one possibility for the underlying map γ with endpoint a ﬁxed point Q by tracing the broken line back from Q. Each such underlying map γ supports only a ﬁnite number of broken lines modulo I · k[P ]. This yields the ﬁniteness of (1). MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 97 The argument for the ﬁniteness statement in (2), once the ﬁrst part of (2) is es- tablished, is the same. For the ﬁrst part of (2), consider a broken line γ contributing to Lift (q).Wetake Q ∈ σ ,inthe notation of Lemma 2.19.Write Mono(γ ) = c z .If Q + L −1 r(q ) ∈ ρ σ then the statement follows from Lemma 2.19. Otherwise (by the deﬁnition L + of broken line) γ crosses ρ , which is the last ray of and the last ray of Supp (D) it crosses before reaching Q. Now the result follows from Lemma 2.19 and the deﬁnition of broken line. Deﬁnition 2.26. — Assume that κ ∈ J for at least one ray ρ ∈ . We say a scattering ρ,ϕ diagram D is consistent if for all ideals I ⊂ P with I = J and for all q ∈ B (Z), the following holds. Let Q ∈ B be chosen so that the line joining the origin and Q has irrational slope, and Q ∈ B 0 0 similarly. Then: (1) If Q, Q ∈ σ ∈ , then we can view Lift (q) and Lift (q) as elements of R ,and max Q Q σ,I as such, we have Lift (q) = θ Lift (q) Q γ,D Q for γ a path contained in the interior of σ connecting Q to Q . (2) If Q ∈ σ and Q ∈ σ with σ ∈ and ρ = σ ∩ σ a ray, and furthermore − − + + ± max + − Q and Q are contained in connected components of B\ Supp (D) whose closures contain − + ρ , then Lift (q) ∈ R are both images under ψ of a single element Q σ ,I ρ,± ± ± Lift (q) ∈ R . ρ ρ,I Of course the deﬁnition is introduced so that the following construction works: Construction 2.27 (Construction of ϑ ). — Suppose D ⊂ Yhas n ≥ 3 irreducible com- ponents, and that D is a consistent scattering diagram for data (B, ),P, ϕ and J. Assume further that κ ∈ Jfor all ρ ∈ , so that we may apply Construction 2.14.Wenow con- ρ,ϕ struct for any I with I = Ja function ϑ ∈ (X , O ) for q ∈ B(Z) = B (Z)∪{0}. q X 0 I,D I,D We deﬁne ϑ = 1. Next, let q ∈ B (Z).For each ray ρ ∈ contained in σ ∈ , 0 0 ± max choose two points Q ∈ B, one each in the two connected components of B\ (Supp (D)∪ ρ I + − ρ) which are adjacent to ρ,with Q ∈ σ and Q ∈ σ . + − ρ ρ We ﬁrst note that Lift (q) is a well-deﬁned element of R , independent of the Q σ ,I ± + particular choice of Q : given a choice say of Q = Q and another choice Q ,we ρ ρ take a path γ connecting Q and Q wholly contained in the connected component of B \ (Supp (D) ∪ ρ) containing Q and Q . By Deﬁnition 2.26, (1), it then follows that Lift (q) = Lift (q). Q Q By Deﬁnition 2.26,(2),wehaveanelement Lift (q) ∈ R whose image under ρ ρ,I ψ is Lift ± (q). It then follows via another application of Deﬁnition 2.26, (1), applied ρ,± to the path of Figure 2,thatif ρ , ρ are adjacent rays in , then Lift (q) and Lift (q) ρ ρ glue under the identiﬁcation of open subsets of U and U given by θ .Thusall ρ,I ρ ,I γ,D 98 MARK GROSS, PAUL HACKING, AND SEAN KEEL these elements of the rings R for ρ ∈ glue to give a regular function on X ,by ρ,I I,D construction of this latter space. This regular function is what we call ϑ . Theorem 2.28. —Suppose D has n ≥ 3 irreducible components, and let ϕ be a multivalued piecewise linear function on B such that κ ∈ J for all rays ρ ∈ .Let D be a consistent scattering ρ,ϕ diagram and I ⊂ P an ideal with I = J.Set X := Spec X , O . I X I,D I,D Since X has the structure of a scheme over Spec R ,sodoes X , which we write as I I I,D f : X → Spec R . I I I Then (1) X contains X as an open subset and f is ﬂat with ﬁbre over a closed point x of Spec R I I I I,D isomorphic to the n-vertex V . (2) For each q ∈ B(Z), there is a section ϑ ∈ (X , O ),and theset q I X ϑ | q ∈ B(Z) is a free R -module basis for (X , O ). I I X Proof. — Construction 2.27 constructs regular functions ϑ on X ,hence by def- I,D inition of X ,weobtain ϑ ∈ (X , O ). I q I X o o Now note that X = X as deﬁned in Section 2.1.Indeed,for any (d, f ) ∈ D with J,D J dim τ = 2we have f ≡ 1 mod J, so the open sets U ,U are glued trivially. Similarly, d d ρ ,J ρ ,J i i+1 if dim τ = 1 then since κ ∈ J, the rings R as given in (2.7)and (2.5) coincide and are d ρ,ϕ ρ,I glued trivially. Thus with I = J, we see the gluing constructions 2.7 and 2.14 coincide. Note that with the assumption that κ ∈ Jfor all rays ρ , ρ,ϕ X = Spec R [ ] \ (Spec R )×{0} J J by Lemma 2.12. We see that the canonical map R · ϑ → X , O J q X q∈B(Z) is an isomorphism. Indeed, by Lemma 2.10, (X , O ) R [ ]. Furthermore, under X J this isomorphism, ϑ is clearly taken to z ∈ R [ ]. This is because the only broken lines q J contributing to Lift (q) modulo J for any Q is the straight line with endpoint Q, and this provides a contribution only if Q lies in the same maximal cone as q. It also follows that X := Spec (X , O ) = Spec R [ ] is ﬂat over Spec R and J X J J the ﬁber over a closed point x is given by Spec k[ ]. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 99 Now let I be an ideal with I = J. Let i: X ⊂ X be the inclusion. Deﬁne a ringed space X with underlying topological space X by O := i O . Then the natural map J X ∗ X I,D O → O is surjective by the existence of the lifts ϑ .Thus X / Spec R is a ﬂat defor- X X q I J I mation of X / Spec R by Lemma 2.29 below. Now since X is afﬁne it follows that X is J J J also afﬁne, so X = X := Spec (X , O ). I X I I,D I,D We showed above that the ϑ form an R -module basis of (X , O ). Now since q J J X X / Spec R is a ﬂat inﬁnitesimal deformation of X / Spec R it follows that the ϑ form a I I J J q R -module basis of (X , O ), see Lemma 2.30 below. I I X Lemma 2.29. —Let X /S be a ﬂat family of surfaces such that the ﬁbres satisfy Serre’s 0 0 condition S .Let i: X ⊂ X be the inclusion of an open subset such that the complement has ﬁnite 2 0 ﬁbres. Note that i O = O by Lemma 2.10. ∗ X X o o Let S ⊂ S be an inﬁnitesimal thickening of S and let X → S be a ﬂat deformation of X /S 0 0 0 over S. Deﬁne a family of ringed spaces X → S by O := i O . X ∗ X Then X/S is a ﬂat deformation of X /S (that is, X/S is ﬂat and X = X × S )ifand 0 0 0 S 0 only if the map (2.12) O := i O o → i O = O X ∗ X ∗ X X is surjective. Proof. — The condition is clearly necessary. Conversely, suppose (2.12) is surjective. Let I ⊂ O be the nilpotent ideal deﬁning S ⊂ S. Let X /S denote the nth order inﬁnitesimal thickening of X /S determined by 0 n 0 0 o n+1 n+1 X /S, that is, O o = O o /I · O o and O = O /I .Deﬁne X /S by O := i O o . X X X S S n n X ∗ X n n n n Note that O → O is surjective because O → O is surjective by assumption. We X X X X n 0 0 show by induction on n that X /S is a ﬂat deformation of X /S .For n = 0 there is n n 0 0 nothing to prove. Suppose the induction hypothesis is true for n. Since X /S is ﬂat n+1 n+1 (being the restriction of the ﬂat family X /StoS )wehaveashortexact sequence n+1 n+1 n+2 o o 0 → I /I ⊗ O → O → O o → 0. X X X 0 n+1 Applying i we obtain an exact sequence n+1 n+2 0 → i I /I ⊗ O → O → O . ∗ X X X n+1 n n+1 n+2 By Lemma 2.10 the ﬁrst term is equal to I /I ⊗ O .Moreover, thelastarrow is surjective because O → O is surjective, O /I · O = O by the induction X X X X X n+1 0 n n 0 hypothesis, and I is nilpotent, as in Theorem 8.4 of [Ma89] (where the module M need not be ﬁnitely generated for the argument given there to work). So we have an exact sequence n+1 n+2 (2.13) 0 → I /I ⊗ O → O → O → 0. X X X 0 n+1 n 100 MARK GROSS, PAUL HACKING, AND SEAN KEEL n+1 It follows that O /I · O = O . (Indeed, consider the map X X X n+1 n+1 n n+1 α: I ⊗ O → O ,α(f ⊗ g) = fg. X X n+1 n+1 We claim that α is equal to the composition of the map n+1 n+1 n+2 β : I ⊗ O → I /I ⊗ O X X n+1 0 given by the natural maps on the factors and the ﬁrst map γ of the exact sequence (2.13). Since O = i O by deﬁnition, it sufﬁces to check the equality after restriction to X ∗ X n+1 n+1 X , where it is obvious. The map β is surjective because O → O is surjective. So X X n+1 0 n+1 the image of γ is equal to the image of α,namely I · O .) Now by [Ma89], Theo- n+1 rem 22.3, p. 174, the exact sequence (2.13)shows that X /S is a ﬂat deformation of n+1 n+1 X /S . 0 0 Lemma 2.30. —Let A → B be a ﬂat homomorphism of Noetherian rings and I ⊂ A a nilpotent ideal. Suppose given a set S of elements of B such that the reductions of the elements of S form an A/I-module basis of B/IB.Then S is an A-module basis of B. Proof. — Since I is nilpotent and S generates B/IB it is clear that S spans B by Theorem 8.4 of [Ma89]. So we have an exact sequence 0 → K → A → B → 0. Tensoring with A/I we obtain an exact sequence 0 → K/IK → (A/I) → B/IB → 0 using ﬂatness of B over A. We deduce that K/IK = 0 by our assumption, hence K = 0 because I is nilpotent. Proposition 2.31. —Let X /S := Spec R be the family of Theorem 2.28. Then the relative I I I dualizing sheaf ω is trivial. It is generated by the global section givenonlocal patches U by X /S ρ ,I I I i dlog X ∧ dlog X = dlog X ∧ dlog X . Here we take the rays ρ in counter-clockwise order, i−1 i i i+1 j after choosing an orientation on B, to obtain a consistent choice of signs. Proof. — By the adjunction formula for the closed embedding U ⊂ A × G × S , ρ m,X I i X ,X i i−1 i+1 the dualizing sheaf ω is freely generated over U by the local section in the statement. X /S ρ I I i These sections patch to give a generator of ω because the scattering automor- X /S I,D phisms preserve the torus invariant two-forms. Both ω and O satisfy the relative X /S X I I I ∗ o S property i i F = F where i: X ⊂ X is the inclusion ([H04], Appendix, where the 2 ∗ I hypothesis of slc is not needed), hence ω is freely generated by . X /S I I MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 101 2.4. The algebra structure. — In the previous section, we saw that the R -algebra A = X , O I X I,D I,D deﬁning the ﬂat deformation X has an R -module basis of theta functions {ϑ | m ∈ I I m B(Z)}. Here we derive a description of the multiplication rule on R using the geometry of the integral afﬁne manifold B. Besides being an attractive combinatorial description of the multiplication rule, we will use this (in the case of the canonical scattering diagram can D )inSection 6 to prove that our deformation extends over completions of larger strata of Spec k[P], as well as for the case that D has 1 or 2 irreducible components. Deﬁnition 2.32. — For a broken line γ with endpoint Q ∈ τ ∈ ,deﬁne s(γ ) ∈ ,c(γ ) ∈ k[P] by demanding that ϕ (s(γ )) Mono(γ ) = c(γ ) · z . Write Limits(γ ) = (q, Q) if γ is a broken line for q and has endpoint Q ∈ B. Remark 2.33. —Recallthat B in fact has the structure of an integral linear mani- fold. One feature of such manifolds is that for any simply connected set U ⊂ B , there is a canonical linear immersion U → , compatible with parallel transport inside U. R,U In particular, if q is a point of B with q ∈ σ ∈ ,and τ ⊂ σ , then the canonical embedding of a neighbourhood of τ \{0} in identiﬁes q with a point of . τ,R τ,R Theorem 2.34. —Let q , q ∈ B(Z). In the canonical expansion 1 2 ϑ · ϑ = α ϑ , q q q q 1 2 q∈B(Z) where α ∈ R for each q, we have q I α = c(γ )c(γ ) q 1 2 (γ ,γ ) 1 2 Limits(γ )=(q ,z) i i s(γ )+s(γ )=q 1 2 Here z ∈ B is a point very close to q contained in a cell τ , and we identify q with a point of using 0 τ Remark 2.33. Proof. — To identify the coefﬁcient of ϑ , choose a point z ∈ B very close to q,and q q 1 2 describe the product using the lifts of z , z at z: Lift (q ) Lift (q ) = α Lift q . z 1 z 2 q z q 102 MARK GROSS, PAUL HACKING, AND SEAN KEEL Now observe ﬁrst that there is only one broken line γ with endpoint z and s(γ ) = q ∈ : this is the broken line whose image is z + R q. Indeed, the ﬁnal segment of such a γ is ≥0 on this ray, and this ray meets no scattering rays, so the broken line cannot bend. Thus the coefﬁcient of Lift (q) on the right-hand side of the above equation can be read off by ϕ (q) looking at the coefﬁcient (in R )of z . This gives the desired description. 3. The canonical scattering diagram can 3.1. Deﬁnition. — Here we give the precise deﬁnition of D . As explained in the introduction, it is, roughly speaking, deﬁned in terms of maps A → Y \ D, which are algebro-geometric analogues of the holomorphic disks used for instanton corrections in the symplectic heuristic. We begin by recalling necessary facts about relative Gromov- Witten invariants used to count these curves. ˜ ˜ ˜ Deﬁnition 3.1. —Let (Y, D) be a non-singular rational surface with D an anti-canonical ˜ ˜ cycle of rational curves, and let C be an irreducible component of D. Consider a class β ∈ A (Y, Z) such that k D = C β i (3.1) β · D = 0 D = C for some k > 0.Let F be the closure of D \ C,and let o o ˜ ˜ Y := Y \ F, C := C \ F. o o Let M(Y /C ,β) be the moduli space of stable relative maps of genus zero curves representing the class β with tangency of order k at an unspeciﬁed point of C .(See[Li00], [Li02] for the algebraic deﬁnition for these relative Gromov-Witten invariants, and [LR01], [IP03] for the original symplectic deﬁnitions.) We refer to β informally as an A -class. The virtual dimension of this moduli space is −K · β + (dim Y − 3) − (k − 1) = 0. ˜ β Here the ﬁrst two terms give the standard dimension formula for the moduli space of stable rational curves in Y representing the class β,and theterm k − 1 is the change in dimension given by imposing the k -fold tangency condition. The moduli space carries a virtual fundamental class. Furthermore, we have o o Lemma 3.2. — M(Y /C ,β) is proper over k. Proof. — This follows as in the proof of [GPS09], Theorem 4.2. In brief, let R be a valuation ring with quotient ﬁeld K, with S = Spec R, T = Spec K. We would like to o o ˜ ˜ extend a morphism T → M(Y /C ,β) to S. We know that the moduli space M(Y/C,β) is proper, so we obtain a family of relative stable maps C → Sto Y. We just need to show MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 103 that in fact the image of the closed ﬁbre C lies in Y . However, the argument in the proof of [GPS09], Theorem 4.2 shows that if the image of C intersects F, then C must be of 0 0 genus at least 1, which is not the case. Given this, we deﬁne N := 1. o o vir [M(Y /C ,β)] ˜ ˜ Morally, one should view N as counting maps from afﬁne lines to Y \ D whose closures represent the class β . In what follows, we ﬁx as usual the pair (Y, D), with tropicalisation (B, ),and ϕ the function given by Example 2.3 for some choice of η : NE(Y) → P. We can assume here that D has an arbitrary number of irreducible components. Deﬁnition 3.3. —Fix aray d ⊂ B with endpoint the origin, with rational slope. If d coincides with a ray of ,set := ; otherwise, let be a reﬁnement of obtained by adding the ray d and a number of other rays chosen so that each cone of is integral afﬁne isomorphic to the ﬁrst quadrant of R . −1 This gives a toric blow-up π : Y → Y (the identity in the ﬁrst case) by Lemma 1.6.Let C ⊂ π (D) be the irreducible component corresponding to d. Let τ ∈ be the smallest cone containing d.Let m ∈ be a primitive generator of the d d τ tangent space to d, pointing away from the origin. Deﬁne η(π (β))−ϕ (k m ) ∗ τ β d f := exp k N z . d β β Here the sum is over all classes β ∈ A (Y, Z) satisfying (3.1). Note that if N = 0, then necessarily 1 β o o ˜ ˜ M(Y /C ,β) is non-empty, and thus β ∈ NE(Y),so π (β) ∈ NE(Y). We note the numbers N ∗ β do not depend on the particular choice of reﬁnement . Indeed, further reﬁning does not change the o o pair Y /C , and hence does not change the numbers N . We deﬁne can D := (d, f )| d ⊂ B a ray of rational slope . We call a class β ∈ A (Y, Z) an A -class if N = 0. 1 β Note that all rays of the canonical scattering diagram are outgoing. Remark 3.4. — In theory, one should be able to use logarithmic Gromov-Witten invariants ([GS11]or[AC11]) to deﬁne N without the technical trick of blowing up and working on an open variety. This would be done by working relative to D, and counting rational curves of class β with one point mapping to the boundary with speciﬁed orders of tangency with each boundary divisor, with non-zero order of tangency with either 104 MARK GROSS, PAUL HACKING, AND SEAN KEEL one divisor D or two adjacent divisors D ,D . However, some additional arguments i i i+1 are required to compare logarithmic invariants with the invariants described above as developed in [GPS09], and we do not wish to do this here. Lemma 3.5. —Let J ⊂ P be an ideal with J = J.Suppose themap η: NE(Y) → P satisﬁes the following conditions: (1) For any ray d ⊂ B of rational slope, let π : Y → Y be the corresponding blow-up. We re- quire that if dim τ = 2 or dim τ = 1 and κ ∈ J then for any A -class β contributing d d τ ,ϕ to f , we have η(π (β)) ∈ J. d ∗ (2) For any ideal I with I = J, there are only a ﬁnite number of d and A -classes β such that η(π (β)) ∈ I. can Then D is a scattering diagram for the data (B, ), P,ϕ, and J. Proof. — Note that η(π (β))−ϕ (k m ) ∗ τ β d z ∈ Ik[P ] if and only if η(π (β)) ∈ I. So the hypotheses of the lemma imply conditions (2)–(4) of Deﬁnition 2.13. Example 3.6. —Let σ ⊂ A (Y) ⊗ R be a strictly convex rational polyhedral cone 1 Z containing NE(Y). (This can be obtained as the dual of a strictly convex rational polyhe- dral cone in Pic(Y) ⊗ R which spans this latter space and is contained in the nef cone.) Let P = σ ∩ A (Y). Since σ is strictly convex, P = 0. For any m-primary ideal I, P \ I is a ﬁnite set. Let η : NE(Y) → P be the inclusion. Then the ﬁniteness hypotheses of the above Lemma hold for J = m (note that the conditions (3.1) determine β ∈ A (Y) given π (β)). Example 3.7. — We return to the example (Y, D) of a del Pezzo surface together with a cycle of 5 (−1)-curves studied in Example 1.9.Let P = NE(Y) and η be the can identity. Let J = m ⊂ Pand I ⊂ P be an ideal with I = J. Then D consists of ﬁve rays: can [E ]−ϕ (v ) i ρ i D = ρ , 1 + z | 1 ≤ i ≤ 5 . Here E is the unique (−1)-curve in Y which is not contained in D and meets D transver- i i sally, and v is the primitive generator of the ray ρ corresponding to D . To derive this i i i formula from the above deﬁnition of the canonical scattering diagram one needs to can show that the only possible stable relative maps contributing to D are multiple cov- ers of the E ’s, and that a k-fold multiple cover contributes a Gromov-Witten invariant of k−1 2 (−1) /k . It is easier to compute this using the main result of [GPS09], which is done by way of Theorem 3.25. See Example 3.26. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 105 FIG. 3. — The different types of broken lines in Example 3.7 can If we accept this description of D , then we can describe all broken lines and the multiplication law given by this diagram. We ﬁrst note that no broken line can wrap around 0 ∈ B, i.e., if a broken line leaves acone σ ∈ , it will never return to that cone. It is enough to check this for a straight max line (as the bending in any broken line for D is always away from the origin), and this can is easily veriﬁed, using e.g., Figure 1. Next, since the only scattering rays are the rays ρ ∈ ,if q, Q ∈ σ ∈ , then the max obvious straight line is the unique broken line for q with endpoint Q. Thus if we describe o ϕ (q) ϑ in the open subset of X corresponding to σ , ϑ is just the monomial z .It can q q I,D follows that a b ϑ ϑ = ϑ av +bv v v i i+1 i i+1 for a, b ≥ 0. In particular, the ϑ ’s generate the k[P]/I-algebra (X , O ),and thealge- v I X i I bra structure is determined once we compute ϑ · ϑ . v v i i+2 We consider a broken line for v . One checks the following, using Figure 1 and the can above description of D : The broken line can cross at most two rays of , and it bends at most once, at the last ray of that it crosses. See Figure 3. From this one deduces using Theorem 2.34: [D ] [E ] i i (3.2) ϑ ϑ = z ϑ + z . v v v i−1 i+1 i [D ] The term z · ϑ corresponds to two straight broken lines for v ,v , with endpoint v i−1 i+1 [D ] [E ] i i the point v of ρ .The term z · z is the coefﬁcient of 1 = ϑ . To compute this we use i i 0 the invariance of broken lines, and so choose a generic point Q near 0 and compute the coefﬁcient α of ϑ using pairs γ as in Theorem 2.34 whose ﬁnal directions are opposite, 0 0 i i.e., s(γ ) + s(γ ) = 0. If we take Q ∈ σ , then there is exactly one term contributing 1 2 i,i+1 to α : γ will bend once where it crosses ρ ,and γ is straight. Alternatively, one can 0 1 i 2 106 MARK GROSS, PAUL HACKING, AND SEAN KEEL use the explicit expressions for Lift (v ), j = i − 1, i and i + 1, and see they satisfy the Q j relation (3.2). Onecan checkthatthe ﬁveequations(3.2)deﬁne X . These equations are alge- braic, and in fact deﬁne a ﬂat family over Spec k[NE(Y)]. (This is always the case in the non-negative semi-deﬁnite case, see Corollary 6.11). can Our goal now is to prove consistency of D , as stated in the following (the ﬁnal step in the construction of our mirror family): Theorem 3.8. — Suppose that we are given a map η : NE(Y) → P such that ϕ is deﬁned as in Example 2.3 by κ = η([D ]). Suppose furthermore the following conditions hold: ρ,ϕ ρ (I) For any A -class β , η(π (β)) ∈ J; (II) For any ideal I with I = J, there are only a ﬁnite number of A -classes β such that η(π (β)) ∈ I. (III) η([D ]) ∈ J for at least one boundary component D ⊂ D. i i can Then D is a consistent scattering diagram. We include here an observation we will need later showing that the canonical scat- tering diagram only depends on the deformation class of (Y, D). Lemma 3.9. —Let (Y , D) → S be a ﬂat family of pairs over a connected base S, with each ﬁbre (Y , D ) being a non-singular rational surface with anti-canonical cycle. Suppose further that there s s is a trivialization D = D × S and the restriction map Pic(Y ) → Pic(Y ) is an isomorphism for any s ∈ S. This in particular gives a canonical identiﬁcation A (Y , Z) with A (Y , Z) for any s, s ∈ S. 1 s 1 s Then for any s, s ∈ S, (Y , D ) and (Y , D ) induce the same canonical scattering diagram. s s s s Proof. — It is enough to show that the numbers N are deformation invariants ˜ ˜ in the above sense, i.e., if we are given a family π : (Y , D) → Swith each ﬁbre as in Deﬁnition 3.1, with an irreducible component C ⊂ D, then the number N := 1 β,s ˜ o o vir [M(Y /C ,β)] s s is independent of s. Indeed, once this is shown, then if N = 0, necessarily β deﬁnes a β,s class in NE(Y ),aswellasin NE(Y ), under the chosen identiﬁcation. This invariance s s follows from the standard argument that (relative) Gromov-Witten invariants are defor- mation invariants, with a little care because our target spaces are open. For this, one o o o o ˜ ˜ considers the moduli space M(Y /C ,β) of stable maps to Y relative to C and whose o o composition with π is constant. Then one has a map ψ : M(Y /C ,β) → Swhose ﬁbre o o o o ˜ ˜ over s is M(Y /C ,β). Letting ξ be the inclusion of this ﬁbre in M(Y /C ,β),deforma- s s tion invariance will follow if we know that vir vir ! o o o o ˜ ˜ ξ M Y /C ,β = M Y /C ,β s s MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 107 and ψ is proper. The ﬁrst statement is standard in Gromov-Witten theory, see e.g. the argument of Theorem 4.2 of [LT98] in the non-relative case, which carries over to the relative case. The second point, the properness of ψ , follows exactly as in the proof of Lemma 3.2. 3.2. Consistency: overview of the proof. — We will describe in detail the intuition be- hind each step of the proof of consistency. In the next subsection, we will work somewhat more generally with a more general scattering diagram D for certain steps, as this will be needed in [K3] for the K3 case. However, for the discussion here let us assume we are only studying the consistency of the canonical scattering diagram. Step I. We can replace (Y, D) with a toric blow-up of (Y, D). This is straightforward— toric blowups just correspond to reﬁnements of , but do not change broken lines or scattering diagrams. Step II. We can assume that (Y, D) has a toric model and P is a ﬁnitely generated submonoid of A (Y, Z) containing NE(Y), with η the inclusion.ByStepIandProposition 1.3,wecan η¯ ψ assume (Y, D) has a toric model. We can then always factor η as NE(Y)−→P−→P where P is a ﬁnitely generated submonoid of A (Y, Z) containing NE(Y) with η¯ the inclusion. In this case there are two canonical scattering diagrams, D and D deﬁned ¯ ¯ using η¯ : NE(Y) → Pand η : NE(Y) → P respectively. Then D can be obtained from D essentially just by applying ψ to each exponent appearing in each function f . In this case we show that if consistency holds for D then it holds for D. The idea is that given a broken line γ for D, we can get something like a broken line for D by applying ψ to the exponents of monomials attached to γ . However, this isn’t necessarily a broken line for D. Indeed, there might be two different broken lines for D,say γ and γ , which after we apply ψ give broken lines with the same sequence of attached exponents. These should not arise as distinct broken lines for D, and we have to combine the monomials attached to these broken lines. This requires a certain amount of book- keeping. Step III. Reduction to the Gross-Siebert locus. By Step II we can assume we have a toric ¯ ¯ model p : Y → Y. Let H be an ample divisor on Y. Shrinking P if necessary, we can ∗ ⊥ assume that P has a face of the form P ∩ (p H) . Let G be the monomial ideal which is gp the complement of this face, E the subgroup of P generated by P \ G. The main work in this step is to show that we can replace P by P+ E. This requires a bit of analysis of the can rays (ρ , f ) of D . In particular, we need to understand the contribution to f coming i ρ ρ i i from the exceptional curves of p meeting D . × o After doing this, we have P = E, so now X lives over the thickening of a torus I,D gs T we call the Gross-Siebert locus. Step IV. Pushing the singularities to inﬁnity. This is the crucial step, and we explain carefully the intuition here. In [GS07], Gross and Siebert considered a smoothing con- struction associated to an integral afﬁne manifold with singularities where (in the two- dimensional case) the singularities occurred only in the interior of edges of a polyhedral 108 MARK GROSS, PAUL HACKING, AND SEAN KEEL decomposition of B, rather than at the vertices. The case at hand, with one singularity at the origin, does not ﬁt into that framework. In particular, in the Gross-Siebert world, the 1 k singularities must have monodromy of the form for some k > 0, with the tangent line to the edge containing the singularity being the invariant direction; in analogy with the Kodaira classiﬁcation, we call this an I singularity. Indeed, one expects a cycle of k two-spheres as ﬁbre over such a point in the SYZ picture. Here, we can view such a surface as being obtained by factoring the complicated sin- gularity 0 ∈ B into I singularities along the edges of . We should have an I singularity k k on the ray ρ where k is the number of exceptional divisors of p : Y → Y intersecting D . i i i ¯ ¯ ¯ ¯ This process can be described as follows. Let (B, ) be the fan associated to (Y, D). There is a piecewise linear isomorphism ν : B → B which identiﬁes each cone in with the corresponding cone in . This is an isomor- phism of integral afﬁne manifolds outside of ρ , but it is not afﬁne along ρ . There is a i i natural one-parameter family of integral afﬁne manifolds interpolating between the two structures by a process Kontsevich and Soibelman [KS06]call moving worms. Precisely, choose points y ∈ ρ \{0}.Let := {y | 1 ≤ i ≤ n},B := B \ . Put a new afﬁne struc- i i i ture on B compatible with the afﬁne structures on the interior of each maximal cell by deﬁning a -piecewise linear function to be linear if its restriction to a small neighbour- hood of (y ,+∞) ⊂ ρ in B is B-linear, and its restriction to a small neighbourhood of i i [0, y ) ⊂ ρ in B is B-linear. Call the resulting integral afﬁne manifold with singulari- i i ties B .The map ν : B → B is a linear isomorphism near 0. This new manifold can be seen to have an I singularity at y , with invariant direction ρ . k i i Now if we were to apply the algorithm of Gross and Siebert [GS07]to B ,one would ﬁnd roughly that one obtains a scattering diagram which initially has two rays em- anating from each singularity. The rays emanating from ρ are initially contained in ρ ; i i one of these goes out to inﬁnity and the other passes through the origin and then to in- ﬁnity. Where all these rays meet at the origin, one must follow a procedure of Kontsevich and Soibelman [KS06] and add some additional rays to ensure that the composition of automorphisms associated to the rays about a loop centered at the origin is the identity. We then obtain a scattering diagram which can be shown to be very close to the canoni- cal scattering diagram, the only difference being the segments of the rays between the y and the origin. We do not actually work with this afﬁne manifold with singularities. Rather, we instead push the singularities y to inﬁnity. In doing so, we replace B with B. We transfer the canonical scattering diagram D to a scattering diagram D on B, differing from D essentially only by changing the rays supported on the ρ ’s in a simple way motivated by the above description. Once this is done, we show consistency of D is equivalent to consistency of D. Now we no longer have to deal with any singularities. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 109 It is much easier to determine consistency when there are no singularities. In par- ticular, we appeal to a result in [CPS], which shows that D is consistent provided that the composition of automorphisms associated to the rays about a loop centered at the origin is the identity. We say such a scattering diagram is compatible. The important point is that we can now make sense of such a statement: when we had a singularity at the origin, there was no common ring which the automorphisms associated to rays could act on. However, without a singularity at the origin, there are such rings, as appeared in [GS07]. Step V. D satisﬁes the required compatibility condition. This step is really the punch-line, explaining why the particular choice of the canonical scattering diagram D gives a dia- gram D which is compatible. We make use of [GPS09] to link the enumerative deﬁnition of D to the notion of compatibility. Indeed, the deﬁnition of the canonical scattering dia- gram was originally obtained by working backwards from the enumerative description of [GPS09]. This connection is worked out in Section 3.4. 3.3. Consistency: reduction to the Gross-Siebert locus. — We now begin the proof of The- orem 3.8, following the outline given in Section 3.2. We will, however, prove a number of lemmas in a slightly more general context, as we will need some more general consis- tency results in [K3]. We assume we are given (Y, D), η : NE(Y) → Pand ϕ deﬁned as in Example 2.3, and a radical ideal J ⊆ P. Suppose we are given a scattering diagram D can for this data; the application in this paper will be D = D . In particular, the hypotheses can of Theorem 3.8 imply D is a scattering diagram for this data. Step I. Replacing (Y, D) with a toric blowup. ˜ ˜ Proposition 3.10. —Let p : (Y, D) → (Y, D) be a toric blowup. Then if we take η˜ := η ◦ p : NE(Y) → P, then D can also be viewed as a scattering diagram for B , P.Furthermore, ˜ ˜ (Y,D) if D is consistent for this latter data, it is consistent for the data B , P. (Y,D) ˜ ˜ Proof. — Decorate notation, writing for example B, for the singular afﬁne mani- ˜ ˜ fold with subdivision into cones associated to (Y, D). By Lemma 1.6, we have a canonical ˜ ˜ identiﬁcation of the underlying singular afﬁne manifolds B = B, and is the reﬁnement of obtained by adding one ray for each p-exceptional divisor. We have multivalued piecewise linear functions ϕ on B and ϕ˜ on B. We can in fact choose representatives so ˜ ˜ ˜ that ϕ˜ = ϕ. Indeed, κ = η(p ([D ])) where D is the irreducible component of Dcor- ρ,ϕ˜ ∗ ρ ρ ˜ ˜ responding to ρ.But p ([D ]) = 0if ρ ∈ ,and p ([D ]) =[D ] if ρ ∈ .Thus ϕ˜ in ∗ ρ ∗ ρ ρ fact has the same domains of linearity as ϕ, and the same bending parameters, so we can choose representatives which agree. As a consequence, we note that the sheaves P and P on B deﬁned using ϕ and ϕ˜ coincide. Furthermore, if τ˜ ⊂˜ σ are cones in ,with τ ∈ the smallest cone containing τ˜ and σ ∈ the smallest cone containing σ˜ , there is a canonical identiﬁcation of P with P and a canonical isomorphism ϕ˜ τ˜ (3.3) R R ; τ, ˜ I τ,I 110 MARK GROSS, PAUL HACKING, AND SEAN KEEL note the slightly non-trivial case when dim τ˜ = 1 but dim τ = 2, in which case we use the fact that κ = 0. τ, ˜ ϕ˜ Using these identiﬁcations, we can view D as living on B, and as such, one sees from the deﬁnition that D is a scattering diagram for the data B, P, ϕ˜ . Now suppose I = J. One observes that the set of broken lines contributing to ˜ ˜ Lift (q) are the same whether we are working in B or B. Thus if Q∈˜ σ ∈ ,Lift (q) ∈ Q max Q ˜ ˜ R , deﬁned using B, coincides under the isomorphism (3.3) with Lift (q) ∈ R .From σ, ˜ I Q σ,I this one sees easily that if D is consistent for Y, it is consistent for Y. Corollary 3.11. —Given Y, P,η, J satisfying the hypotheses of Theorem 3.8, then Theo- rem 3.8 holds for this data if it holds for the data Y, P, η, ˜ J. Proof. — By the proposition, one just needs to check that the canonical scattering diagrams deﬁned using Y or Y are identical. Indeed, given a ray d ⊂ B, we can choose ˜ ˜ a reﬁnement of which is also a reﬁnement of , giving maps π˜ : Y → Yand π : Y → Y. Then for an A -class β ∈ A (Y , Z), η(π (β))=˜ η(π˜ (β)),and so f is the 1 ∗ ∗ d same for Yand Y. Step II. Changing the monoid P. We would like to change the monoid P, which was fairly arbitrary, to one with better properties. For this step, assume we are given monoid homomorphisms η¯ ψ NE(Y)−→P−→P with η = ψ ◦¯ η.Then η and η¯ induce multivalued piecewise linear functions ϕ and ϕ¯ respectively, via Example 2.3,with ϕ = ψ◦¯ ϕ. The monoids P,P and functions ϕ, ϕ¯ yield ¯ ¯ ¯ sheaves P and P over B .The map ψ : P → P induces a map of sheaves ψ : P → P using ϕ = ψ ◦¯ ϕ, and hence it also induces monoid homomorphisms ψ : P → P for ϕ¯ ϕ τ τ any τ ∈ \{0}. ¯ ¯ ¯ Suppose D is a scattering diagram for the data B, P, m¯ = P \ P .For each ray (d, f ), f ∈ k[P ].Now we cantry to deﬁne ψ(f ) by applying ψ to each exponent of d d ϕ¯ d f , but in general, this need not make sense even formally since ψ may take an inﬁnite number of exponents occurring in f to a single element of P. However, we shall write ψ(f ) for such an expression if it does make sense as an element of k[P ].If ψ(f ) makes d ϕ d sense for each (d, f ) ∈ D, we write ¯ ¯ ψ(D) = d,ψ(f ) | (d, f ) ∈ D . d d Proposition 3.12. — In the above situation, suppose D is a scattering diagram for the data ¯ ¯ ¯ ¯ B, P, m¯ = P\ P , such that D = ψ(D) makes sense and is a scattering diagram for the data B, P, J, where J is a radical ideal in P. Assume that κ ∈ J for at least one ray ρ ∈ .If D is consistent for ρ,ϕ P, η, ¯ m¯ , then D is consistent for P,η, J. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 111 Proof.—Let q ∈ B (Z).Thenif γ¯ is a broken line for q with endpoint Q with respect to the barred data, i.e., P, P etc., we can construct what we shall call ψ(γ) ¯ . This will be the data required for deﬁning a broken line for the unbarred data. The underlying map of ψ(γ) ¯ coincides with that of γ¯ . For the attached monomials, we simply apply ψ to the monomial m (γ) ¯ attached to a domain of linearity L of γ¯ to get the attached monomial for ψ(γ) ¯ . This is not a broken line for the unbarred data, as condition (3) of Deﬁnition 2.16 need not hold. Indeed, when a broken line bends at a ray, the attached monomial will be replaced by a term in (2.9). However, there might be several different ¯ s terms c z appearing in (2.9)with ¯ s ∈ P such that ψ(¯ s ) all coincide with some s ∈ P . i i ϕ i ϕ τ τ ¯ s Each choice c z leads to a different broken line γ¯ ,but ψ(γ¯ ) is not a broken line because i i i ψ(¯ s ) s c z = c z is not a term in the formula (2.9) for the monoid P. Rather, one needs to i i replace the collection of broken lines γ¯ with a single one which has monomial c z i i attached after the bend. To deal with this, we need to do a certain amount of book- keeping. Fix an ideal I ⊂ Pwith I = J, Q ∈ σ ∈ ,and let B be the set of broken lines γ¯ for the barred data with endpoint Q such that ψ(Mono(γ) ¯ ) ∈ I · k[P ].The same ﬁniteness argument of Lemma 2.25 shows that B is a ﬁnite set. Note this uses the facts (1) at least one κ ∈ J and (2) all but a ﬁnite number of monomials appearing in D lie ρ,ϕ in I. We deﬁne an equivalence relation on B by saying γ¯ ∼¯ γ provided ψ(γ¯ ) and 1 2 1 ψ(γ¯ ) coincide except possibly for the k-valued coefﬁcients of the monomials attached to the domains of linearity. Given an equivalence class ξ ⊂ B with respect to this equiv- alence relation, we will show there is at most one broken line γ for the unbarred data such that (3.4) ψ Mono(γ) ¯ = Mono(γ ), γ¯∈ξ with there being no such broken line precisely if the above quantity is zero. Furthermore, every broken line γ for the unbarred data with Mono(γ ) ∈ I · k[P ] arises in this way. Deﬁne γ to be the broken line with underlying piecewise linear map given by any element of ξ , with the following attached monomials. For any domain of linearity L=[s, t] for γ , choose a maximal subset ξ ⊂ ξ of broken lines such that the attached ξ L monomials for γ¯ and γ¯ on (−∞, t] do not coincide for any γ¯ , γ¯ ∈ ξ . Then deﬁne 1 2 1 2 L m (γ ) = m ψ(γ) ¯ . L ξ L γ¯∈ξ Assuming that the ﬁnal monomial attached to γ is not zero, one checks easily that γ is a ξ ξ broken line, now satisfying (3) of Deﬁnition 2.16,and (3.4) is satisﬁed since for L the last domain of linearity of γ , one takes ξ = ξ . Furthermore, it is easy to see that any broken ξ L line for the unbarred data with the same underlying map and attached monomials at 112 MARK GROSS, PAUL HACKING, AND SEAN KEEL most differing by their coefﬁcients from γ must in fact coincide with γ . This shows the ξ ξ claim. ¯ ¯ ¯ Since B is ﬁnite, there is some k > 0such that for any γ¯ ∈ B,Mono(γ) ¯ ∈ k[P ] k −1 ¯ ¯ does not lie in m¯ · k[P ].Ifwetake I = ψ (I),thenitisclear from (3.4)that (3.5) ψ Lift (q) = Lift (q), Q Q where Lift (q) is the lift deﬁned with respect to the ideal I and the other barred data, and Lift (q) is deﬁned with respect to the unbarred data and the ideal I. Now D is consistent for m¯ , which implies (1) and (2) of Deﬁnition 2.26 hold for the ideal m¯ + I. Since any ¯ ¯ ¯ ¯ monomial in P \ I appearing in Lift (q) is in P \ (m + I),wecan use(3.5) to deduce consistency of D from consistency of D. Step III. Reduction to the Gross-Siebert locus. As a consequence of Proposition 1.3 and can Corollary 3.11, in order to prove Theorem 3.8 (i.e., with D = D ), we may assume we ¯ ¯ ¯ ¯ ¯ have a toric model p : (Y, D) → (Y, D) with D = D +···+ D . Furthermore, by replac- 1 n ing (Y, D) with a deformation equivalent pair and using Lemma 3.9, we can assume that p is the blowup at distinct points x ,1 ≤ j ≤ ,along D , with exceptional divisors E . ij i i ij Assume D is the proper transform of D , corresponding to the ray ρ ∈ . i i i By Proposition 3.12, we can replace P with a better suited choice of monoid. We can shall do this as follows in the case that D = D .AsinExample 3.6, the nef cone K(Y) ⊂ 1 ∨ A (Y, R) contains a strictly convex rational polyhedral cone σ,so σ ⊂ A (Y, R) is a strictly convex rational polyhedral cone containing NE(Y).The map η : NE(Y) → P gp induces a map η : A (Y, R) → P . Since P is toric, there is some rational polyhedral gp gp cone σ ⊂ P such that P = σ ∩ P . In addition, let H be an ample divisor on Y, so that P P ∗ ⊥ NE(Y) ∩ (p H) is a face of NE(Y), generated by the classes [E ].Now take ij −1 ∨ ∗ σ = η (σ ) ∩ σ ∩ q ∈ A (Y, R)| p H · q ≥ 0 , P 1 and take P = σ ∩ A (Y, Z). ¯ ¯ ¯ As σ is strictly convex, (P) ={0}, m¯ = P\{0}, and if Iis an m¯ -primary ideal, ¯ ¯ ¯ P \ I is ﬁnite. Thus the hypotheses of Theorem 3.8 trivially hold for η¯ : NE(Y) → P. By Proposition 3.12, we can replace P with Pto prove Theorem 3.8. The above discussion shows that in order to complete a proof of consistency of can D , (i.e., Theorem 3.8), we can operate under the following assumptions: Assumptions 3.13. • There is a toric model ¯ ¯ p : (Y, D) → (Y, D) which blows up distinct points x on D , with exceptional divisors E . ij i ij MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 113 • η : NE(Y) → P is an inclusion, and P ={0}. Via Example 2.3, this gives the function ϕ. ∗ ⊥ • There is a face of P whose intersection with NE(Y) is NE(Y) ∩ (p H) .Let G be the prime monomial ideal given by the complement of this face. Note that G = m unless p is an isomorphism. • J = m = P\{0}. • D is a scattering diagram for the data P,ϕ and J. gs Deﬁnition 3.14. —The Gross-Siebert locus is the open torus orbit T of Spec k[P]/G. We now want to work not with the maximal ideal m but with the ideal G, effec- tively extending the families X with I = m to inﬁnitesimal neighbourhoods of the I,D toric boundary stratum of Spec k[P] associated to G. We will then ﬁnd it easier to check the explicit equalities of Deﬁnition 2.26 after restricting to these neighbourhoods of the Gross-Siebert locus. To do so requires showing that the diagram D we are working with can can (D in this paper) is also a scattering diagram for the data P,η, G. In the case of D , this requires analyzing elements of this scattering diagram supported on the ρ . can We ﬁrst perform this analysis for D ; we will then continue our proof assuming that the elements of D supported on the rays ρ take the same form as the corresponding can elements of D modulo G. can For each ray ρ in , we have a unique ray (ρ , f ) ∈ D with support ρ .The i i ρ i following describes f mod G. can Lemma 3.15. — Given Assumption 3.13 with D = D ,viewing f as an element of k[P ]⊗ R with I = m,wehave ϕ I −1 f = g 1 + b X ρ ρ ij i i i j=1 η([E ]) 1 ij where b = z and g ≡ 1mod G. The jth term of the product is the contribution from A -classes ij ρ coming from multiple covers of the p-exceptional divisor E ,and g is the product of contributions from ij ρ all other A -classes. Proof. — Note that in deﬁning f using the deﬁnition of the canonical scattering diagram, we take Y = Y. Now the only terms that contribute to f mod G will involve classes β ∈ NE(Y) ⊂ A (Y) with η(β) ∈ G, so in particular, such a β must be a linear combination c [E ],with k = c . Furthermore, if f : C → Y contributes to N , j ij β j β j=1 f (C) must be contained in E .Indeed,if f (C) has an irreducible component C ij i,j not contained in this set, then η([C ]) ∈ G, so η(f ([C])) ∈ G, as G is an ideal. But η(f ([C])) = η(β), which we have assumed is not an element of G. Since f (C) is connected and intersects D , we now see that the image of f is E for i ij some j , and in particular, f is a degree k cover of E .ThenTheorem 6.1of[GPS09] β ij 114 MARK GROSS, PAUL HACKING, AND SEAN KEEL k −1 2 tells us that the contribution from k -fold multiple covers of E is (−1) /k . From this β ij we conclude that k−1 (−1) −1 f = exp h + k b X ρ ij k=1 j=1 −1 = exp(h) 1 + b X ij j=1 where h ≡ 0mod G. We take g = exp(h). can Corollary 3.16. — D is a scattering diagram for the data (B, ), P, ϕ and G. Proof. — Fixing an I ⊂ Pwith I = G, there exists a bound n such that q ∈ P \ I ∗ 1 implies q · p H < n, where H is a ﬁxed ample divisor on Y. Thus if β is an A -class with η(π (β)) ∈ P\ I, there are only a ﬁnite number of choices for p π β . We need to examine ∗ ∗ ∗ the possible choices for π β . Given a choice for α = p π β,wehave π β = p α + a E ∗ ∗ ∗ ∗ ij ij for some collection of a ∈ Z. Clearly the a are bounded below by the requirement that ij ij (π β) · D ≥ 0for each i. On the other hand, if a > 0for some i, j,then (π β) · E < 0, ∗ i ij ∗ ij so if f : C → Yis an A -curve with f [C]= π β then its reduced image C must contain ∗ ∗ E . (Technically a relative stable map in Y is a map to an expanded degeneration of ij ◦ ◦ ◦ ˜ ˜ ˜ Y , but we compose with the projection to Y and then the natural map Y → Y.) Write C = C ∪ E ,with C a reduced divisor distinct from E . Suppose C is non-empty. ij ij Necessarily either C ∩ Dis empty or C intersects D only at E ∩ D; otherwise C cannot ij be the image of a relative stable map with one point of tangency with D. In either case there is an integer k such that O (C + kE )| is the trivial sheaf. However, by [GHK12], Y ij D Proposition 4.1, for a general deformation (Y , D) of (Y, D), the kernel of the restriction map Pic Y → Pic D is trivial. Thus by Lemma 3.9,N = 0. We conclude that there are only a ﬁnite number of choices of π β,exceptwhen π β is a multiple of some E . This ∗ ∗ ij shows condition (4) in Deﬁnition 2.13 of scattering diagrams, as well as condition (2). Note that κ =[D]∈ Gfor each i so condition (3) is vacuous for dim τ = 1. If dim τ = 2, ρ ,ϕ i d d any contributing A -class β satisﬁes π β ∈ G, so (3) holds. can Theorem 3.17. — We follow the above notation. If D is consistent as a scattering diagram for (B, ), P, ϕ,and G, then Theorem 3.8 is true. Proof. — This just follows from the series of reductions of Theorem 3.8 already made and the observation that if I is an m-primary ideal, then since G ⊂ m one can ﬁnd some k such that kG ⊂ I . To show consistency holds for the ideal I , we use the assumed consistency to observe consistency holds for the ideal I = kG, and this gives the desired result. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 115 Remark 3.18. — Given a consistent scattering diagram D for (B, ),P, ϕ,and G, and κ ∈ Gfor all rays ρ ∈ ,Theorem 2.28 shows that with I = G, ρ,ϕ X := Spec X , O I X I,D I,D is ﬂat over Spec k[P]/I, and X = V × Spec k[P]/G. G n gs gs Let T ⊂ Spec k[P]/G be the Gross-Siebert locus, Deﬁnition 3.14.Note T de- termines open subschemes of the thickenings Spec k[P]/I, which we will shall denote gs by T . gs gp We can describe the subscheme T of Spec k[P]/I as follows. Let E ⊂ P be the gs lattice generated by the face P \ G. Then as a subset of Spec k[P]/G, T Spec k[E]. gs Furthermore, if we take the localization P + EofPalong the face P \ G, then T as a subscheme of Spec k[P]/IisSpec k[P + E]/(I + E). Note that m = (P + E) \ E, and G = P ∩ m , so we can write k[E]= k[P + P+E P+E E]/m . P+E We can now view ϕ as a multivalued strictly (P + E)-convex function. Then we have the following obvious Lemma 3.19. — Suppose D is a consistent scattering diagram for the data (B, ), P + E, ϕ, m , and a scattering diagram for the data (B, ), P, ϕ,and G. Then D is also consistent as P+E can a scattering diagram for the latter data. In particular, by Theorem 3.17,Theorem 3.8 holds if D is consistent as a scattering diagram for P + E, m . P+E Proof. — Since P ∩ m = G, the equalities in Deﬁnition 2.26 can be tested for P+E √ √ an ideal I of P with I = G by choosing some ideal I ⊆ P + Ewith I = m and P+E I ∩ P ⊂ I. Then the equalities of Deﬁnition 2.26 hold for the data P + Eand I by the assumed consistency, and hence also for P and I. ¯ ¯ ¯ Let I ⊂ m be an ideal with I = m .Set I = I ∩ P. Then X is ﬂat over P+E P+E I,D o o Spec k[P]/I. Restricting X to the open set Spec k[P + E]/Igives the ﬂat family X . I,D ¯ I,D We now replace P by P + Eand Jby m in what follows. We now summarize P+E our current situation with the following assumptions: Assumptions 3.20. • There is a toric model ¯ ¯ p : (Y, D) → (Y, D) which blows up distinct points x on D , 1 ≤ j ≤ , with exceptional divisors E . ij i i ij • η : NE(Y) → P is an inclusion. Via Example 2.3, this gives the function ϕ. E = P = ∗ ⊥ P ∩ (p H) is generated by the classes of exceptional curves of p. Let G = P \ E = m . P 116 MARK GROSS, PAUL HACKING, AND SEAN KEEL • D is a scattering diagram for the data P,ϕ and G. Furthermore, for each ray ρ ∈ ,the unique outgoing ray (ρ , f ) ∈ D satisﬁes i ρ −1 f = g 1 + b X ρ ρ ij i i i j=1 [E ] ij with g ≡ 1mod G and b = z . ρ ij can We note we have shown that D = D achieves these assumptions. Step IV. Pushing the singularities to inﬁnity. We work with Assumptions 3.20.Consider ¯ ¯ ¯ ¯ ¯ the tropicalisation (B, ) of (Y, D). By Example 1.7, B in fact has no singularity at the 2 2 origin, and is afﬁne isomorphic to M = R (with M = Z ), while is precisely the fan ¯ ¯ ¯ for Y. In order to distinguish between constructions on (Y, D) and (Y, D),wedecorate all existing notation with bars. For example, if τ ∈ , denote the corresponding cone of gp ¯ ¯ by τ¯.Let ϕ¯ be the multivalued P -valued function on Bsuch that (3.6) κ = p [D ]. ρ, ¯ ϕ¯ ρ¯ Note that by Lemma 1.13, we can assume ϕ¯ is in fact a single-valued function on M . This single-valuedness will be important to be able to apply the method of Kontsevich and Soibelman, Theorem 3.23. We now have sheaves P on B and P on B , induced by the two functions ϕ¯ and 0 0 ϕ respectively. Note that since ϕ¯ is single-valued and B has no singularities, P is the constant sheaf gp with ﬁbre P ⊕ M. There is a canonical piecewise linear map ν : B → B which restricts to an integral afﬁne isomorphism ν| : σ →¯ σ ,where σ ∈ and σ¯ ∈ σ max ¯ ¯ ¯ is the corresponding cell of . Note this map identiﬁes B(Z) with B(Z). max For each maximal cone σ ∈ , the derivative ν of ν induces a canonical iden- max ∗ tiﬁcation of with ¯ . This then gives an induced isomorphism of monoids: B,σ B,σ (3.7) ν˜ : P → P σ ϕ ϕ¯ σ σ¯ given by ϕ (m) + p → ϕ¯ ν (m) + p, σ σ¯ ∗ for p ∈ Pand m ∈ . This identiﬁes the k[P]-algebras k[P ] and k[P ],and thecom- σ ϕ ϕ¯ σ σ¯ pletions k[P ] and k[P ]. ϕ ϕ¯ σ σ¯ MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 117 Because the map ν is only piecewise linear around rays ρ ∈ , there is only a piecewise linear identiﬁcation of P with P and hence no identiﬁcation of the corre- ϕ ϕ¯ ρ ρ¯ sponding rings. However, ν is still deﬁned on the tangent space to ρ,and thereisan identiﬁcation ν˜ : ϕ (m)+ p| m is tangent to ρ, p ∈ P → ϕ¯ (m)+ p| m is tangent to ρ, ¯ p ∈ P ρ ρ ρ¯ given by ϕ (m) + p → ϕ¯ ν (m) + p. ρ ρ¯ ∗ We now explain the Kontsevich-Soibelman lemma. This has to do with scattering diagrams on the smooth afﬁne surface M = R (such as B = B ). For this general ¯ ¯ R (Y,D) discussion, we ﬁx the data of a monoid Q which comes along with a map r : Q → M. Let m = Q\ Q ,and let k[Q] denote the completion of k[Q] with respect to the monomial ideal m . (In our application we take Q = P as deﬁned in (1.5).) Q ϕ¯ We can then consider a variant of the notion of scattering diagram: Deﬁnition 3.21. —Wedeﬁne a scattering diagram for the pair Q, r : Q → M.Thisis aset D = (d, f ) where • d ⊂ M is given by d=−R m ≥0 0 if d is an outgoing ray and d = R m ≥0 0 if d is an incoming ray,for some m ∈ M\{0}. • f ∈ k[Q]. • f ≡ 1mod m . d Q • f = 1 + c z for c ∈ k,r(p) = 0 a positive multiple of m . d p p 0 • For any k > 0, there are only a ﬁnite number of rays (d, f ) ∈ D with f ≡ 1mod m . d d Deﬁnition 3.22. — Given a loop γ in M around the origin, we deﬁne the path ordered product θ : k[Q]→ k[Q] γ,D 118 MARK GROSS, PAUL HACKING, AND SEAN KEEL as follows. For each k > 0,let D[k]⊂ D be the subset of rays (d, f ) ∈ D with f ≡ 1mod m . d d This set is ﬁnite. For d ∈ D[k] with γ(t ) ∈ d,deﬁne k k k θ : k[Q]/m → k[Q]/m γ,d Q Q by n ,r(q) k q q d θ z = z f γ,d for n ∈ M primitive satisfying, with m a non-zero tangent vector of d, n , m= 0, n ,γ (t ) < 0. d d 0 Then, if γ crosses the rays d ,..., d in order with D[k]={d ,..., d }, we can deﬁne 1 n 1 n k k k θ = θ ◦···◦ θ . γ,D γ,d γ,d n 1 We then deﬁne θ by taking the limit as k →∞. γ,D The following is a slight generalisation of a result of Kontsevich and Soibelman which appeared in [KS06]. Theorem 3.23. —Let D be a scattering diagram in the sense of Deﬁnition 3.21. Then there is another scattering diagram Scatter(D) containing D such that Scatter(D) \ D consists only of outgoing rays and θ is the identity for γ a loop around the origin. γ,Scatter(D) For a proof of this theorem essentially as stated here, see [GPS09], Theorem 1.4. The result is unique if Scatter(D) \ D has at most one ray in each possible direction; we shall assume Scatter(D) has been chosen to have this property. This can always be done. We apply this in the following situation. We take Q to be the monoid P which ϕ¯ ¯ ¯ ¯ yields the Mumford degeneration associated to the data (B, ), ϕ¯ (recalling B = M ), deﬁned by gp P = m, ϕ( ¯ m) + p | m ∈ M, p ∈ P ⊂ M × P . ϕ¯ This comes with a canonical map r : P → M by projection. ϕ¯ Deﬁnition 3.24. — Suppose we are in the situation of Assumptions 3.20. We deﬁne a scattering diagram ν(D) on B as follows. For every ray (d, f ) ∈ D not equal to (ρ , f ) for some i, ν(D) d i ρ contains the ray (ν(d), ν˜ (f )), and for each ray (ρ , f ), ν(D) contains two rays, (ρ¯ , ν˜ (g )) and τ d i ρ i τ ρ d i d i i −1 (ρ¯ , (1 + b X )). i i j=1 ij We note that ν(D) may not actually be a scattering diagram in the sense of Deﬁnition 3.21,as it is possible that f ∈ k[P ]:if p ∈ P , then ν˜ (p) ∈ P but need not lie in P . d ϕ¯ ϕ τ ϕ ϕ τ MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 119 can In the case of D = D , we can use the Kontsevich-Soibelman lemma to describe can can ν(D ). This will both show that ν(D ) is a scattering diagram in the sense of Deﬁni- tion 3.21 and that it satisﬁes an important additional property, namely the condition that θ can is equal to the identity. This will allow us to prove consistency. Let γ,ν(D ) −1 ¯ ¯ (3.8) D = ρ¯ , 1 + b X 1 ≤ i ≤ n . 0 i i ij j=1 ¯ ¯ Let m = P \ P as usual. Then by the strict convexity of ϕ¯ , X ∈ m so that D is a ϕ¯ ϕ¯ i ϕ¯ 0 ϕ¯ scattering diagram for the pair P , r in the sense of Deﬁnition 3.21.Now deﬁne ϕ¯ ¯ ¯ D := Scatter(D ) ¯ ¯ where we require D\ D to have only one outgoing ray in each direction (and no incom- ing rays). The following will be Step V, which we defer until Section 3.4. can can Theorem 3.25. — D = ν(D ). In particular, ν(D ) is a scattering diagram in the sense of Deﬁnition 3.21 and θ ≡ 1 for a loop γ around the origin. γ,D Example 3.26. — Continuing with Example 3.7, note that the pair (Y, D) can be ¯ ¯ ¯ obtained from the toric pair (Y, D) deﬁned by the fan with rays generated by (1, 0), ¯ ¯ (1, 1), (0, 1), (−1, 0) and (0,−1), corresponding to D ,..., D , by blowing up one point 1 5 ¯ ¯ ¯ ¯ on each of D and D . This description determines D and hence D. One can check this 4 5 0 can description agrees with that given in Example 3.7 for D , see e.g. [GPS09], Example 1.6 for a similar computation. Returning to the situation of Assumptions 3.20, suppose in addition that ν(D) is a scattering diagram in the sense of Deﬁnition 3.21.(Forexample,byTheorem 3.25, can D = D satisﬁes these assumptions.) For I ⊂ P an ideal with I = J, we now have o o deformations X and X . The latter scheme is glued from open sets I,D I,ν(D) U = Spec R ρ, ¯ I ρ, ¯ I along open sets identiﬁed with Spec R . Here we are decorating the rings coming from σ, ¯ I the data on B with bars as before, while we maintain the notation R , etc., for those ρ,I rings coming from the data on B. Lemma 3.27. — Given Assumptions 3.20, assume also that ν(D) is a scattering diagram in the sense of Deﬁnition 3.21. Then there are isomorphisms p : R → R i ρ ,I ρ¯ ,I i i 120 MARK GROSS, PAUL HACKING, AND SEAN KEEL and p : R → R i−1,i σ ,I σ¯ ,I i−1,i i−1,i for all i such that the diagrams ψ ψ ρ ,− ρ ,+ i i R R R R ρ ,I σ ,I ρ ,I σ ,I i i−1,i i i,i+1 p p p p i i−1,i i i,i+1 R R R R ρ¯ ,I σ¯ ,I ρ¯ ,I σ¯ ,I i i−1,i i i,i+1 ψ ψ ρ¯ ,− ρ¯ ,+ i i and γ,D R R σ ,I σ ,I i−1,i i−1,i p p i−1,i i−1,i R R σ¯ ,I σ¯ ,I i−1,i i−1,i γ, ¯ ν(D) are commutative, where γ is any path in σ for which θ is deﬁned, and γ¯ = ν ◦ γ . i−1,i γ,D Consequently, the maps p and p induce an isomorphism i i−1,i o o p : X → X I,D I,ν(D) over Spec k[P]/I. Proof.—Recall that R [X , X , X ] I i−1 i+1 (3.9) R = , ρ,I −D i −1 η([D ]) (X X − z X g (1 + b X )) i−1 i+1 ρ ij i i j=1 i ¯ ¯ ¯ R [X , X , X ] I i−1 i+1 (3.10) R = . ρ,I −D ∗ ¯ −1 i i η(p [D ]) ¯ ¯ i ¯ ¯ (X X − z X g¯ (1 + b X )) i−1 i+1 ρ i i i ij j=1 We simply deﬁne p to be the identity on R and p (X ) = X . This makes sense i I i j j 2 2 ∗ i ¯ ¯ since D = D − and [D]= p [D]− E ,sothat i i i ij i i j=1 i i i 2 2 −D ∗ −D η([D ]) i −1 η(p [D ]) i −1 −1 i i ¯ ¯ ¯ p z X 1 + b X = z X b X 1 + b X i ij i ij i i i ij i j=1 j=1 j=1 ∗ −D η(p [D ]) i −1 ¯ ¯ = z X 1 + b X . ij j=1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 121 The map p is induced by ν˜ deﬁned in (3.7). It is then straightforward to check the i−1,i σ i−1,i commutativity of the three diagrams. Lemma 3.28. — Given Assumptions 3.20,suppose ν(D) is a scattering diagram in the sense of Deﬁnition 3.21.For Q ∈ σ , we distinguish between i−1,i Lift (q) ∈ R Q σ ,I i−1,i for the lift of q ∈ B (Z) and Lift ν(q) ∈ R ν(Q) σ¯ ,I i−1,i the lift of ν(q). Then (1) p (Lift (q)) = Lift (ν(q)). i−1,i Q ν(Q) gp gp (2) Under the natural identiﬁcations (P ) = (P ) ,for τ ∈ \{0}, P ⊂ P ,and for ϕ¯ ϕ¯ ϕ¯ ϕ¯ τ τ any broken line γ for q, Mono(γ ) ∈ k[P ]. ϕ¯ (3) ν induces a bijection between broken lines: If γ : (−∞, 0]→ B is a broken line in B , 0 0 ¯ ¯ then ν ◦ γ is a broken line in B , and conversely, if γ¯ : (−∞, 0]→ P is a broken line in −1 B , then ν ◦¯ γ is a broken line in B . 0 0 Proof. — (3) implies (1). For (3), clearly it is enough to compare bending and at- tached monomials of broken lines near a ray ρ . Consider a broken line γ in B passing from σ to σ ,and let cz be the 0 i−1,i i,i+1 monomial attached to the broken line before it crosses over ρ ,sothat q ∈ P .Let θ , i ϕ ρ σ i i−1,i θ be deﬁned by ρ¯ p p n,r(p) θ z := z f i ρ ¯ n,r¯(p) p p −1 ¯ ¯ θ z := z g¯ 1 + b X ρ¯ ρ i i i ij j=1 where (ρ¯ , g¯ ) ∈ ν(D) is the outgoing ray with support ρ¯ .Here n, n¯ are primitive cotan- i ρ i gent vectors vanishing on tangent vectors to ρ , ρ¯ and positive on σ , σ¯ respectively. i i i−1,i i−1,i Then we need to show that q q (3.11) p θ cz = θ p cz i,i+1 ρ ρ¯ i−1,i i i to get the correspondence between broken lines. Note that ¯ ¯ ¯ p (X ) = X , p (X ) = X , p (X ) = X , i−1,i i−1 i−1 i,i−1 i i i,i+1 i i but to compute p (X ), we need to use the relation (see Proposition 2.5) i,i+1 i−1 −D η([D ]) X X = z X i−1 i+1 i 122 MARK GROSS, PAUL HACKING, AND SEAN KEEL in k[P ] to write −D η([D ]) i −1 X = z X X . i−1 i i+1 On the other hand, one has the relation −D η(p [D ]) i i ¯ ¯ ¯ X X = z X i−1 i+1 in k[P ],so ϕ¯ ρ¯ 2 2 −D +D η([D ]−p [D ]) i i i i ¯ ¯ p (X ) = z X X i,i+1 i−1 i−1 i −1 ¯ ¯ = X X b . i−1 ij j=1 Thus using Assumptions 3.20 for the form of f ,wehave p θ (X ) = p (X f ) i,i+1 ρ i−1 i,i+1 i−1 ρ i i i i i −1 −1 ¯ ¯ ¯ = X X b 1 + b X g¯ i−1 ij ρ ij i i j=1 j=1 −1 ¯ ¯ = X g¯ 1 + b X i−1 ρ i ij j=1 = θ p (X ) ρ¯ i−1,i i−1 as desired. Also, ¯ ¯ ¯ p θ (X ) = X = θ p (X ) . i,i+1 ρ i i ρ¯ i−1,i i i i Thus (3.11) holds. This shows (3). For (2), the statement that P ⊂ P is obvious. For q ∈ σ ∈ , by deﬁnition the ϕ¯ ϕ¯ ϕ (q) monomial attached to the ﬁrst domain of linearity of a broken line for q is z , which (ν(q),ϕ( ¯ ν(q))) is identiﬁed under ν˜ with z ∈ k[P ].For any (d, f ) ∈ ν(D), f ∈ k[P ] by σ ϕ¯ d d ϕ¯ assumption, and hence all monomials associated to broken lines in B lie in k[P ], 0 ϕ¯ hence (2). Deﬁnition 3.29. —Let D be a scattering diagram in the sense of Deﬁnition 3.21 for the pair P, r : P → M for some toric monoid P.Let I ⊂ P be an ideal with I = m .Wedeﬁne for q ∈ B (Z) P 0 and Q ∈ B , Lift (q) = Mono(γ ) ∈ k[P]/I Q MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 123 where the sum is over all broken lines γ for q with endpoint Q in B with respect to the scattering diagram D. One sees easily as in Lemma 2.25 that this is a ﬁnite sum. The last crucial result we need for consistency is the following result of [CPS]. Theorem 3.30. — With the assumptions of Deﬁnition 3.29, suppose furthermore that θ ≡ 1 γ,D for a loop γ around the origin. Fix an ideal I ⊂ P with I = m and q ∈ B (Z).If Q, Q ∈ P 0 M \ Supp(D ) are general, and γ is a path connecting Q and Q for which θ is deﬁned, then R I γ,D Lift (q) = θ Lift (q) Q Q γ,D as elements of k[P]/I. Proof. — This is shown in [CPS] in a rather more general setup. For a version of the argument closer to the current setup, see the proof of Theorem 5.35 of [G11]. Proof of Theorem 3.8. — By Lemmas 3.15 and 3.19, we can assume we are in the can can situation of Assumptions 3.20 with D = D . In checking (1) of Deﬁnition 2.26 for D in this situation, we want to check equalities can Lift (q) = θ Lift (q) Q γ,D Q for Q, Q ∈ σ . By Lemmas 3.27 and 3.28, it is sufﬁcient to show that i−1,i (3.12)Lift ν(q) = θ Lift ν(q) . ν(Q ) γ, ¯ ν(D) ν(Q) To check this equality we can compare coefﬁcients of monomials, and given any mono- mial z appearing on the left- or right-hand sides, we can apply Theorems 3.25 and 3.30,where we take P = P ,I = m for sufﬁciently large k so that p ∈ I. The hypothesis ϕ¯ ϕ¯ θ ≡ 1of Theorem 3.30 holds by Theorem 3.25. γ,ν(D) To show (2) of Deﬁnition 2.26,wecan take Q = Q and Q = Q on opposite − + sides of a ray ρ .If γ is a short path joining Q and Q ,westill have (3.12) after inverting f .Wehaveamap ψ := (ψ ,ψ ) : R → R × R . ρ¯ ,− ρ¯ ,+ ρ ,I σ ,I σ ,I i i i−1,i i,i+1 i −1 If f =¯ g (1 + b X ) then ψ is given by i ρ i i j=1 ij ϕ (v ) ϕ (v ) i−1 i−1 ρ ρ i i X → z , f z , i−1 i ϕ (v ) ϕ (v ) ρ i ρ i i i X → z , z , ϕ (v ) ϕ (v ) i+1 i+1 ρ ρ ¯ i i X → f z , z . i+1 i 124 MARK GROSS, PAUL HACKING, AND SEAN KEEL One checks easily that this map is injective. Furthermore, the image is described as fol- lows. Let I ⊂ P be the monoid ideal ρ¯ ϕ¯ i ρ¯ I = q ∈ P | q − ϕ r(q) ∈ Ior q − ϕ r(q) ∈ I . ρ¯ ρ σ σ i i−1,i i,i+1 Then the image consists of those elements (g , g ) such that every monomial of g and − + − g has exponent in P ⊂ P , P , and the images g¯ of g in (k[P ]/I ) satisfy + ϕ¯ ϕ¯ ϕ¯ ± ± ρ ρ¯ f ρ¯ σ σ i i i i i−1,i i,i+1 θ (g¯ )=¯ g , where this makes sense as we have localized at f . (See e.g., the proof of γ,ν(D) − + i Lemma 2.34 in [GS07] for a similar statement.) Thus by (3.12) and Lemma 2.25,(2), thereisan α ∈ R such that ρ ,I ψ (α) = Lift ν(q),ψ (α) = Lift ν(q) . ρ ,− ν(Q) ρ ,+ ν(Q ) i i −1 Thus we may take Lift (q) = p (α), and by Lemma 3.28, ψ (Lift (q)) = Lift (q), ρ ρ ,± ρ Q i i i i ± giving consistency. 3.4. Step V: the proof of Theorem 3.25 and the connection with [GPS09]. — Here we derive Theorem 3.25 from the main result of [GPS09]. We will need to review one form of this result, which gives an enumerative interpretation for the output of the Kontsevich- Soibelman lemma. Fix M = Z as usual. Suppose we are given positive integers ,..., and prim- 1 n itive vectors m ,..., m ∈ M. Let = and Q = M ⊕ N ,with r : Q → Mthe 1 n i i=1 projection. Denote the variables in k[Q] corresponding to the generators of N as t ,for ij 1 ≤ i ≤ n and 1 ≤ j ≤ . Consider the scattering diagram for the data r : Q → M(in the sense of Deﬁnition 3.21) D = R m , 1 + t z | 1 ≤ i ≤ n . ≥0 i ij j=1 We wish to interpret (d, f ) ∈ Scatter(D) \ D. Choose a complete fan in M which d d R contains the rays R m ,..., R m as well as the ray d (which may coincide with one of ≥0 1 ≥0 n the other rays). Let X be the corresponding toric surface, and let D ,..., D , D be the d 1 n out divisors corresponding to the above rays. Choose general points x ,..., x ∈ D ,and let i1 i i ν : X → X d d ˜ ˜ ˜ be the blow-up of all the points {x }.Let D ,..., D , D be the proper transforms of the ij 1 n out divisors D ,..., D , D and E the exceptional curve over x . 1 n out ij ij Now introduce the additional data of P = (P ,..., P ),where P denotes a se- 1 n i quence p ,..., p of non-negative numbers. We will use the notation P = p +···+ i1 i i i i1 p and call P an ordered partition.Deﬁne i i |P|= p . i ij j=1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 125 We shall restrict attention to those P such that (3.13) − |P |m = k m i i P d i=1 where m ∈ M is a primitive generator of d and k is a positive integer. d P Given this data, consider the class β ∈ A (X , Z) speciﬁed by the requirement 1 d that, if D is a toric divisor of X with D ∈ {D ,..., D , D },then D · β = 0; if D d 1 n out out {D ,..., D }, 1 n D · β =|P |, D · β = k ; i i out P while if D = D for some j,then out j |P | i = j, D · β = |P |+ k i = j. i P That such a class exists follows easily from (3.13) and Lemma 1.13. It is also unique. We can then deﬁne β = ν (β) − p [E ]∈ A (X , Z). P ij ij 1 d i=1 j=1 ˜ ˜ ˜ ˜ ˜ We deﬁne N := N as in Deﬁnition 3.1, using (Y, D) = (X , D),where Dis the proper P β d transform of the toric boundary of X , and using C = D . Then one of the main theo- d out rems of [GPS09] (see Section 5.7 of that paper) states Theorem 3.31. P −k m P d (3.14)log f = k N t z , d P P ij where the sum is over all P satisfying (3.13)and t denotes the monomial t . ij ij We can adapt this theorem for our purposes as follows. Fix a fan in M deﬁning ¯ ¯ a complete non-singular toric surface Y, with D = D +··· + D the toric boundary. 1 n Choose points x ,..., x ∈ D , and deﬁne a new surface Y as the blow-up ν : Y → Yat i1 i i the points {x }.Let E be the exceptional curve over x . ij ij ij ¯ ¯ Let P = NE(Y); because Y is toric, this is a ﬁnitely generated monoid with P = gp {0}.Let ϕ¯ : M → P be the -piecewise linear strictly P-convex function given by Lemma 1.13. We will need the following, an immediate corollary of Lemma 1.13, using the no- tation of that lemma applied to the fan for Y: 126 MARK GROSS, PAUL HACKING, AND SEAN KEEL Lemma 3.32. —If a t ∈ ker s, then the corresponding element of ker s = A (Y, Z) is ρ ρ 1 gp a ϕ( ¯ m ) ∈ P . ρ ρ Let E ⊂ A (Y, Z) be the lattice spanned by the classes of the exceptional curves of ν,sothat A (Y, Z) = ν A (Y, Z) ⊕ E. We then obtain a map 1 1 ∗ ∗ gp ϕ = ν ◦¯ ϕ : M → ν P ⊕ E. Let Q = (m, p) ∈ M ⊕ A (Y, Z)|∃p ∈ ν P ⊕ Esuchthat p = p + ϕ(m) . Thereisanobvious projection r : Q → M, and by strict convexity of ϕ¯,Q = E. We consider the scattering diagram, D ,over k[Q] given by (m ,ϕ(m )−E ) i i ij D = R m , 1 + z 1 ≤ i ≤ n . 0 ≥0 i j=1 Then we have ¯ ¯ Theorem 3.33. —Let (d, f ) ∈ Scatter(D ) \ D ,assuming that thereisatmostone ray d 0 0 ¯ ¯ ¯ of Scatter(D ) \ D in each possible outgoing direction. (Note by deﬁnition of Scatter(D ), (d, f ) 0 0 0 d cannot be incoming.) Then, following the notation of Deﬁnition 3.1 and 3.3, (−k m ,π (β)−ϕ(k m )) β d ∗ β d (3.15)log f = k N z . d β β ˜ ˜ Here π : Y → Y is the toric blow-up of Y determined by d and C ⊂ Y is the component of the boundary determined by d.If d is not one of the rays R m , then we sum over all A -classes β ∈ A (Y, Z) ≥0 i 1 satisfying (3.1), and if d = R m we sum over all such classes except for classes given by multiple ≥0 i covers of one of the exceptional divisors E . ij Proof.—Let Q be the submonoid of M ⊕ N generated by elements of the form (m , d ),where d is the (i, j)-th generator of N .Notethat Q itself is freely generated by i ij ij these elements. Thus we can deﬁne a map α : Q → Q by (m , d ) → (m ,ϕ(m ) − E ). The scattering diagram i ij i i ij (m ,d ) i ij D := R m , 1 + z | 1 ≤ i ≤ n ≥0 i j=1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 127 then has image under the map α (applying α to each f ) the scattering diagram D .Thus d 0 if we apply α to each element of Scatter(D ), we must get Scatter(D ),as θ 0 γ,Scatter(D ) being the identity on k[Q ] implies that θ is the identity on k[Q]. γ,α(Scatter(D )) To obtain the result, we now note that the set of possible A -classes in Y occurring in the expression (3.15) are precisely the classes {β } where P runs over all partitions satisfying (3.13). Now applying α to a term appearing in (3.14)ofthe form ij P −k m |P |m d i i i=1 k N t z = k N t z , P P β β P P ij we get (−k m , |P |ϕ(m )− p E ) P d i i ij ij i,j i=1 k N z . β β P P But by Lemma 3.32 and (3.13), |P |ϕ(m ) − p E = π (β ) − ϕ(k m ), i i ij ij ∗ P P d i=1 i,j hence the result. A direct comparison of the formula of the above theorem and the formula in the deﬁnition of the canonical scattering diagram then yields Theorem 3.25. 4. Smoothness: around the Gross-Siebert locus Next we prove that our deformation of V is indeed a smoothing. The main the- orem of this section (Theorem 4.6) will show this in the situation of Theorem 0.1 when (Y, D) has a toric model. The full smoothness statement of Theorem 0.1 will require some more work,whichwillbecarried outinSection 6. We prove smoothness by working over the Gross-Siebert locus (Deﬁnition 3.14). Here our deformation (when restricted to one-parameter subgroups associated to p A, A an ample divisor on Y) agrees with the construction of [GS07]. This is important here because the deformations of [GS07] come with explicit charts that cover all of V ,from which it is clear that they give a smoothing. So conceptually, the smoothing claim is clear. Because we work with formal families the actual argument is a bit more delicate. First we make rigorous the notion of a smooth generic ﬁbre for a formal family: Deﬁnition-Lemma 4.1. —Let f : Z → W be a ﬂat ﬁnite type morphism of schemes of relative dimension d . Then Sing(f ) ⊂ Z is the closed embedding deﬁned by the d th Fitting ideal of . Z/W Sing(f ) is empty if and only if f is smooth. Formation of Sing(f ) commutes with all base extensions of W. 128 MARK GROSS, PAUL HACKING, AND SEAN KEEL Proof. — For the deﬁnition of the Fitting ideal, see e.g., [E95], 20.4. The fact that it commutes with base-change follows from the fact that commutes with base-change Z/W and [E95], Cor. 20.5. That Sing(f ) is empty if and only if f is smooth follows from [E95], Prop. 20.6 and the deﬁnition of smoothness. Now for a formal family, smoothness of the generic ﬁbre is measured by the fact that Sing(f ) does not surject scheme-theoretically onto the base. More precisely: Deﬁnition 4.2. —Let S be a normal variety, V ⊂ S a connected closed subset, and S the formal completion of S along V.Let f: X → S be an adic ﬂat morphism of formal schemes of pure relative dimension and Z ⊂ X the scheme theoretic singular locus of f. Then we say the generic ﬁber of f is smooth if the map O → f O is not injective. S ∗ Z For the statement of Proposition 4.3, we ﬁx our usual setting of a surface (Y, D), and assume given Assumptions 3.20 and that ν(D) is a scattering diagram in the sense of Deﬁnition 3.21. Suppose furthermore that θ ≡ 1 for a loop γ around the origin. γ,ν(D) Thus by Theorem 3.30, D is consistent. These hypotheses on D apply in particular when can D = D . gs gs Let T be the Gross-Siebert locus; we have T = Spec k[P]/G. Consistency of D gives a ﬂat family f : X → Spec k[P]/I I I gs over a thickening of T whenever I = G. ¯ ¯ ¯ On the other hand, letting be the fan for Yin B = M , we have the piecewise gp ¯ ¯ linear function ϕ¯ : B → P with κ = p [D ],asin(3.6). This now determines the ρ, ¯ ϕ¯ ρ¯ Mumford family ¯ ¯ f : X → Spec k[P]/I. I I gs gs Our goal is to compare these two families. Note that both X → T and X → T G G gs gs gs are the trivial family V × T → T . Thus either family contains a canonical copy of T , gs i.e., {0}× T , where 0 is the vertex of V . Proposition 4.3. — In the above situation, ﬁx an ideal I with I = G. There are open afﬁne gs ¯ ¯ sets U ⊂ X , U ⊂ X , both sets containing the canonical copy of T , and an isomorphism I I I I μ : U → U I I I of families over Spec k[P]/I. Moreover, there is a non-zero monomial y ∈ k[P] whose pullback to X is in the stalk at any gs gs point x∈{0}× T ⊂ V × T of the ideal of Sing for all I. I MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 129 Proof. — The generic ﬁbre of the Mumford family over Spec k[P] is smooth: indeed the family is trivial over the open torus orbit of Spec k[P], with ﬁbre an algebraic torus. It follows that there is a non-zero monomial y ∈ k[P] in the ideal of Sing for the global Mumford family f : Spec k[P ]→ Spec k[P]. ϕ¯ Of course its restriction then lies in the stalk at any point x of the ideal sheaf of Sing for all I. Thus once we establish the claimed isomorphisms, the ﬁnal statement follows. ¯ ¯ Recall from Section 3.3 the construction of D := ν(D) and the scheme X from I,D o o ¯ ¯ D. By Lemma 3.27,X X ,soweinfacthaveanisomorphism I,D I,D ¯ ¯ X = Spec X , O ¯ =: X ¯ . ¯ X I,D I,D I,D So we can work with X instead of X . On the other hand, the Mumford family I,D Spec k[P ]/Ik[P ] over Spec k[P]/I can be described similarly. Using the empty scat- ϕ¯ ϕ¯ tering diagram instead of the scattering diagram D, one has by Lemma 2.9 ¯ ¯ X = Spec X , O o . I,∅ I,∅ X I,∅ Now deﬁne an ideal I ⊂ P as follows. For σ ∈ ,let ϕ¯ denote the linear 0 ϕ¯ max σ extension of ϕ¯| . We set I := (m, p) ∈ P | p−¯ ϕ (m) ∈ Ifor some σ ∈ . 0 ϕ¯ σ max Note that I = m . By assumption, D is a scattering diagram for P , and hence there 0 P ϕ¯ ϕ¯ are only a ﬁnite number of (d, f ) ∈ D for which f ≡ 1mod I . Furthermore, modulo I , d d 0 0 each f is a polynomial. ¯ ¯ Let D be the scattering diagram obtained from D by, for each outgoing ray (d, f ), I d truncating each f by throwing out all terms which lie in I . The incoming rays remain d 0 unchanged. Thus D can be viewed as a ﬁnite scattering diagram. Let h := f . d∈D This is an element of k[P ]. Note that necessarily h ≡ 1mod m .Thus h = 0deﬁnes ϕ¯ P ϕ¯ gs ¯ ¯ ¯ ¯ an open subset U ⊂ Spec k[P ]/Gk[P ]= X ¯ = X = V × T . Furthermore, U ϕ¯ ϕ¯ G,∅ n G,D gs ¯ ¯ contains the canonical copy of T . Since X ¯ and X both have underlying topological I,∅ I,D ¯ ¯ ¯ ¯ ¯ space X , this deﬁnes open sets U of X and U of X . We shall show these two ¯ ¯ G I,∅ I,∅ I,D I,D open subschemes are isomorphic. ¯ ¯ ¯ First the following claim shows that X X ,asfor any τ ∈ \{0},the auto- ¯ = ¯ I,D I,D morphisms involved will have the same effect modulo I . As a consequence, we can work with the scattering diagram D . I 130 MARK GROSS, PAUL HACKING, AND SEAN KEEL Claim 4.4. —Let τ ∈ , and suppose (m, p) ∈ P satisﬁes −m ∈ τ . Then (m, p) ∈ I if ϕ¯ 0 and only if (m, p) ∈ I ,where I := (m, p) ∈ P | p−¯ ϕ (m) ∈ I for some σ ∈ with τ ⊂ σ . τ ϕ¯ σ max Proof of claim. — Clearly I ∩ P ⊂ I , so one implication is clear. Conversely, sup- τ ϕ 0 pose that (m, p) ∈ I ,sothat p − ϕ (m) ∈ Ifor some σ ∈ .If τ ⊂ σ ∈ ,let 0 max max ρ ,...,ρ be the sequence of rays traversed in passing from σ to σ , chosen so that all 1 n ρ ,...,ρ lie in a half-plane bounded by the line Rm.Then 1 n ϕ (m) = ϕ (m) + n , mκ , σ σ ρ ρ ,ϕ i i i=1 with n primitive, vanishing on ρ ,and positive on ρ . Note that since −m ∈ τ ,wemust ρ i i+1 have n , m≤ 0for each i, and hence p− ϕ (m) = p− ϕ (m)+ p for some p ∈ P. Hence i σ σ (m, p) ∈ I . ¯ ¯ To show that U and U are isomorphic, let us describe these open subschemes I,D I,∅ explicitly away from the origin. Recall that X is obtained by gluing together schemes I,D which are spectra of rings R for τ ∈ . However in the case that dim τ = 1, this ring τ,I depends on the scattering diagram, so we write R for D = D or ∅. τ,I,D If dim τ = 2, then R = k[P ]/Ik[P ]. Since h ∈ k[P ]⊂ k[P ], h deﬁnes an τ,I,D ϕ¯ ϕ¯ ϕ¯ ϕ¯ τ τ τ element of R in this case. τ,I,D If dim τ = 1, then τ = ρ for some i, and we have a surjection ±1 ¯ ¯ ¯ ¯ ¯ R → R = R X , X , X /(X X ) = k[P ]/Gk[P ], ρ ,I,D ρ ,G,D G i−1 i+1 i−1 i+1 ϕ¯ ϕ¯ i i ρ ρ i i so that h ∈ k[P ]⊂ k[P ] deﬁnes an element of R . Choosing any lift of h to R , ϕ¯ ϕ¯ ρ ,G,D ρ ,I,D i i we note the localization (R ) is independent of the lift since the kernel of the above ρ ,I,D h surjection is nilpotent. We can then deﬁne regardless of dim τ , S := (R ) . τ,I,D τ,I,D h Note there is an isomorphism ψ : S → S i ρ ,I,D ρ ,I,∅ given by −1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ X → X g¯ 1 + b X , X → X , X → X . i−1 i−1 ρ i i i i+1 i+1 i ij j=1 This has an inverse because of the localization at h. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 131 Given a path γ in M \{0}, note that by construction of h, θ makes sense as γ,D an automorphism of the localization k[P ] , since to deﬁne the automorphism associated ϕ¯ h with crossing a ray (d, f ), we only need f to be invertible. However since by construction d d h is divisible by f , f is invertible. In particular, θ also makes sense as an automorphism d d γ,D of (k[P ]/Ik[P ]) = S for any τ ∈ \{0}. Thus using the equality S = S ¯ ϕ¯ ϕ¯ h τ,I,∅ σ,I,∅ τ τ σ,I,D for dim σ = 2 we see that θ ¯ also makes sense as an automorphism of S ¯ . γ,D σ,I,D Choose an orientation on M , labelling the rays ρ ,...,ρ of in a counterclock- R 1 n wise order, with σ as usual the maximal cone containing ρ and ρ . For two distinct i−1,i i−1 i points p, q on the unit circle in M not contained in Supp(D ),let γ be a counterclockwise R I p,q path from p to q, and write θ for θ acting on any of the rings S . p,q τ,I,∅ γ ,D p,q I For each ρ ,let p be a point on this unit circle contained in the connected com- i i,+ ponent of σ \ Supp(D ) adjacent to ρ ,and p a point in this unit circle contained in i,i+1 I i i,− the connected component of σ \ Supp(D ) adjacent to ρ . i−1,i I i Choose a base-point q on the unit circle not in Supp(D ). Recall in the construction of X , the open sets Spec R and Spec R are ρ ,I,D ρ ,I,D i+1 i I,D glued together along the common open set Spec R using the trivial automorphism σ ,I,D i,i+1 or the automorphism θ in the cases D=∅ or D = D respectively. After localizing p ,p i,+ i+1,− at h, we have a commutative diagram θ ◦ψ p ,q i i,+ S S ρ ,I,∅ ρ ,I,D i ρ ,+ θ ◦ψ p ,p ρ ,+ i,+ i+1,− i p ,q i+1,− S S σ ,I,∅ σ ,I,D i,i+1 i,i+1 ρ ,− i+1 ρ ,− i+1 S S ρ ,I,∅ ρ ,I,D i+1 i+1 θ ◦ψ p ,q i+1 i+1,+ Here the maps ψ are the ones deﬁned in Proposition 2.5 and (2.8). This shows that ρ,± the isomorphisms θ ◦ ψ between Spec S ¯ and Spec S are compatible with the p i ρ ,I,∅ ρ ,I,D i,+ i i gs gs ¯ ¯ gluings, and hence give an isomorphism between U \ ({0}× T ) and U \ ({0}× T ). I,D I,∅ Now V satisﬁes Serre’s condition S . Since X and X are ﬂat deformations of n 2 I I gs V × T , by Lemma 2.10 the above isomorphism extends across the codimension two set gs {0}× T , giving the desired isomorphism between U and U . I I We now need to use the above observations along the Gross-Siebert locus to obtain results about deformations away from the Gross-Siebert locus. For the remainder of the section, we work with data (Y, D), η, P, but now as in Assumptions 3.13. Furthermore, √ √ can we take D = D . Thus if we take I an ideal with either I = Gor I = m,weobtain 132 MARK GROSS, PAUL HACKING, AND SEAN KEEL aﬂat family X → Spec R =: S , and in the former case, X → S restricts to the open I I I I I gs subscheme of Spec R whose underlying open subset is T , giving the family over the thickening of the Gross-Siebert locus. With J = m or G, let f : X → S denote the formal deformation determined by J J J N+1 the deformations X N+1 → Spec R N+1 for N ≥ 0. Thus S = Spf(lim k[P]/J ) is the J J J ←− formal spectrum of the J-adic completion of k[P], X is a formal scheme, and X → S J J J is an adic ﬂat morphism of formal schemes. We refer to [G60] for background on formal schemes. Let Z := Sing(f ) ⊂ X denote the singular locus of f : X → S .Thus Z ⊂ X is I I I I I I I I a closed embedding of schemes. Since the singular locus is compatible with base-change, n n the singular loci Z ⊂ X determine a closed embedding Z ⊂ X which we refer to as J J J J the singular locus of f : X → S . J J J Again, with J = m or G, we have a section s: S → X = S × V given by s(t) = J J J n o o o t ×{0} for t ∈ S . We write X := X \ s(S ) ⊂ X and X ⊂ X , X ⊂ X for the induced J J J J I J J I J open embeddings. Lemma 4.5. — In the above situation, there exists 0 = g ∈ k[P] such that Supp(g · O ) is contained in s(S ). In particular, f (g · O ) is a coherent sheaf on S . J J∗ Z J Proof. — We can write an explicit open covering {U } of X in the two cases J = m i,J [D ] 2 or J = G, as follows. Write a = z and m =−D .Inthe case J = m, i i (4.1) U = V X X − a X ⊂ A × (G ) × S . i,J i−1 i+1 i m X J i X ,X i i−1 i+1 In the case J = G, i −1 2 U = V X X − a X 1 + b X ⊂ A × (G ) × S , i,J i−1 i+1 i ij m X J i i i X ,X i−1 i+1 [E ] ij with b = z as usual. ij We now use the charts U to compute the singular locus explicitly. In the case i,J J = m, the singular locus Z of U /S is given by i,J i,J J Z = V(X , X , a ) ⊂ U . i,J i−1 i+1 i i,J Hence if we deﬁne g = a ··· a then Supp(g · O ) is contained in s(S ). 1 n Z J Similarly, if J = G, the structure sheaf of the singular locus of U is annihilated by i,J g := a (b − b ).(Here (b − b ) is the discriminant of the polynomial f (X ) := i i ij ik ij ik i =k j =k (X + b ). It is a linear combination of f and f with coefﬁcients in k[{b }][X ].See i ij ij i [L02], p. 200–204.) So we can take g = g ··· g . 1 n The support of g · O is a closed subset of s(S ), hence proper over S . It follows Z J J that f (g · O ) is coherent by [G61], 3.4.2. J∗ Z J MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 133 can Theorem 4.6. —Let (Y, D), η, P satisfy Assumptions 3.13,and take D = D . Then the maps k[P]→ f O and O → f O are not injective, so the generic ﬁbre of f is m Z S m Z m ∗ m m ∗ m N+1 smooth in the sense of Deﬁnition 4.2. This also implies that for I = m and N $ 0, the map k[P]/I → f O is not injective. I∗ Sing(f ) Proof. — By Lemma 4.5 there exists 0 = g ∈ k[P] such that Supp(g · O ) ⊂ s(S ). Z G gp Let E be the subgroup of P generated by P \ G, so that U = Spec k[P + E] is an open subset of Spec k[P].Denoteby S the open subset of S isomorphic to the completion of U along the subscheme deﬁned by G + E. This is the formal thickening of the Gross- gs Siebert locus T .ByProposition 4.3 there then exists 0 = h ∈ k[P+ E] such that Supp(h· O −1 o ) is disjoint from s(S ∩ U). By multiplying h by a monomial whose exponent Z ∩f (S ) G G lies in P \ G, we can assume that h ∈ k[P].Thus gh · O has support in the closed subset s(S \ (S ∩ U)). Since this sheaf is coherent, there exists a non-zero element k ∈ m ⊂ G G k[P] such that ghk · O = 0. Noting by construction that ghk ∈ k[P],wehave k[P]→ (Z , O ) is not injective, hence the composition k[P]→ (Z , O ) → (Z , O ) G Z G Z m Z G G m is not injective. Since k[P]⊂ (S , O ), O → f O is not injective. m S S m∗ Z m m m 5. The relative torus The ﬂat deformations X can → Spec k[P]/I produced by the canonical scatter- I,D ing diagram have a useful special property: there is a natural torus action on the total space X can compatible with a torus action on the base. The meaning of this action I,D will be clariﬁed in Part II, where we will prove that our family extends naturally, in the positive case, to a universal family of Looijenga pairs (Z, D) together with a choice of isomorphism D → D ,where D is a ﬁxed n-cycle. The torus action then corresponds to ∗ ∗ changing the choice of isomorphism. D n Fixing the pair (Y, D) as usual, D = D +···+ D ,let A = A be the afﬁne space 1 n D D with one coordinate for each component D .Let T be the diagonal torus acting on A , i.e., the torus T whose character group D D χ T = Z is the free module with basis e ,..., e . D D 1 n Deﬁnition 5.1. — We deﬁne a canonical map w : A (Y) → χ(T ) given by C → (C · D )e . i D Suppose P ⊂ A (Y) is a toric submonoid containing NE(Y).Wethenget an action of T on Spec k[P],aswellason Spec k[P]/I for any monomial ideal I, and hence also on Spf(k[P]) for any completion of k[P] with respect to a monomial ideal. 134 MARK GROSS, PAUL HACKING, AND SEAN KEEL We can also deﬁne a unique piecewise linear map w : B → χ T ⊗ R with w(0) = 0and w(v ) = e ,for v the primitive generator of the ray ρ . i D i i Theorem 5.2. —Let I be an ideal for which X can → Spec k[P]/I is deﬁned. Then T I,D acts equivariantly on X can → Spec k[P]/I. Furthermore, each theta function ϑ ,q ∈ B(Z),isan I,D q eigenfunction of this action, with character w(q). Proof. — It’s enough to check this on the open subset X ⊂ X can.Wehavea can I,D I,D cover of X by open sets the hypersurfaces can I,D U ⊂ A × (G ) × Spec R ρ ,I m X I i X ,X i i−1 i+1 given by the equation −D [D ] i i X X = z X f , i−1 i+1 ρ i i can where f is the function attached to the ray ρ in D .Ifweact on X with weight w(v ) ρ i j j p can and on z with weight w(p) (for p ∈ P),thenwenotethatfor every (d, f ) ∈ D ,every monomial in f has weight zero by the explicit description of f in Deﬁnition 3.3.Inpar- d d ticular, the equation deﬁning U is clearly T -equivariant, and each of the monomials ρ ,I is an eigenfunction. Now X is obtained by gluing U ⊂ U with U ⊂ U ,us- can ρ ,σ ,I ρ ,I ρ ,σ ,I ρ ,I I,D i i,i+1 i i+1 i,i+1 i+1 can ing scattering automorphisms of D , and these open sets are naturally identiﬁed with 2 D (G ) × Spec R . The scattering automorphisms commute with the T action, by m I X ,X i i+1 the fact that the scattering functions have weight zero. Thus T acts equivariantly on can X → Spec k[P]/I. I,D Now we check our canonical global function ϑ is an eigenfunction, with charac- ter w(q). By construction, given a broken line γ , the weights of monomials attached to adjacent domains of linearity are the same, since the functions in the scattering diagram are of weight zero. Thus the weight of Mono(γ ) only depends on q. This weight can be determined by ﬁxing the base point Q in a cone σ which contains q, in which case the broken line for q which doesn’t bend and is wholly contained in σ yields the monomial ϕ (q) z , which has weight w(q).Thus ϑ is an eigenfunction with weight w(q). 6. Extending the family over boundary strata Here we prove Theorem 0.1 and Theorem 0.2. Let us review what we know so far. can For any pair (Y, D), we know that D is consistent by Theorem 3.8. Thus, if the number n of irreducible components of D satisﬁes n ≥ 3, Theorem 2.28 gives the construction of MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 135 f : X → Spec R of Theorem 0.1. The algebra structure on A has structure constants I I I given by counts of broken lines as in Theorem 2.34.The T equivariance is given by Theorem 5.2. If furthermore (Y, D) has a toric model, then the smoothness statement follows from Theorem 4.6. We will give a proof of Theorem 0.2 and the remaining cases of 0.1 by ﬁrst proving Theorem 0.2 in the case that we know that we have the desired algebra structure on A , and then bootstrap to the general case for both theorems. 6.1. Theorem 0.2 in the case that (Y, D) has a toric model. — As usual, let P be the toric monoid associated to a strictly convex rational polyhedral cone σ ⊂ A (Y) which P 1 R contains the Mori cone NE(Y) .Wehave m = P\{0}. For a monomial ideal I ⊂ Pwe deﬁne A := R · ϑ I I q q∈B(Z) can where R = k[P]/I. We take throughout D = D . Assumptions 6.1. — For any monomial ideal I with I = m, the multiplication rule of The- orem 2.34 deﬁnes an R -algebra structure on A ,sothat A ⊗ R = H (V , O ). I I I R m n V I n Note we have already shown that Assumptions 6.1 hold if n ≥ 3by Theorems 2.28, 2.34 and 3.8. Let ⊂ B(Z) be a ﬁnite collection of integral points such that the corresponding functions ϑ generate the k-algebra H (V , O ).(Then the ϑ , q ∈ generate A as an q n V q I R -algebra if I = m and Assumptions 6.1 hold.) Note for n ≥ 3we can take for the points {v },and for n = 1, 2, one can make a simple choice for , see Section 6.2. Lemma 6.2. — For any monomial ideal J ⊂ P, (J + m ) = J. k>0 Proof. — The inclusion ⊃ is obvious. For the other direction, as the intersection is a monomial ideal, it’s enough to consider a monomial in the intersection. But notice that k k k a monomial is in J + m iff it is either in J or in m . The result follows since m = 0. Assuming 6.1,let A be the collection of monomial ideals J ⊂ P with the following properties: (1) There is an R -algebra structure on A such that the canonical isomorphism of J J R -modules A ⊗ R = A is an algebra isomorphism, for all I = m. I+J J R I+J I+J (2) ϑ , q ∈ generate A as an R -algebra. q J J By the lemma, the algebra structure in (1) is unique if it exists. The algebra struc- √ √ ture on all A determines such a structure on A := lim A , A := lim A . Also, I I J I+J I=m I=m ←− ←− 136 MARK GROSS, PAUL HACKING, AND SEAN KEEL there are canonical inclusions A ⊂ R · ϑ q∈B(Z) A ⊂ R · ϑ J J q q∈B(Z) ˆ ˆ where R is the completion of R at m and R = lim R/(I + J) the inverse limit over all ←− ˆ ˆ ideals I with I = m. Here the direct products are viewed purely as R, R modules. We can also view A := R · ϑ ⊂ R · ϑ . J J q J q q∈B(Z) q∈B(Z) It is clear that A ⊂ A (as submodules of the direct product). Thus (1) holds if and only if J J the following holds: (1 )For each p, q ∈ B(Z), at most ﬁnitely many z ϑ with C ∈ J appear in the product expansion of Theorem 2.34 for ϑ · ϑ ∈ A . p q J Lemma 6.3. —If J ∈ A and J ⊂ J , then J ∈ A. In addition, A is closed under ﬁnite intersections. Proof. — The ﬁrst statement is clear. Now assume J , J ∈ A. It’s clear that (1 ) holds 1 2 for J ∩ J ,so A is an algebra. Moreover we have an exact sequence of k-modules 1 2 J ∩J 1 2 0 → A → A × A → A → 0 J ∩J J J J +J 1 2 1 2 1 2 exhibiting A as the ﬁbre product A × A =: A × A =: A. We now show J ∩J J A J 1 B 2 1 2 1 J +J 2 1 2 this ﬁbre product is a ﬁnitely generated k-algebra. Indeed, note that since the maps A , A → B are surjective, so are the maps A → A .Let {u } be a generating set for 1 2 i i the ideal ker(A → B). Since A is Noetherian, one can ﬁnd a ﬁnite such set. Note 2 i that u˜ := (0, u ) ∈ A. In addition, choose ﬁnite sets {x }, {y } generating A and A i i i j 1 2 as k-algebras. For each of these elements, choose a lift to A, giving a ﬁnite set of lifts {˜ u , x˜ , y˜ }, which we claim generate A. Indeed, given (x, y) ∈ A, one can subtract a poly- i i i nomial in the x˜ ’s to obtain (0, y ). Necessarily y ∈ ker(A → B), and hence we can write i 2 y = f u with f a polynomial in the y ’s. Let f be the same polynomial in the y˜ ’s. Then i i i i i i f u˜ = (0, y ), showing generation. i i Thus A is also a ﬁnitely generated R -algebra. Now the generation state- J ∩J J ∩J 1 2 1 2 ment follows from Lemma 6.4, taking R = R ,S = A ,I = J /J ∩ J ,J = J /J ∩ J , J ∩J J ∩J 1 1 2 2 1 2 1 2 1 2 ={q ,..., q },and themap R[T ,..., T ]→ Sgiven by T → ϑ . 1 m 1 m i q i MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 137 Lemma 6.4. —Let I, J ⊂ R be ideals in a Noetherian ring, with I · J = 0,and let S be a ﬁnitely generated R-algebra, and R[T ,..., T ]→ S an R-algebra map which is surjective modulo 1 m I and J. Then the map is surjective. Proof. — The associated map Spec S → A × Spec R is proper, as can be easily checked using the valuative criterion for properness. Indeed, any map S → Kfor a ﬁeld K factors through either S/IS or S/JS. Since this is a map of afﬁne schemes, S is a ﬁnite R[T ,..., T ]-module. Now we can apply Nakayama’s lemma. 1 m Proposition 6.5. — There is a unique minimal radical monomial ideal I ⊂ P such that (1) min and (2) hold for any monomial ideal J with I ⊂ J. min Proof. — Certainly any ideal J with J = m lies in A. Note that a radical monomial ideal is the complement of a union of faces of P, so there are only a ﬁnite number of such ideals. Suppose I , I are two radical ideals such that J ∈ A for any J with I ⊂ J .Note 1 2 i i i i that any ideal J with I ∩ I ⊂ J can be written as J ∩ J ,with I ⊂ J .(Indeed,we 1 2 1 2 i i can use the primary decomposition of J. If J = p is an intersection of primary ideals, necessarily the prime ideal p contains either I or I for each k.Thenlet J be the k 1 2 1 intersection of those p whose radical contains I and J be the intersection of those p k 1 2 k whose radical contains I .) Thus by Lemma 6.3,J ∈ A. This shows the existence of I . 2 min Proposition 6.6. — Suppose Assumptions 6.1 hold. (1) Suppose the intersection matrix (D · D ) is not negative semi-deﬁnite. Then I = (0) ⊂ i j min k[P]. (2) Suppose F ⊂ σ is a face such that F does not contain the class of every component of D. Then I ⊂ P \ F. min Proof. — We prove both cases simultaneously, writing F := Pin case (1). We claim there exists an effective divisor W = a D with support D such that W· D > 0for allD i i j j contained in F and a > 0for all i. For case (1), see Lemma 6.9.Incase(2),say [D ] ∈ / F. i 1 Then we can take a $ a $···$ a > 0. 1 2 n The algebra structure depends only on the deformation type of (Y, D).ByPropo- sition 4.1 of [GHK12], we may replace (Y, D) by a deformation equivalent pair such that any irreducible curve C ⊂ Y intersects D. Let NE(Y) ⊂ A (Y, R) denote the closure of NE(Y) .Let F := NE(Y) ∩ F, R 1 R R afaceof NE(Y) .Deﬁne = D− W, 0 < 1. Then (Y, ) is KLT (Kawamata log terminal). We claim K + ∼−Wis negative onF \{0}. By construction (K + )· D < 0 Y Y j for [D]∈ F and (K + ) · C < 0for C ⊂ D. Let N be a nef divisor such that F = j Y NE(Y) ∩ N .Then aN − (K + ) is nef and big for a $ 0, and thus some multiple of R Y N deﬁnes a birational morphism g by the basepoint-free theorem [KM98], Theorem 3.3. 138 MARK GROSS, PAUL HACKING, AND SEAN KEEL Thus F is generated by exceptional curves of g. We deduce that (K + ) ∩ F ={0} and (K + ) is negative on F \{0} as claimed. Now by the cone theorem [KM98], Theorem 3.7, NE(Y) is rational polyhedral near F and there is a contraction p: Y → Ysuch thatF is generated by the classes of curves contracted by p. It follows that we can ﬁnd NE(Y) ⊂ σ ⊂ σ such that F is a R P P face of σ . Now the algebra structure for P comes from P by base extension, so (replacing PbyP ) we can assume F = F , and thus that W is positive on F\{0}. Now let J be a monomial ideal with J = P \ F. Consider condition (1 ). By the D C T -equivariance of Theorem 5.2,any z ϑ that appears in ϑ · ϑ has the same weight s p q for T . Thus it is enough to show that the map w: B(Z) × (P \ J) → χ T ,(q, C) → w(q) + w(C) has ﬁnite ﬁbres. It is enough to consider ﬁbres of σ(Z) × (P \ J) → χ(T ) for each σ ∈ .Notethat σ(Z)× P is the set of integral points of a rational polyhedral cone, and w max is linear on this set. Thus it is enough to check that ker(w)∩ (σ (Z)× F) = 0. So suppose we have q ∈ σ(Z),C ∈ Fwith w(q) + w(C) = 0. Say σ = σ .Then q = av + bv , i,i+1 i i+1 for a, b ∈ Z .Wehave ≥0 w(q) + w(C) = ae + be + (C · D )e ; D D j D i i+1 j thus if this is zero, we have C· D ≤ 0for all j.Inparticular, W· C ≤ 0. Since W is positive on F\{0},C = 0. Now necessarily a = b = q = 0. This proves (1 ). For (2), let A ⊂ A be the subalgebra generated by the ϑ , q ∈ .Fix aweight J q w ∈ χ(T ).Toshow A = A it is enough to show that the ﬁnite set z ϑ ∈ A | (q, C) ∈ B(Z) × (P \ J) of weight w q J is contained in A (since the z ϑ give a k-basis of A ). We argue by decreasing induction q J on ord (C) (see Deﬁnition 2.21). Since the set of possible (q, C) is ﬁnite, there is an upper bound on the possible ord ’s. So the claim is vacuously true for large ord .Consider m m z · ϑ ,with ord (C) = h. Since the ϑ generate A modulo m,wecan ﬁnd a ∈ A such p m q J that ϑ = a + m with m ∈ m · A . Moreover, we can assume a,and thus m, is homogeneous for the T action. Now C C C z ϑ = z a + z m. p MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 139 C D C Clearly z m is a sum of terms z ϑ of weight w and ord (D)> h,so z m ∈ A by induc- q m tion. Remark 6.7. — Suppose p: Y → Y is a contraction such that some component of D is not contracted by p.Let F bethe face of NE(Y) generated by classes of curves contracted by p.Then NE(Y) is rational polyhedral near F. (This follows from the cone theorem, cf. the proof of Proposition 6.6.) In particular there exists a rational polyhedral cone σ ⊂ A (Y, R) such that NE(Y) ⊂ σ and σ coincides with NE(Y) near F. P 1 R P P R Corollary 6.8. —Theorem 0.2 holds if D has n ≥ 3 irreducible components. Proof. — Immediate from Proposition 6.6. 6.2. Proof of Theorems 0.1 and 0.2 in general. — We now consider an arbitrary Looijenga pair (Y, D), along with a toric monoid P with NE(Y) ⊂ P ⊂ A (Y, Z).Let τ : (Y , D ) → (Y, D) be a toric blowup such that (Y , D ) has a toric model p : (Y , D ) → ¯ ¯ (Y, D).Wehavethe map τ : A (Y , Z) → A (Y, Z). We can ﬁnd a strictly convex ratio- ∗ 1 1 nal polyhedral cone σ with NE Y ⊂ σ ⊂ A Y , R P 1 which has a face F spanned by the τ -exceptional curves, and which surjects under τ onto σ ⊂ A (Y, R). For any monomial ideal I ⊂ Pwith I = m,let I ⊂ P be the in- P 1 verse image of I under τ .Then I is the prime monomial ideal associated to the face F. Since the exceptional curves are a proper subset of D we have I ∈ A(Y ) by Proposi- tion 6.6.Notethat Spec k[P]/I is naturally a closed subscheme of Spec k[P ]/I ,via the map induced by the surjection τ : P → P. Now restrict the family X → Spec k[P ]/I ∗ I to Spec k[P]/I. This gives an algebra structure on A := k[P]/I ϑ . I q q∈B(Z) We now verify Assumptions 6.1. First, we show that the multiplication rule of this algebra is the one described in Theorem 2.34. The argument is just as in the proof of Proposi- can can tion 3.12:Wehave B = B and take ψ := τ : P → P. Note ψ(D ) = D (Y ,D ) (Y,D) ∗ (Y ,D ) (Y,D) (i.e., the rays are the same, and we apply ψ to the decoration function). This does not literally give a bijection on broken lines (because different exponents in the decoration of can aray in D could map to the same exponent under ψ ). However, by Equation (3.4), (Y ,D ) 140 MARK GROSS, PAUL HACKING, AND SEAN KEEL with z a point close to q, ⎛ ⎞ ⎛ ⎞ ⎝ ⎠ ⎝ ⎠ c(γ )c(γ ) = ψ c γ ψ c γ 1 2 1 2 (γ ,γ ) (γ ,γ ) 1 2 1 2 γ ∈ξ γ ∈ξ γ γ 1 1 2 2 Limits(γ )=(q ,z) Limits(γ )=(q ,z) i i i i s(γ )+s(γ )=q s(γ )+s(γ )=q 1 2 1 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ψ c γ c γ , 1 2 ⎜ ⎟ ⎜ ⎟ (γ ,γ ) 1 2 ⎝ ⎠ Limits(γ )=(q ,z) s(γ )+s(γ )=q 1 2 can where ξ denotes the set of all broken lines γ for D such that ψ ◦ γ = γ as paths γ i i i (Y ,D ) i and the monomials attached to ψ(γ ) differ from those attached to γ only in the k-valued coefﬁcients (see the proof of Proposition 3.12). This implies the claim. Next we need to check that the ﬁbre over the zero stratum of Spec k[P] is V .In case n ≥ 3, this is straightforward from the multiplication rule. Indeed, modulo m,every broken line contributing to the multiplication rule is a straight line, and furthermore it cannot cross any ray of . From this one sees that A = R [ ]. m m The cases n = 1 and 2 require special attention. We will do the case of n = 1, as n = 2 is similar (and simpler). We cut B = B along the unique ray ρ = ρ ∈ ,and (Y,D) 1 consider the image under a set of linear coordinates ψ on B \ ρ . This identiﬁes B \ ρ with a strictly convex rational cone in R .Let w, w be the primitive generators of the two boundary rays. Modulo m the decoration on every scattering ray is trivial, so every broken line is straight. Moreover, no line can cross ρ (or the attached monomial becomes trivial modulo m by the strict convexity of ϕ). Now it follows for any x ∈ B(R)\ ρ and any q ∈ (B\ ρ)(Z) there is a unique (straight) broken line with Limits = (q, x), while there are exactly two (straight) broken lines with Limits = (v, x), v = v —under ψ these become two distinct straight lines with directions w, w . Performing a toric blowup of (Y, D) to get n = 3 can be accomplished by subdividing the cone generated by w and w along the rays generated by w + w and 2w + w . Then by Theorem 0.2 in the case n = 3, we see that A is generated over k by ϑ = ϑ = ϑ ,ϑ ,ϑ v w w w+w 2w+w where we abuse notation and use the same symbol for an integer point in the convex cone generated by w and w , and the corresponding point in B(Z). Now applying the multiplication rule of Theorem 2.34 one checks easily the equalities: ϑ · ϑ = ϑ + ϑ v w+w 2w+w w+2w ϑ · ϑ = ϑ = ϑ . 2w+w w+2w 3w+3w w+w MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 141 It follows that 2 3 ϑ · ϑ · ϑ = ϑ + ϑ 2w+w v w+w 2w+w w+w 2 3 and thus A = k[x, y, z]/(xyz − x − z ), which is isomorphic to the ring of sections H C, O(m) m≥0 for a line bundle O(1) of degree one on an irreducible rational nodal curve C of arith- metic genus 1. Thus Spec A = V . m 1 Combining this with Propositions 6.5 and 6.6, this proves Theorems 0.1 and 0.2 hold for all (Y, D) except for the smoothness statement of Theorem 0.1. To show smoothness, note that if m denotes the maximal monomial ideal of P , X → S the formal deformation provided by Theorem 0.1 for the pair (Y , D ) with the toric model, we know that k[P]→ H (Z , O ) is not injective by Theorem 4.6. m Z Now choose (see the beginning of the proof of Proposition 6.6) a divisor A = a D with i i a ≥ 0for all i and A relatively τ -ample, so that A · D > 0for anyD contracted by τ . j j A D D This determines a one-parameter subgroup T = G of T via the map χ(T ) → Z given by e → a . D i Let J = P \ F, so that [C]∈ J if and only if C is not contracted by τ .Thusif [C]∈ F, A [C] A T acts on z with weight C.A > 0, and for q ∈ B(Z),T acts with non-negative weight since a ≥ 0for all i. It then follows that the map 0 0 H (Z , O ) → H (Z , O ) J Z m Z is injective because every component of Z has a limit point in Z under the T action. J m So we conclude that k[P]→ H (Z , O ) is not injective. J Z gp F Now F is generated by the classes of the D contracted by τ.Let T := gp D Hom(F, G ). The composition F ⊂ A (Y , Z) → χ(T ) is a primitive embedding, be- m 1 cause the intersection matrix of F ⊂D ,..., D is unimodular, where D ,..., D are 1 r 1 r the irreducible components of the boundary of Y . So the corresponding composition D F F D F T → Hom(A (Y , Z), G ) → T admits a splitting T → T .By T -equivariance, 1 m the restriction of the family X /S to the open subscheme of S deﬁned by T ⊂ S is J J J J isomorphic to a direct product of X /S (coming from (Y, D), P) with T . In particular, m m X /S has smooth generic ﬁbre. m m 6.3. The case that (Y, D) is positive. Lemma 6.9. — The following are equivalent for a Looijenga pair (Y, D): (1.1) There exist integers a ,..., a such that ( a D ) > 0. 1 n i i (1.2) There exist positive integers b ,..., b such that ( b D ) · D > 0 for all j . 1 n i i j 142 MARK GROSS, PAUL HACKING, AND SEAN KEEL (1.3) Y \ D is the minimal resolution of an afﬁne surface with (at worst) Du Val singularities. (1.4) There exist 0 < c < 1 such that −(K + c D ) is nef and big. i Y i i If any of the above equivalent conditions hold, then so do the following: (2.1) The Mori cone NE(Y) is rational polyhedral, generated by ﬁnitely many classes of rational curves. Every nef line bundle on Y is semi-ample. (2.2) The subgroup G of Aut(Pic(Y),·,·) ﬁxing the classes [D ] is ﬁnite. (2.3) The union R ⊂ Y of all curves disjoint from D is contractible. Deﬁnition 6.10. — We say a Looijenga pair (Y, D) is positive if it satisﬁes any of the equivalent conditions (1.1)–(1.4) of the above lemma. Proof.—We have K + c D = (K + D) − (1 − c )D =− (1 − c )D Y i i Y i i i i so (1.2) implies (1.4), and (1.2) obviously implies (1.1). ⊥ ⊥ If (1.1) holds then (D ,·,·),where D ={H ∈ Pic Y| H· D = 0 ∀i},isnegative deﬁnite, by the Hodge Index Theorem, and this implies (2.2) and (2.3). Suppose (1.4) holds. By the basepoint-free theorem [KM98], 3.3, the linear system m b D = −m K + c D i i Y i i deﬁnes a birational morphism for m ∈ N sufﬁciently large, with exceptional locus the union R of curves disjoint from D. Adjunction shows R is a contractible conﬁguration of (−2)-curves, which gives (1.3). (2.1) follows from the cone theorem [KM98], 3.7. We show (1.1) implies (1.2). By the Riemann-Roch theorem, if W is a Weil divisor (on any smooth surface) and W > 0 then either W or −W is big (i.e., the rational map given by |nW| is birational for sufﬁciently large n). So, possibly replacing the divisor by its negative, we may assume W = a D is big. Write i i W = a D = W + (−a )D . i i i i a >0 −a >0 i i Thus W is big, and replacing W by nW , we may assume all a ≥ 0and |W| deﬁnes a birational (rational) map. Subtracting off the divisorial base-locus (which does not affect the rational map) we may further assume the base locus is at most zero dimensional. Now W = b D is effective, nef and big, and supported on D. We show we may assume that i i in addition b > 0and W · D > 0for each i.If W · D > 0, then we may assume b > 0 i i i i (by adding D to W if necessary). Now consider the set S ⊂{1,..., n} of components D of D such that W · D = 0. By connectedness of D we ﬁnd b > 0for each i ∈ S. i i i Thus Supp(W) = D. By the Hodge index theorem the intersection matrix (D · D ) is i j i,j∈S MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 143 negative deﬁnite. Hence there exists a linear combination E = α D ,with α ∈ Z for i i i i∈S each i ∈ S, such that E · D < 0for each i ∈ S. Now replacing W by W − E, we obtain W · D > 0for each i = 1,..., n. Finally we show (1.3) implies (1.1). Since U = Y \ D is the resolution of an afﬁne variety U with Du Val singularities, we have U = Y \ D where Y is a normal projective surface and D is a Weil divisor such that D is the support of an ample divisor A. Let −1 π : Y → Y be a resolution of singularities such that π| −1 : π (U ) → U is the reso- π (U ) lution U → U . Furthermore, we can assume the inclusion U ⊂ Y extends to a birational ˜ ˜ ˜ ˜ morphism f : Y → Y. Let D be the inverse image of D under π,so Y \ D = U. The ∗ ∗ ∗ divisor π A has support D. So we can write π A = f ( a D ) + μ E where the E i i j j j 2 ∗ 2 2 are the f -exceptional curves and a ,μ ∈ Z.Then ( a D ) ≥ (π A) = A > 0. i j i i Corollary 6.11. —Let (Y, D) be a positive Looijenga pair. Let P = NE(Y). The multi- can D plication rule Theorem 2.34 applied with D = D determines a ﬁnitely generated T -equivariant R = k[P]-algebra structure on the free R-module A = R · ϑ . q∈B(Z) Furthermore, Spec A → Spec R is a ﬂat afﬁne family of Gorenstein semi-log canonical (SLC) surfaces with central ﬁbre V , and smooth generic ﬁbre. Any collection of ϑ whose restrictions generate A/m = n q H (V , O ) generate A as an R-algebra. In particular the ϑ generate for n ≥ 3. n V v n i Proof. — Everything but the singularity statement follows from Theorem 0.2.The Gorenstein SLC locus in the base is open, and T -equivariant. Taking a big and nef divi- sor H = a D with a > 0and H· D > 0for all i, we obtain a one-parameter subgroup i i i i D D of T given by the map χ(T ) → Z, e → a . By deﬁnition of the weights, the weights D i [C] of z for C ∈ NE(Y) and ϑ for p ∈ B(Z) are all non-negative. Thus the corresponding torus T = G gives a contracting action. In particular, since V is Gorenstein and SLC, H m n all ﬁbres are Gorenstein and SLC. Moreover, the map 0 0 H (Z, O ) → H (Z , O ) Z m Z is injective, where Z is the singular locus of the family X → S. But since R → H (O , O ) is not injective, as shown in the proof of Theorem 0.1, we deduce that Z Z m m the map R → H (Z, O ) is not injective. Letting f be in the kernel of the map, the ﬁbres over Spec R \ V(f ) are smooth. In Part II we will prove that when D is positive, our mirror family admits a canon- ical ﬁbrewise T -equivariant compactiﬁcation X ⊂ (Z, D). The restriction (Z, D) → T := Pic(Y) ⊗ G comes with a trivialization D → D × T . We will show that (Z, D) Y m ∗ Y 144 MARK GROSS, PAUL HACKING, AND SEAN KEEL is the universal family of Looijenga pairs (Z, D ) deformation equivalent to (Y, D) to- gether with a choice of isomorphism D → D . Now for any positive pair (Z, D ) to- Z ∗ Z gether with a choice of isomorphism φ: D → D , our construction equips the com- Z ∗ plement U = Z \ D with canonical theta functions ϑ , q ∈ B (Z). We will give a Z q (Z,D) characterisation in terms of the intrinsic geometry of (Z, D ). Changing the choice of isomorphism φ changes ϑ by a character of T = Aut (D ), the identity component of q ∗ Aut(D ). Here we illustrate with two examples: Example 6.12. — Consider ﬁrst the case (Y, D) a5-cycle of (−1)-curves on the (unique) degree 5 del Pezzo surface, Example 3.7.Inthis case T = T = Pic(Y)⊗ G , Y Z m andthusbythe T -equivariance, all ﬁbres of the restriction X → T are isomorphic. We consider the ﬁbre over the identity e ∈ T , thus specializing the equations of Example 3.7 by setting all z = 1. It’s well known that these equations deﬁne an embedding of the 5 5 original U = Y\ D into A —ifwetakethe closurein P (for the standard compactiﬁcation 5 5 A ⊂ P ) one checks easily we obtain Y with D the hyperplane section at inﬁnity. Now it is easy to compute the zeroes and poles: (ϑ ) = E + D − D − D v i i i+2 i−2 (indices mod 5). In particular {ϑ = 0}= E ∩ U ⊂ U, which characterizes ϑ up to v i v i i scaling. Example 6.13. —Now let (Y, D = D + D + D ) be (the deformation type of) 1 2 3 a cubic surface together with a triangle of lines. Let X ⊂ Spec(k[NE(Y)]) × A be the canonical embedding given by ϑ := ϑ , i = 1, 2, 3. In this case, as we shall see in Part II, i v the scattering diagram is particularly beautiful, with every ray d of rational slope occur- ring, with precisely six curves on the cubic surface contributing to f . We will also show in Part II that the mirror is given by the equation D 2 E D π H D +D +D i ij i 1 2 3 ϑ ϑ ϑ = z ϑ + z z ϑ + z − 4z . 1 2 3 i i i j π Here the E are the interior (−1)-curves meeting D ,and thesum over π is the sum over ij i ¯ ¯ ¯ all possible toric models π : Y → Yof (Y, D) to a pair (Y, D) isomorphic to P with its toric boundary. (Such π are permuted simply transitively by the Weyl group W(D ) by [L81], Prop. 4.5, p. 283.) The same family, in the same canonical coordinates, was discov- ered by Oblomkov [Ob04]. As we learned from Dolgachev, after a change of variables (in A ), and restricting to T (the locus over which the ﬁbers have at worst Du Val singular- ities) this is identiﬁed with the universal family of afﬁne cubic surfaces (the complement to a triangle of lines on projective cubic surface) constructed by Cayley in [C1869]. The 3 3 universal family of cubic surfaces with triangle is obtained as the closure in A ⊂ P .In particular, as in the ﬁrst example, our mirror family compactiﬁes naturally to the universal MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 145 family of Looijenga pairs deformation equivalent to the original (Y, D). There is again a geometric characterisation of ϑ (up to scaling): The linear system |−K − D|=|D + D | i Y i j k (here {i, j, k}={1, 2, 3}) is a basepoint free pencil. It deﬁnes a ruling π : Y → P which 1 1 restricts to a double cover D → P .Let {a, b}⊂ P be the branch points of π| .Let i D 1 1 p = π(D + D ) ∈ P . There is a unique point q ∈ P \{a, b, p} ﬁxed by the unique in- j k 1 ∗ interchanging a and b and ﬁxing p.Let Q = π (q) ∈|D + D | be the volution of P j k corresponding divisor. The curve Q ⊂ P is a smooth conic. In Part II we will show Proposition 6.14. — (ϑ ) = Q − D − D . i j k 7. Looijenga’s conjecture In this section we apply the main construction of this paper to give a proof of Looijenga’s conjecture on smoothability of cusp singularities, Theorem 0.5. The simple conceptual idea is explained in the introduction. Here we give the rather involved details. 7.1. Duality of cusp singularities. — We review the notion of dual cusp singularities. By deﬁnition, a cusp is a normal surface singularity for which the exceptional locus of the minimal resolution is a cycle of rational curves. The self-intersections of these ex- ceptional curves determine the analytic type of the singularity, see [L73]. Cusps have a quotient construction due to Hirzebruch [Hi73] which we explain here. See also [L81], III, Section 2 for this point of view. Let M = Z ,and let T ∈ SL(M) be a hyperbolic matrix, i.e., T has a real eigen- value λ> 1. Then T determines a pair of dual cusps as follows: Let w ,w ∈ M be 1 2 R eigenvectors with eigenvalues λ = 1/λ, λ = λ, chosen so that w ∧ w > 0 (in the stan- 1 2 1 2 dard counter-clockwise orientation of R ). Let C, C be the strictly convex cones spanned by w ,w and w ,−w ,and let C, C be their interiors, either of which is preserved by 1 2 2 1 T. Let U , U be the corresponding tube domains, i.e., C C U := z ∈ M | Im(z) ∈ C /M ⊂ M /M = M ⊗ G . C C C m T acts freely and properly discontinuously on U , U .Write Y , Y for the holomor- C C C C phic hulls of U /, U /,where is the group generated by T. These each have one C C additional point, p ∈ Y , p ∈ Y ,and (Y , p), (Y , p ) are normal surface germs of C C C C cusps. Deﬁnition 7.1. — (Y , p) and (Y , p ) are dual cusp singularities. C C All cusps (and their duals) arise this way. Remark 7.2. — If M is identiﬁed with its dual by choosing an isomorphism M = Z, the cone C is identiﬁed with the dual cone of C. In this way C / and C/ are dual 146 MARK GROSS, PAUL HACKING, AND SEAN KEEL integral afﬁne manifolds, which suggests that the duality between the corresponding cusps is a form of mirror symmetry. To resolve the cusp singularities of Y , say, one considers the convex hull of integral points in C, and let v , i ∈ Z, be the integral points in ∂, listed so that v i i−1 and v are the integral points adjacent to v on ∂.Let be the inﬁnite fan with two- i+1 i dimensional cones generated by v ,v for i ∈ Z. Note that T acts on , and thus acts by i−1 i translation T(v ) = v for some integer n, which we can take to be positive by reversing i i+n the ordering of the v if necessary. Then X is a toric variety with an inﬁnite chain of P ’s, and T acts on X . There is a tubular neighbourhood N of this inﬁnite chain of P ’s on which the group generated by T acts properly discontinuously, see [AMRT75], p. 48. Then N/ is a minimal resolution of singularities of a neighbourhood of the singularity of Y . Note the exceptional divisors D , i taken modulo n,with D corresponding to the C i i ray of generated by v ,satisfy (1.2). Thus if there is a Looijenga pair (Y, D ) with 2 2 D = D + ··· + D and (D ) = D for each i, the corresponding afﬁne manifold with 1 n i i singularities is precisely B=| |/ by Example 1.10,and B = C/. In fact, the dual cusp singularity can be described directly from the cone C and the polyhedron : Lemma 7.3. —Let T, C, and the v be as above, giving a cusp singularity p ∈ Y .Let Z i C be the toric variety (only locally of ﬁnite type) associated to .Let E ⊂ Z be the toric boundary of Z,an inﬁnite chain of smooth rational curves corresponding to the boundary of . Then there exists a tubular neighbourhood E ⊂ N ⊂ Z such that the action on induces a properly discontinuous action on N.Let F ⊂ X denote the quotient of E ⊂ N by .So F is a cycle of smooth rational curves. Then F ⊂ X can be contracted to a singularity p ∈ X, which is a copy of the dual cusp p ∈ Y .Moreover, X is obtained from the minimal resolution of p ∈ X by contracting all the (−2)-curves. Proof.—Let be the normal fan for the polytope and C the closure of its support. We observe that C coincides with the dual of C, together with the induced -action. By Remark 7.2, it follows that X is a partial resolution of a copy of the dual cusp. The surface X has Du Val singularities of type A. Indeed, v is a vertex of iff m := −D > 2. The corresponding point of Z is smooth if m = 3 and a singularity of i i type A if m > 3 (by direct calculation using v + v = m v ). Also K is relatively m −3 i i−1 i+1 i i ˜ ample over X by Lemma 7.4 below. Indeed, the vectors u := v − v are the primitive i i i−1 integral vectors in the direction of the edges of ,and u − u = v + v − 2v = (m − 2)v , i+1 i i+1 i−1 i i i so the lines v + R· (u − u ) = R· v all meet at the origin. We deduce that X is obtained i i+1 i i from the minimal resolution of X by contracting all (−2)-curves as claimed. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 147 Lemma 7.4. —Let P ⊂ R be a rational convex polygon and X the associated toric surface. Fix an orientation of the boundary of P and let e , e , e denote oriented consecutive edges of the boundary 0 1 2 of P.Let C ⊂ X denote the component of the toric boundary associated to the bounded edge e ⊂ P.Let v = e ∩ e and v = e ∩ e be the vertices of e .Let u , u , u ∈ Z denote the primitive integral 0 0 1 1 1 2 1 0 1 2 vectors in the direction of e , e , e . Then K · C > 0 if and only if the lines v + R(u − u ) and 0 1 2 X 0 1 0 v + R(u − u ) meet on the opposite side of e to P. 1 2 1 1 2 2 Proof.—Write M = Z ⊂ R for the lattice of characters of the torus of X. Choose an orientation M ' Z and use it to identify M with its dual lattice N. Let U ⊂ X denote the union of the two toric afﬁne open subsets corresponding to the vertices v and v of P. Then U is a toric open neighbourhood of C ⊂ X. Then, under this identiﬁcation anduptoasign,the fan of U in N consists of the two cones u , u , u , u R 0 1 R 1 2 R ≥0 ≥0 and their faces. The condition on the lines v + R(u − u ) and v + R(u − u ) in the 0 1 0 1 2 1 statement is equivalent to the condition that the primitive generator u of the central ray of the fan lies on the same side of the afﬁne line spanned by the primitive generators u and u of the two outer rays of as the origin 0 ∈ N. Now by [R83], 4.3, this condition is equivalent to K · C > 0. Because this quotient construction is analytic we will have to deal with convergence issues to show that our construction extends to this analytic situation. 7.2. Cusp family. — In this subsection, we ﬁx the following. Let (Y, D) be a rational surface with anti-canonical cycle, now over the ﬁeld k = C.Weobtain (B, ),with having one-dimensional cones ρ and two-dimensional cones σ as usual. i i,i+1 We assume that the intersection matrix (D · D ) is negative deﬁnite. Let i j 1≤i,j≤n f : Y → Y be the contraction (in the analytic category) of D ⊂ Y to a cusp singular- ity q ∈ Y . We assume that f is the minimal resolution of Y ,thatis, D ≤−2for all i.We further assume that n ≥ 3 to avoid additional technical issues of the ﬂavour dealt with in Section 6.2. The case of Looijenga’s conjecture with n ≤ 2 is in fact trivial, see the proof of Theorem 7.13. Let L be a nef divisor on Y such that NE(Y) ∩ L =D ,..., D . R 1 n R ≥0 ≥0 Here the subscript R denotes the real cone in A (Y, R) generated by the given elements ≥0 1 or set. (Indeed, if Y is projective we can take L = h A for A an ample divisor on Y . In general, let A be an ample divisor on Y. There exist unique a ∈ Q such that L := A + a D is orthogonal to D for each j . By negative deﬁniteness of D ,..., D ,we i i j 1 n have a > 0for each i. It follows that L is nef.) Let σ ⊂ A (Y, R) be a rational polyhedral cone containing NE(Y),P = σ ∩ P 1 P A (Y, Z) the associated toric monoid. We assume that σ is strictly convex and σ ∩ L 1 P P is a face of σ .Let m = P \{0} and J = P \ P ∩ L ⊂ P, the radical monomial ideal P 148 MARK GROSS, PAUL HACKING, AND SEAN KEEL associated to the face σ ∩ L of σ . We will write S = Spec k[P], and for any monomial P P ideal I, we write S = Spec k[P]/I. We take the multivalued piecewise linear function ϕ as usual to have bending pa- rameter κ =[D ]∈ P. We wish to build a deformation of the n-vertex over S with ρ,ϕ ρ I I = J. However, this is already a problem over S because none of the κ lie in J. Thus J ρ,ϕ we can’t apply the results of Section 2 directly as the standard open sets U will not glue ρ,J compatibly because of issues involving triple intersections. To deal with this, we need to shrink these open sets. This procedure is carried out as follows. Theorem 7.5. —Fix R > 1. There exists an analytic open neighbourhood S of 0 ∈ S and an analytic ﬂat family f : X → S together with a section s: S → X satisfying the following properties: J J J J J (1) The general ﬁbre X of f is a Stein analytic surface with a unique singularity s(t) ∈ X J,t J J,t isomorphic to the dual cusp to q ∈ Y . (2) For each ray ρ ∈ there is an open analytic subset V ⊂ X and open analytic embed- i ρ ,J J dings V ⊂ (X , X , X ) ∈ U ||X | < R|X |, |X | < R|X | ⊂ U ρ ,J i−1 i i+1 ρ ,J i−1 i i+1 i ρ ,J i i i where −D [D ] 2 ρ i U := V X X − z X ⊂ A × (G ) × S ρ ,J i−1 i+1 m X J i i X ,X i i−1 i+1 such that (a) X := X \ s(S ) = V . J ρ,J J J ρ∈ (b) V ∩ V =∅ unless ρ = ρ or ρ and ρ are the edges of a maximal cone σ ∈ . ρ,J ρ ,J (3) The restriction of X /S to S N+1 is identiﬁed with an analytic neighbourhood of the vertex J J+m can in the restriction of the family X N+1 /S N+1 given by Theorem 2.28, (1) with D = D , m m for each N ≥ 0. Proof. — (1) We use the notation of Example 1.10, so that the pair (Y, D) deter- mines an inﬁnite fan in M with the primitive generators of the rays being the v for R i ˜ ˜ i ∈ Z.Wealsohave T ∈ SL(M) acting on the fan .Wehave B=| |/,where is the group generated by T. Now as in Section 7.1,let ⊂ M be the convex hull of the points v ∈ M. Thus R i ˜ ˜ is an inﬁnite convex polytope. Let be the subdivision of induced by .Inwhat follows, we will build a Mumford degeneration Z/S with special ﬁbre Z the stable toric J 0 variety associated to .Inother words, Z will be the union of the toric surfaces associ- ated to the maximal polytopes in . S is the afﬁne toric variety associated to the face σ := σ ∩ L of σ ,and P := J bdy P P bdy σ ∩ A (Y, Z) contains the classes of the components of the boundary of Y. We deﬁne bdy 1 gp a piecewise linear convex function ϕ˜:| |→ P ⊗ R by restriction of a piecewise bdy MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 149 linear convex function ϕ˜ on | |. This function is the single-valued representative for ϕ on the universal cover of B , and as such, is deﬁned up to an integral linear function by specifying its bending parameter κ =[D ]∈ P ρ ,ϕ˜ i mod n bdy if ρ = R v . i ≥0 i Then ϕ˜ determines a Mumford degeneration: this is a slight generalization of Sec- tion 1.3.One deﬁnes gp := (m, r)| m ∈ , r∈˜ ϕ(m) + σ ⊂ M ⊕ P ⊗ R . bdy R Z bdy Let C() be the closure of gp (sm, sr, s)| (m, r) ∈ , s ≥ 0 ⊂ M ⊕ P ⊗ R ⊕ R. R Z bdy gp Then C[C() ∩ (M ⊕ P ⊕ Z)] has a natural grading given by the last coordinate, and bdy the degree zero part of this ring is easily seen to contain C[P ].Thusweobtainthe bdy Mumford family determined by ϕ˜ as gp Z := Proj C C() ∩ M ⊕ P ⊕ Z → Spec C[P ]= S . bdy J bdy One sees easily that the ﬁbre over 0 ∈ S of Z → S has inﬁnitely many compo- J J nents indexed by the 2-cells of the subdivision of ,eachofwhichisacopy of the blowup of A at the origin. The general ﬁbre is a toric surface (only locally of ﬁnite type) containing an inﬁnite chain of smooth rational curves, which specializes to the union of the exceptional curves of the blowups in Z . By construction acts on Z over S (because 0 J ϕ˜ is -invariant modulo integral afﬁne functions). Let E ⊂ Z/S be the family of curves described above (the relative toric boundary). The group acts properly discontinuously on a tubular neighbourhood N of E ⊂ Z(cf. [AMRT75], p. 48). Let p: (F ⊂ X) → S denote the quotient of (E ⊂ N) → S by . The divisor F ⊂ X is Cartier and the dual of its normal bundle is relatively ample over a neighbourhood of 0 ∈ S . Indeed, the special ﬁbre X is a union of n irreducible J 0 components each isomorphic to a tubular neighbourhood of the exceptional locus in the blowup of A ,and F ⊂ X is the cycle of n smooth rational curves formed by the 0 0 exceptional curves of the blowups. Hence the normal bundle of F in X has degree 0 0 1 ⊗k −1 on each component of F . Moreover, we have R p (N ) = 0for each k > 0. 0 ∗ F/X 1 ∨ ⊗k Indeed, by cohomology and base change it sufﬁces to show that H ((N ) ) = 0, F /X 0 0 and this follows from Serre duality. Now by a relative version of Grauert’s contractibility criterion [F75], Thm. 2, taking global sections of the structure sheaf deﬁnes a contraction p: X → X /S to a family of Stein analytic spaces with exceptional locus F. The general J J ﬁbre X of X /S is the dual cusp by Lemma 7.3. The section s : S → X takes x ∈ S to J,t J J J J J the cusp of X . J,x 150 MARK GROSS, PAUL HACKING, AND SEAN KEEL We now show that X /S is ﬂat and the special ﬁbre is the neighbourhood of the J J n-vertex obtained by contracting F ⊂ X . The key point is that R p O is a locally free 0 0 ∗ ˜ O -module, cf. [W76], Theorem 1.4(b). Indeed, we have R p O (−F) = 0 by cohomology and base change, the theorem on formal functions, and the vanishing 1 ∨ ⊗k H ((N ) ) = 0for k > 0 used above. So, pushing forward the exact sequence F /X 0 0 0 → O (−F) → O → O → 0 ˜ ˜ X X we obtain 1 1 R p O = R p O ' O . ∗ ˜ ∗ F S X J Recall that S is a toric variety, so in particular Cohen-Macaulay. Let t ,..., t be a regular J 1 r sequence at 0 ∈ S of length dim S and write J J S = V(t ,..., t ) ⊂ S , 1 i J i i i i ˜ ˜ X = X| i ,and let X /S be the family over S deﬁned by O i = p O i . Arguing as above S X ∗ X J J J J J we ﬁnd that R p O i ' O . Pushing forward the exact sequence ∗ ˜ X S ·t i+1 0 → O i−→O i−→O i+1−→0 ˜ ˜ ˜ X X X we deduce that the natural map (7.1) O i /t O i → O i+1 i+1 X X X J J is an isomorphism. Hence by the local criterion of ﬂatness [Ma89], Ex. 22.3, p. 178, it r r r sufﬁces to show that X /S is ﬂat with special ﬁbre the n-vertex. But S is the spectrum of J J J an Artinian C-algebra, so this follows from [W76], Theorem 1.4(b). o [D ] i mod n (2) Write Z = Z \ E, and m =−D , a = z for i ∈ Z.Wehaveanopen i i i mod n covering Z = U , i,J i∈Z where i 2 U = V x x − a x ⊂ A × (G ) × S . i,J i−1 i+1 i m x J i i x ,x i−1 i+1 Similarly, we have an open covering Z = U , i,J i∈Z MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 151 where m −2 U = V x x − a x ⊂ A × S , i,J i J i−1 i+1 i x ,x ,x i−1 i+1 ¯ ¯ E ∩ U = V(x ) ⊂ U , i,J i i,J and U \ E = U via x = x x , x = x x . (Note that m =−D ≥ 2 by assump- i,J i,J i−1 i i+1 i i i−1 i+1 i tion.) Recall that the inﬁnite cyclic group acts on Z/S , there is a -invariant tubular neighbourhood N ⊂ ZofE ⊂ Z on which the action is properly discontinuous, and F ⊂ X is obtained as the quotient of E ⊂ Nby . In terms of the open covering above the action is given by U → U , x → x . Note that the map U ∩ N → Xis not an open i,J i+n,J j j+n i,J embedding (because, for example, U contains the general ﬁbre of Z /S ). Fix R > 1. i,J J We deﬁne W ⊂ U ∩ Nby i i,J W = (x , x , x ) ∈ U ∩ N||x | < R|x |, |x | < R|x | i i−1 i i+1 i,J i−1 i i+1 i and similarly deﬁne W ⊂ U ∩ Nby i i,J W = x , x , x ∈ U ∩ N | x < R, x < R . i i i,J i−1 i+1 i−1 i+1 Then W \ E = W . i i The W cover the special ﬁbre E of E/S (using R > 1). The open set W ⊂ i 0 J i ˜ ˜ Nis -invariant, the quotient (N, E) → (X, F) by is a covering map, and p: X → X is proper with exceptional locus F. Hence we may assume (passing to an analytic neighbourhood S of 0 ∈ S and s(0) ∈ X )that N = W . J J i By Lemma 7.6 below there exists δ> 0such that W ∩ (x , x , x )||x | <δ ⊂ |x | < 1 ∀j i i−1 i i+1 i j for each i. We replace W by W ∩{|x | <δ},and modify W similarly. Then as above we i i i i may assume that N = W ,and N ⊂{|x | < 1 ∀i}.Weclaim that W ∩ W =∅ for all i i i j j > i + 1. It sufﬁces to work on the general ﬁbre of Z /S , which is an algebraic torus. The coordinate functions x are characters of this torus (up to a multiplicative constant). By construction we have |x | < 1for each i on N. Hence, shrinking the base S ,wemay m −2 i 2 assume that |a x | < 1/R for each i.The relation x x = a x i−1 i+1 i gives the inequality |x /x | < 1/R |x /x |. i+1 i i i−1 152 MARK GROSS, PAUL HACKING, AND SEAN KEEL Combining such inequalities we obtain j−i (7.2) |x /x | < 1/R |x /x | for j > i. j+1 j i+1 i Now |x /x | < RonW and |x /x | < RonW ,so |x /x | >(1/R )|x /x | on W ∩ i+1 i i j−1 j j j j−1 i+1 i i W .For j > i + 1 this contradicts the inequality (7.2), hence W ∩ W =∅ as claimed. j i j It follows that W embeds in X ,using thefactthatwehaveassumed n ≥ 3. Let i J V denote its image, with indices now understood modulo n.Thus X = V and the i i inverse image of V is the (disjoint) union of W such that j ≡ i mod n.Wehave V ∩ V = i j i j ∅ for j = i − 1, i, i + 1 by our claim above. So, writing V := V for ρ the ray of ρ,J i corresponding to D , the condition (2)(b) is satisﬁed. (3) We have the open covering X = V and open embeddings V ⊂ U ,and i i i,J an open covering X = U N+1 . The restrictions of U /S and U N+1 /S N+1 to N+1 i,m i,J J i,m m N+1 S are identiﬁed, and the gluing maps coincide. It follows that the restriction of J+m N+1 N+1 X /S is identiﬁed with a neighbourhood of the vertex in the restriction of X /S J J m m using Lemma 2.10. Lemma 7.6. — We use the notation of the proof of Theorem 7.5(2). Let x be the coordinate functions on Z = U . There exists δ> 0 such that on each open set U if |x | <δ, |x | < i,J i,J i i−1 R|x |, |x | < R|x |,and |z | < 1 for all p ∈ P then |x | < 1 for all j . i i+1 i bdy j Proof. — The points v , v ,and v are consecutive integral points on the bound- i−1 i i+1 ary of the inﬁnite convex polytope with asymptotic directions w ,w . It follows that 1 2 w = α v + β (v − v ) and w = α v + β (v − v ) for some α ,β ,α ,β ∈ R . 1 i1 i i1 i−1 i 2 i2 i i2 i+1 i i1 i1 i2 i2 >0 Note that the ratios β /α ,β /α only depend on i modulo n (because w ,w are eigen- i1 i1 i2 i2 1 2 vectors of T and T(v ) = v ). Let μ be the maximum of the ratios β /α ,β /α for i i+n i1 i1 i2 i2 −μ i = 1,..., n.Let δ = R .If j > i then v = αv + β(v − v ) with β/α < β /α . j i i+1 i i2 i2 p α β The coordinate function x can be written as z x (x /x ) on V ,where p ∈ P .Thus j i+1 i i bdy α β p |x | <δ R < 1for |x | <δ and |z | < 1. Thesameistrue for j < i by symmetry. j i 7.3. Thickening of the cusp family. — We continue to work with the setup at the be- ginning of Section 7.2,with (Y, D),L, σ ,P, m and J as given there. Theorem 7.7. —Let p : X → S be the analytic family of Theorem 7.5. Possibly after J J replacing S by a smaller neighbourhood of 0 ∈ S and X by a smaller neighbourhood of s(S ) ⊂ X , J J J J J independent of the choice of I below, the following holds. Let I ⊂ P be a monomial ideal such that I = J and let S ⊂ S denote the induced thickening of S ⊂ S . There is an inﬁnitesimal deformation I J I J N+1 f : X → S of f : X → S such that for each N > 0 the restriction to Spec k[P]/(I + m ) is I I J J I J identiﬁed with an analytic neighbourhood of the vertex in the restriction of the family X N+1 /S N+1 given m m can by Theorem 2.28, (1) applied with D = D . can Proof. — As usual, D is the canonical scattering diagram on B associated to the pair (Y, D). Note that the hypotheses (I) and (II) of Theorem 3.8 are satisﬁed for the MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 153 ideal J (but not (III)). This is because we can take L = A+ a D with A ample on Y and i i a > 0, and any A -class β intersects the D non-negatively. Then for any k there are only i i 1 1 a ﬁnite number of A -classes β such that L · β< k.Inaddition,there areno A -classes β with β · L = 0. can Let D be the scattering diagram obtained by reducing D modulo I as follows. can For each ray d of D , we truncate the attached function f by removing monomial terms lying in I · k[P ], and we discard the ray if the truncated function equals 1. Because of (II), D has only ﬁnitely many rays d and the attached functions f areﬁnitesumsof monomials. Because of (I), D is empty if I = J. We use the scattering diagram D to deﬁne a complex analytic space X /S as I I follows. Recall from (2.7) the description of the schemes U /S for ρ = ρ ∈ : ρ ,I I i −D [D ] ρ 2 U = V X X − z X f ⊂ A × (G ) × S . ρ ,I i−1 i+1 ρ m X I i i X ,X i i−1 i+1 We have open subsets U , U ⊂ U deﬁned by X = 0and X = 0re- ρ ,σ ,I ρ ,σ ,I ρ ,I i−1 i+1 i i−1,i i i,i+1 i spectively. We have canonical identiﬁcations (7.3)U = U = U ρ ,σ ,I σ ,σ ,I ρ ,σ ,I i i,i+1 i,i+1 i,i+1 i+1 i,i+1 where U = (G ) × S . σ ,σ m I i,i+1 i,i+1 X ,X i i+1 Recall that we have an open covering X = V and open analytic embeddings ρ,J J ρ∈ V ⊂ U .Write ρ,J ρ,J V := (V ∩ U ) ∩ (V ∩ U ) ⊂ U σ ,σ ,J ρ ,J ρ ,σ ,J ρ ,J ρ ,σ ,J σ ,σ ,J i,i+1 i,i+1 i i i,i+1 i+1 i+1 i,i+1 i,i+1 i,i+1 where we use the identiﬁcation (7.3). Let V ⊂ V ,V ⊂ V denote ρ ,σ ,J ρ ,J ρ ,σ ,J ρ ,J i i,i+1 i i+1 i,i+1 i+1 the open subsets corresponding to V under (7.3). Let V ,V , etc., be the σ ,σ ,J ρ ,I ρ ,σ ,I i,i+1 i,i+1 i i i,i+1 inﬁnitesimal thickenings of these open sets determined by the thickenings U of U . ρ ,I ρ ,J i i Let θ : U → U be the gluing isomorphism deﬁned as in Section 2.2. γ,D ρ ,σ ,I ρ ,σ ,I i+1 i,i+1 i i,i+1 can Note that as the canonical scattering diagram D is trivial modulo J, θ restricts to γ,D the identiﬁcation (7.3) modulo J, and thus restricts to an isomorphism V → V . ρ ,σ ,I ρ ,σ ,I i+1 i,i+1 i i,i+1 Gluing the V via these isomorphisms we obtain an inﬁnitesimal deformation X /S of ρ,I I I X /S . Note that there are no triple overlaps of the V by Theorem 7.5(2)(b), hence no ρ,J J J compatibility condition for the gluing automorphisms. It is clear from the construction that the families X /S and X N+1 /S N+1 are compatible. m m I I We deﬁne sections ϑ ∈ (X , O ) for q ∈ B(Z), compatible with the sections q X of Theorem 2.28, (2). We proceed as in the algebraic case: we ﬁrst deﬁne a local sec- tion Lift (q) for each choice of basepoint Q ∈ B \ Supp(D) on a corresponding open Q 0 154 MARK GROSS, PAUL HACKING, AND SEAN KEEL patch of X using the broken lines construction. The new difﬁculty here is that the func- tions Lift (q) are not algebraic, even over the unthickened locus S . Indeed, by deﬁnition Lift (q) = Mono(γ ) is a formal sum of monomials corresponding to broken lines γ for q with endpoint Q. Note that with our current choice of ideal J, this sum is always inﬁnite, as is already evident in Example 2.18. So we must prove convergence. This is done in Section 7.3.1, see Propositions 7.10 and 7.11. Once this convergence is proved, we observe that these patch to give well-deﬁned can o global sections. This follows from the consistency of D and compatibility of X /S with I I X /S N+1 for N ≥ 0. N+1 m We deﬁne an inﬁnitesimal thickening X /S of X /S by O = i O where I J X ∗ X I J I i: X ⊂ X is the inclusion. Then X /S is ﬂat by Lemma 2.29 and the existence of the J I J I lifts ϑ . 7.3.1. Convergence of lifts. — Let C ⊂ M be the closure of the support | | of the fan , a closed convex cone. Let w ,w be generators of C. Then w ,w are eigen- 1 2 1 2 −1 vectors of T with eigenvalues λ ,λ for some λ ∈ R. We may assume that λ> 1. Let ˜ ˜ π : B → B denote the universal cover of B .So B is identiﬁed with C = Int(C),with 0 0 0 0 ˜ ˜ deck transformations given by the action of =T on C. Let P , ϕ˜,and B (Z) denote ˜ ˜ the pullbacks of P , ϕ,and B (Z).Let D denote the scattering diagram on B induced 0 0 gp by D. We ﬁx a trivialization of P as the constant sheaf with ﬁbre P ⊕ M. The behaviour of the broken lines γ is best studied by passing to the universal ˜ ˜ ˜ ˜ cover B of B .Let Q ∈ B , q ∈ B (Z), and choose lifts Q ∈ B , q˜ ∈ B (Z). Then a broken 0 0 0 0 0 0 line γ on B for q with endpoint Q lifts uniquely to a broken line γ˜ on B for T (q˜) with 0 0 endpoint Q, for some N ∈ Z, and the attached monomials are identiﬁed via P = π P . Note that T (q˜) approaches R · w as N→∞ and R · w as N→−∞. ≥0 2 ≥0 1 ˜ ˜ ˜ If γ is a broken line for a point q˜ ∈ B (Z) with endpoint Q ∈ B then γ : (−∞, 0] 0 0 ˜ ˜ ˜ → B is a piecewise linear path in B = C with initial direction −˜ q, ending at Q, and 0 0 ˜ ˜ crossing all the rays of D between R q˜ and R Q in order. Let t ,..., t ∈ (−∞, 0) ≥0 ≥0 1 l denote the points where γ is not afﬁne linear. Each point γ(t ) lies on a ray d of D and i i the change γ (t + )− γ (t − ) in the direction of γ as it crosses d is an integral multiple i i i of the primitive generator of d . Moreover this multiple is positive because each ray of the canonical scattering diagram is an outgoing ray in the terminology of Deﬁnition 2.13.So the path γ is “convex when viewed from the origin”. It is convenient for the convergence calculation to decompose the monomials for broken lines as follows. Let γ : (−∞, 0]→ B be a broken line, t ∈ (−∞, 0] a point such that γ is afﬁne linear near t and γ(t) lies in the interior of a maximal cone σ of ,and cz the monomial attached to the domain of linearity of γ containing t.Here c ∈ k and gp q ϕ˜ (m) q−˜ ϕ (m) σ σ q ∈ P ⊂ P ⊕ M. We write cz = az ,where m = r(q) ∈ Mand a = cz is a ϕ˜ monomial in k[P]. We also use the same decomposition for the monomials occurring in ˜ ˜ the scattering functions f for d ∈ D.Let d be a ray in D with primitive integral generator m ∈ Mand τ = τ the smallest cone of containing d.Then f − 1 is a sum of monomials d d MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 155 q q ϕ˜ (m) cz where c ∈ k and q ∈ P ,0 = −r(q) ∈ d. We write cz = az where m = r(q) and ϕ˜ q−˜ ϕ (m) a = cz is a monomial in k[P]. The scattering diagram D on B is ﬁnite (because we have reduced modulo I), and ˜ ˜ ˜ D is its inverse image under π : B → B .Thus D has only ﬁnitely many -orbits of 0 0 rays. Moreover, the -action on the scattering functions f is induced by the given action ϕ˜ (m) on M and the trivial action on k[P] as follows: writing f = 1 + a z as above, R d m ϕ˜ (T(m)) T(τ ) f = 1 + a z . T(d) m ˜ ˜ Lemma 7.8. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal ˜ ˜ ˜ cone σ ∈ . We consider broken lines γ on B for T (q˜),some N ∈ Z, with endpoint Q, such that Mono(γ ) ∈ / I · k[P ].Let k ≥ 0 be such that (k + 1)J ⊂ I. ϕ˜ (1) The number of bends of γ is at most k. N k (2) The number of broken lines for T (q˜) with endpoint Q is O(|N| ). N ϕ˜ (m ) σ γ (3) For γ a broken line for T (q˜) with endpoint Q write Mono(γ ) = a z where a is γ γ a monomial in k[P]. Assume that |z | < < 1 for all p ∈ P. Then |N| [D ] |a |= O z . ρ∈ dim ρ=1 Proof. — (1) At a bend t ∈ (−∞, 0] of γ the attached monomial c z is replaced by i i q q q q i+1 i the monomial c z = cz · c z where cz is a term in a positive power of the scattering i+1 i function f associated to the ray d containing γ(t ). In particular cz ∈ J · k[P ]. Since d i ϕ˜ Mono(γ ) ∈ / I · k[P ] and (k + 1)J ⊂ I it follows that there are at most k bends. ϕ˜ (2) Such a broken line crosses O(|N|) scattering rays. If γ is a broken line for T (q˜) ϕ( ˜ T (q˜)) then the initial attached monomial is speciﬁed, equal to z . At a scattering ray d, let u denote the primitive generator of d, f = f the attached function, and let cz be the monomial attached to the incoming segment of the broken line. Then the possible continuations of the broken line past d correspond to the monomial terms in f ,where d =|r(q) ∧ u| is the index of the sublattice of M generated by r(q) and u.Notethat since f ≡ 1mod J · k[P ] the number of monomial terms in f not lying in I · k[P ] ϕ˜ ϕ˜ τ τ d d is bounded independent of d . Further, since there are a ﬁnite number of -orbits of scattering rays, and the -action preserves monomials, there is a bound on the number of monomial terms independent of the ray d. Thus for a broken line γ for T (q˜) there O(|N|) are = O(|N| ) choices of how it may bend by (1). So the total number of broken lines is O(|N| ). (3) By symmetry we may assume that N ≥ 0. Let d ∈ D be a scattering ray, f = f the attached function, and γ a broken line that crosses d. Suppose ﬁrst that d is contained ϕ˜ (m) in the interior of a maximal cone σ of .Let az be the monomial attached to the incoming segment of γ near d.Let u be the primitive generator of d. Then the outgo- ϕ˜ (m ) ing monomial a z is obtained from the incoming monomial by multiplication by a 156 MARK GROSS, PAUL HACKING, AND SEAN KEEL monomial term in f ,where d =|m∧ u|.Write f = 1+ f +···+ f , a sum of monomials. 1 r Since f ≡ 1mod J · k[P ] we have d i i (7.4) f ≡ f ··· f mod I · k[P ]. 1 r σ i ,..., i 1 r i +···+i ≤k 1 r The multinomial coefﬁcient d d! := i ,..., i i !··· i !(d − i −···− i )! 1 r 1 r 1 r k s is bounded by d . The direction of the scattering ray d is u = T (β) where 0 ≤ s ≤ Nand β ∈ M is chosen from a ﬁnite set. The vector m ∈ Mis of the form N s m = T (q˜) − T α i=1 where l ≤ k,0 ≤ s ≤ Nfor each i,and the α ∈ M are chosen from a ﬁnite set. Indeed, as i i q d in the proof of (2), for the monomial terms cz occurring in the powers f of the function f = f attached to a given scattering ray d,onlyﬁnitelymanyexponents q ∈ P occur d ϕ˜ (working modulo I· k[P ]). So there are only ﬁnitely many possible changes of exponent ϕ˜ q for the attached monomial cz of a broken line at a scattering ray modulo the action of . 2 2 Now identify M = Z and let (·( denote the standard norm on M = R .Then 2N d =|m ∧ u|≤(m(·(u(= O λ . So, the coefﬁcient a ∈ k[P] of the outgoing monomial is given by a = c · z · a where 2kN 2kN p c ∈ C, p ∈ P, and |c|= O(λ ).Thus |a |= O(λ )·|a| for |z | < 1. Next, let ρ ∈ be a ray and σ , σ the maximal cones containing ρ . Suppose γ + − ϕ˜ (m) is a broken line that crosses ρ , travelling from σ to σ .Let a z be the monomial − + − ϕ˜ (m ) attached to the incoming segment of γ near ρ and a z the monomial attached to the outgoing segment. By the deﬁnition of ϕ˜ , −n ,m ϕ˜ (m) [D ] ϕ˜ (m) σ ρ σ − + z = z z , where n ∈ N is primitive, annihilates ρ , and is positive on σ .Write d := −n , m;note ρ + ρ d =|u ∧ m| > 0where u ∈ M is the primitive generator of ρ.If γ does not bend at [D ] d [D ] d ϕ˜ (m ) ρ ρ σ ρ then a = (z ) · a . In general a = (z ) · a where a z is obtained from + − + − − ϕ˜ (m) a z as above (by applying the scattering automorphism associated to ρ and selecting a monomial term). [D ] We need to show the exponent d =|u ∧ m| > 0of z in the previous paragraph 2N is large for some lift ρ˜ of any given ray ρ ∈ . This will allow us to absorb the O(λ ) MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 157 factors coming from bends of γ and obtain the estimate (3). Let γ be a broken line for T (q˜).Then N s m = T (q˜) − T α γ i i=1 as above, where 0 ≤ l ≤ k,N ≥ s ≥··· ≥ s ≥ 0, and the α lie in a ﬁnite set. Write 1 l i s = Nand s = 0. Choose j such that s − s ≥ N/(k + 1). Now consider the exponent 0 l+1 j j+1 N s d =|u ∧ m| for m = T (q˜) − T α given by the monomial attached to the segment i=1 of the broken line between bends j and j + 1. Let ρ˜ ∈ be the lift of ρ ∈ between bends j and j + 1 which is closest to bend j + 1 (such a lift exists if N is sufﬁciently s s −s j+1 j j+1 large). Let u = T β be the primitive generator of ρ˜.Then |u ∧ m|=|β ∧ T m | −s where m = T (m). Writing m = μ w + μ w , we see that |μ | is bounded since w 1 2 1 1 2 1 −1 has eigenvalue λ < 1. Also, μ is bounded away from zero by Lemma 7.9.Now s −s −(s −s ) s −s j j+1 j j+1 j j+1 u ∧ m = β ∧ T m = μ (β ∧ w )λ + μ (β ∧ w )λ , 1 2 1 2 where s − s > N/(k + 1),so j j+1 N/(k+1) |u ∧ m| > c · λ for some constant c > 0. [D ] Combining our results now gives, when |z | < 1for all p ∈ P , the estimate bdy N/(k+1) k c·λ 2kN [D ] a = O λ · z , ρ∈ dim ρ=1 where the ﬁrst factor bounds the contribution associated to bends of γ and the second factor bounds the contribution associated to rays ρ of the fan crossed by γ , as described in the preceding two paragraphs. This implies the estimate (3) in the statement, using [D ] |z | < 1. Indeed, the above expression is of the form bN aN c·λ λ · x [D ] N where a, b, c > 0and λ> 1 are constants, and x = |z |. This is bounded by Cx for 0 ≤ x < < 1, for some constant C (depending on ). (To see this, we may assume x = 0, take logarithms, and establish an inequality bN (7.5) aNlog λ + cλ log x ≤ log C + Nlog x. We have − log x > − log> 0. Rearranging (7.5), we require that, for some choice of C, bN aNlog λ ≤ log C + (− log x) cλ − N 158 MARK GROSS, PAUL HACKING, AND SEAN KEEL for all N. This holds because bN aNlog λ ≤ (− log x) cλ − N for N sufﬁciently large.) Lemma 7.9. —Let A ⊂ R be a ﬁnite set and λ ∈ R, λ> 1.For k ∈ N let S ⊂ R be the setofrealnumbers s ofthe form s = c λ where l ≤ kand c ∈ A,n ∈ Z for each i. Then S i i i ≥0 k i=1 is discrete for each k. Proof. — Proof by induction on k.Wehave S ={0}. Suppose S is discrete. We 0 k have S = λ (S + A). Since λ> 1 we deduce that S discrete. k+1 k k+1 n≥0 For Propositions 7.10 and 7.11 below, the assertions hold after possibly replacing X by a smaller neighbourhood of s(0) ∈ X (independent of I, Q and q). J J Proposition 7.10. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal cone σ of . Each term of the formal sum Lift (q) = Mono(γ ) is an analytic function on V Q σ,σ,I and the sum deﬁnes an analytic function on V . σ,σ,I Proof.—Recall that V is an inﬁnitesimal thickening of the reduced complex σ,σ,I analytic space V .Write σ,σ,J V := (X , X )||X | < R|X |,|X | < R|X | ⊂ (G ) × S. σ,σ 1 2 1 2 2 1 m X ,X 1 2 Then V is a reduced complex analytic space containing V as a locally closed sub- σ,σ σ,σ,I space. We show that the sum Lift (q) converges (uniformly on compact sets) to an ana- lytic function on a neighbourhood of V in V . σ,σ,I σ,σ ˜ ˜ ˜ Let Q ∈ B be a lift of Q and σ˜ the lift of σ containing Q. Let u , u be the primitive 0 1 2 generators of σ˜ (a basis of M) such that the orientation of u , u agrees with that of w ,w . 1 2 1 2 ϕ˜ (u ) σ˜ i Let X = z , i = 1, 2, be the associated coordinate functions on V ,sothat i σ,σ,I V ⊂ (X , X )||X | < R|X |,|X | < R|X | ⊂ (G ) × S . σ,σ,I 1 2 1 2 2 1 m X ,X I 1 2 α α ϕ˜ (m) 1 2 σ˜ For m ∈ M, writing m = α u + α u ,wehave z = X X . 1 1 2 2 1 2 As already noted, broken lines γ on B for q with endpoint Q lift uniquely to ˜ ˜ broken lines on B for T (q˜) with endpoint Q, for some N ∈ Z, and the attached ϕ˜ (m ) σ˜ γ monomials are identiﬁed. Write Mono(γ ) = a z and m = α u + α u . Clearly γ γ 1 1 2 2 α α 2 ±1 ±1 Mono(γ ) = a X X ∈ k[P][X , X ] is an analytic function on V . Also write m = γ σ,σ γ 1 2 1 2 μ w + μ w . By Lemma 7.12,(1), μ and μ are bounded below (using the symmetric 1 1 2 2 1 2 statement interchanging w and w if T (q˜)∈Q,w ). The points u and u are ad- 1 2 1 R 1 2 ≥0 jacent integral points on the boundary of the inﬁnite convex polytope with asymptotic MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 159 directions w ,w . It follows that w = β u + γ (u − u ) and w = β u + γ (u − u ), 1 2 1 1 2 1 1 2 2 2 1 2 2 1 for some β ,β ,γ ,γ > 0. Hence 1 2 1 2 μ μ α α 1 2 ϕ˜ (m ) 1 2 β γ β γ σ˜ γ 1 1 2 2 z = X X = |X | |X /X | |X | |X /X | . 2 1 2 1 2 1 1 2 Now |X /X | < R, |X /X | < RonV . Thus, as we have chosen δ in Lemma 7.6 so 1 2 2 1 σ,σ −γ /β −γ /β ϕ˜ (m ) 1 1 2 2 γ σ˜ that 0 <δ < min(R , R ),if μ ,μ are both positive, |z | is bounded for 1 2 |X |,|X | <δ. On the other hand, suppose μ , say, is negative. Then if |X | >δ > 0, we 1 2 1 2 have μ β μ β μ 1 1 1 1 1 β γ −μ γ −μ γ 1 1 1 1 1 1 |X | |X /X | < δ |X /X | < δ R . 2 1 2 2 1 Since β and γ are ﬁxed and μ is bounded below, the above quantity is bounded. 1 1 1 β γ μ 2 2 2 Similarly, if μ is negative, (|X | |X /X | ) is bounded provided |X | >δ > 0. Thus 2 1 2 1 1 ϕ˜ (m ) σ˜ γ in any event, |z | is bounded for 0 <δ < |X |,|X | <δ. (We will only obtain uniform 1 2 convergence of the series Lift (q) on compact subsets of V .) By Lemma 7.8,(3),if Q σ,σ p |N| |z | < < 1for all p ∈ Pwe have |a |= O( ). By Lemma 7.8, (2), the number of N k broken lines for T (q˜) is O(|N| ). Combining, we deduce that Lift (Q) = Mono(γ ) is convergent on the open analytic subset V of V deﬁned by |X |,|X | <δ and σ,σ 1 2 σ,σ |z | < 1for all p ∈ P, for some δ> 0 (independent of I and q). After replacing X by an analytic neighbourhood of the vertex s(0) ∈ X , we may assume that V ⊂ V . J σ,σ,I σ,σ Proposition 7.11. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal cone σ of and let ρ be an edge of σ . Consider the formal sum Lift (q) = Mono(γ ).For Q sufﬁciently close to ρ each term of the sum is an analytic function on V and the sum deﬁnes an analytic ρ,I function on V . ρ,I Proof.—Write ρ = ρ , without loss of generality assume σ = σ ,sothat σ ∈ i i,i+1 i−1,i is the other maximal cone containing ρ .Let Q, ρ˜ , σ˜ , σ˜ be compatible lifts to i i i,i+1 i−1,i B .Let u , u , u be the primitive generators of ρ˜ and the remaining edges of σ˜ 0 i i−1 i+1 i i−1,i and σ˜ , and write X , X , X for the corresponding coordinates on V .So i,i+1 i i−1 i+1 ρ,I −D [D ] V ⊂ (X , X , X )||X | < R|X |, |X | < R|X | ⊂ V X X − z X f ρ,I i−1 i i+1 i−1 i i+1 i i−1 i+1 ρ ⊂ A × (G ) × S . m X X ,X i I i−1 i+1 Deﬁne −D [D ] V = (X , X , X )||X | < R|X |, |X | < R|X | ⊂ V X X − z X f ρ i−1 i i+1 i−1 i i+1 i i−1 i+1 ρ ⊂ A × (G ) × S. m X X ,X i i−1 i+1 We assume that the orientation of u , u is thesameasthatof w ,w . i−1 i+1 1 2 We ﬁrst consider broken lines γ lying in the cone generated by u and w .Write i 2 ϕ˜ (m ) σ˜ γ i,i+1 Mono(γ ) = a z ,and m = αu + α u = μ w + μ w . By Lemma 7.12,(1), γ γ i + i+1 1 1 2 2 160 MARK GROSS, PAUL HACKING, AND SEAN KEEL |μ | is bounded, and μ > 0 for all but ﬁnitely many γ . By Lemma 7.12,(2), α ≥ 0, so 1 2 + + ±1 Mono(γ ) = a X X ∈ k[P][X , X , X ] is analytic on V for each γ . In particular γ i−1 i+1 ρ i i+1 i we may assume in what follows (discarding ﬁnitely many terms Mono(γ ))that μ > 0. Writing w =−β u + γ u and w = β u + γ (u − u ),wehave β ,β ,γ ,γ > 0 1 1 i+1 1 i 2 2 i 2 i+1 i 1 2 1 2 and μ μ 1 2 ϕ˜ (m ) −β γ β γ γ 1 1 2 2 σ˜ z = |X | |X | |X | |X /X | . i+1 i i i+1 i −γ /β 2 2 Recall that |X /X |,|X /X | < RonV . Note then that for 0 <δ < R ,the i+1 i i−1 i ρ second factor on the right is bounded for |X | <δ as we are taking μ > 0. If μ < 0, i 2 1 then we have −β γ −β μ (−β +γ )μ 1 1 1 1 1 1 1 |X | |X | < R |X | i+1 i i which is bounded for δ < |X | <δ for any small δ > 0. Finally, if μ > 0, we use the i 1 equation for V , which gives −1 D −1 −[D ] |X | =|X |·|X | ·|f | · z . i+1 i−1 i ρ The function f on V restricts to the constant function 1 over S . Hence we may impose ρ ρ J the condition |f | >δ for any small δ > 0. Note that 2 β μ 2 −β μ 1 1 1 1 −β μ D −1 −[D ] β μ β μ (1+D ) −β μ [D ] 1 1 ρ ρ 1 1 1 1 ρ 1 1 ρ |X | = |X ||X | |f | z < R |X | |f | z . i+1 i−1 i ρ i ρ Thus we see that in any event, if δ < |X | <δ, |f | >δ and |z | < < 1for all p ∈ P, i ρ then ϕ˜ (m ) [D ] σ˜ γ ρ z · z is bounded, where c = β · sup({μ }, 0) is a constant. Now by Lemma 7.8, (3), again using 1 1 |z | < < 1for all p ∈ P, then ϕ˜ (m ) |N| σ˜ γ Mono(γ ) = a z = O . N k Recall that the number of broken lines for T (q˜) is O(|N| ) (Lemma 7.8, (2)). We deduce that the sum Mono(γ ) over broken lines γ lying in u ,w is uniformly convergent i 2 R ≥0 on compact sets for |X | <δ, f = 0, and |z | < 1for all p ∈ P, where δ> 0 is independent i ρ of I and q. Symmetrically, if we choose a basepoint Q ∈ σ sufﬁciently close to ρ,wecan i−1,i use the same argument for broken lines lying in u ,w ending at Q . Such a broken i 1 R ≥0 line will have Mono(γ ) = a X X with α ≥ 0. Thesameargumentasabove shows γ − i−1 i that the sum Mono(γ ) over all such broken lines γ is uniformly convergent on com- pact sets for |X | <δ, f = 0, and |z | < 1for all p ∈ P, where δ> 0 is again independent i ρ of I and q. However, the statement we are trying to prove involves the lift at Q, not Q .For MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 161 this purpose, we will omit terms coming from broken lines with α = 0from the above sum, as such terms will also arise in the above analysis at Q. To deal with this issue, note that given such a broken line γ ending at Q with α > 0, we get a ﬁnite number of broken lines γ ,1 ≤ j ≤ s, ending at a point Q ∈˜ σ − j i,i+1 sufﬁciently close to ρ˜ as follows. Extend γ until it reaches ρ˜ (which can be done since i i α > 0). Then applying the automorphism associated to crossing ρ˜ to Mono(γ ) gives − i α s α j − − α α a X X f , which we write as a X X .The j th monomial in the sum gives a γ γ i−1 i ρ j=1 j i−1 i new broken line γ bending at ρ˜ (unless α = α, in which case we just extend the ﬁnal line j i j α j ϕ˜ (m ) − γ σ˜ i,i+1 j segment of γ ), with new attached monomial a X X = a z .Aslongas Q lies γ γ j i−1 i j in the cone generated by ρ and −m , by extending or shortening the last line segment i γ of γ and applying a homothety, one can obtain a broken line γ ending at Q .Thusfor j j ˜ ˜ Q sufﬁciently close to ρ˜ , we obtain broken lines γ ,...,γ ending at Q .However,the i 1 s ˜ ˜ choice of Q may depend on γ . To see there is a choice of Q which works for all γ , note that for all but a ﬁnite number of γ , m lies in the half-space R · w + R · w γ ≥0 1 2 by Lemma 7.12, (1). Further, since γ bends at ρ˜ , m ∈ Z · u + Z · u .Thus m = j i γ i <0 i+1 γ β u + β u for β> 0, β < 0, and m = (β − l)u + β u for some l > 0 with a bound i + i+1 + γ i + i+1 only depending on f mod I. From this it follows that there cannot be a sequence of γ and γ constructed from γ as above so that the cones generated by u and −m get smaller j i γ and smaller. ˜ ˜ Thus we see that taking Q sufﬁciently close to ρ˜ , the broken lines ending at Q contained in w , Q with α > 0 give in the above fashion broken lines ending at 1 R − ≥0 ˜ ˜ ˜ Q contained in w , Q , and every such broken line ending at Q clearly arises in this 1 R ≥0 way. Furthermore, there are no broken lines contained in w , Q with α = 0, and 1 R − ≥0 thus all broken lines ending at Q have been accounted for. Now consider again a broken line γ contained in w , Q with α > 0. By 1 R − ≥0 construction, Mono(γ ) = Mono(γ )f . We wish to understand the contribution of j=1 ρ Mono(γ ) to Lift (q), and to do so, we write Mono(γ ) in terms of X , X using j Q j i i+1 j=1 −D [D ] the relation X X = z X in k[P ] (see Proposition 2.5). So we can write i−1 i+1 ϕ˜ i ρ˜ −α D +α −α − α [D ] − ρ α − ρ − Mono(γ ) = a z X X f . j γ i+1 i ρ This deﬁnes a (possibly rational) function on V . On the other hand, Mono(γ ) = − α a X X deﬁnes a holomorphic function on V , and using the equation γ ρ i−1 i −D [D ] (7.6)X X = z X f i−1 i+1 ρ which is satisﬁed on V ,wehave −α D +α −α − α [D ] − ρ α − ρ − Mono(γ ) = a z X X f i+1 i ρ 162 MARK GROSS, PAUL HACKING, AND SEAN KEEL as a function on V . Thus we see that Mono(γ ) and Mono(γ ) coincide as func- ρ j tions on V , in the above interpretation. (Essentially we are just using the fact that the relation (7.6) encodes the automorphism associated to crossing ρ ). Thus the fact that Mono(γ ) deﬁnes an analytic function on V for broken lines γ ending at Q con- ρ,I tained in u ,w implies that the sum Mono(γ ) over all broken lines γ ending at i 1 R ≥0 γ ˜ ˜ Q and contained in Q,w is analytic. This implies Lift (q) is analytic on V . 1 R Q ρ,I ≥0 ˜ ˜ ˜ Lemma 7.12. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal ˜ ˜ ˜ cone σ˜ ∈ . Consider broken lines γ on B for T (q˜) with endpoint Q, for all N ∈ Z such that N ϕ˜ (m ) ˜ σ˜ T (q˜)∈Q,w . Write Mono(γ ) = a z ,and m = μ w + μ w . 2 R γ γ 1 1 2 2 ≥0 (1) |μ | is bounded, and μ is positive for all but ﬁnitely many γ . In particular, μ ,μ are 1 2 1 2 bounded below. (2) Let u , u be generators of σ˜ with the same orientation as w ,w .Thenfor Q sufﬁciently 1 2 1 2 close to ρ˜ := R · u ,m lies in the half space R · u + R · u for each γ . ≥0 1 γ 1 ≥0 2 Proof.—(1)Wemay assume thelift q˜ of q is chosen so that T (q˜) ∈Q,w 2 R ≥0 if and only if N ≥ 0. Note that the rays spanned by w ,w are irrational so μ ,μ = 0. 1 2 1 2 Suppose for a contradiction that there is an inﬁnite sequence of broken lines γ such that m = μ w + μ w with μ < 0. Each broken line has at most k bends and there γ 1 1 2 2 2 are a ﬁnite number of -orbits of possible changes α ∈ M of the derivative of γ at a bend; see Lemma 7.8 and its proof. So, passing to a subsequence, we may assume that s s 1 l the bends (in order of increasing t ∈ (−∞, 0])ofeach γ are of types T α ,..., T α 1 l for some ﬁxed α ,...,α ∈ M, l ≤ k,and N ≥ s ≥ s ≥ ··· ≥ s ≥ 0 (depending on γ ). 1 l 1 2 l Say N − s is bounded for i ≤ l and unbounded otherwise. Passing to a subsequence, we may assume that N − s is constant for i ≤ l .Let γ be the broken line obtained by truncating γ after the ﬁrst l bends, and moving by a homothety so that (extending its ﬁnal line segment) γ ends at Q. Then m = T (m) for some ﬁxed m ∈ M. Furthermore, m is obtained from m by adding a positive linear combination of w and w . But since γ γ 1 2 −1 μ < 0and w ,w are eigenvectors of T with eigenvalues λ ,λ, it follows that we must 2 1 2 have m = ν w + ν w with ν < 0. But then for sufﬁciently large N, T (m) does not lie 1 1 2 2 2 in the half-space R · Q + R · w . This is a contradiction because m always lies in this ≥0 2 γ half-space. N s To see that |μ | is bounded, recall that m = T (q˜) − T α where 0 ≤ l ≤ k, 1 γ i i=1 −1 0 ≤ s ≤ N, and the α are selected from a ﬁnite set. Now since Tw = λ w it follows i i 1 1 that |μ | is bounded. (2) Let u be the connected component of B \ Supp (D) contained in σ˜ and ˜ ˜ ˜ containing ρ˜ in its closure. Let Q ∈ u be a point such that Q ∈ρ,˜ Q .Thenif ≥0 N N ˜ ˜ T (q˜)∈Q,w and γ is a broken line for T (q˜) with endpoint Q , we obtain a bro- 2 R ≥0 ken line γ for T (q˜) with endpoint Qand m = m as follows. First apply a homothety γ γ ˜ ˜ to obtain a broken line passing through Q, then truncate at Q. This gives an injective MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 163 ˜ ˜ map between the set of broken lines for T (q˜)∈Q,w ending at Q with the set of 2 R ≥0 such broken lines ending at Q. Now suppose γ is a broken line for T (q˜) with endpoint Qand m not lying in the half-space R · u + R · u . Since m lies in the half-space R · Q + R · w we 1 ≥0 2 γ ≥0 2 ﬁnd m lies in the cone generated by −Q,−u . In particular, m does not lie in the half- γ 1 γ space R · w + R w , so by (1) there are only ﬁnitely many such γ .Now by theabove 1 ≥0 2 construction it follows that for Q sufﬁciently close to ρ˜ there are none. 7.4. Smoothness. — We will now complete the proof of: Theorem 7.13 (Looijenga’s conjecture). — Suppose that k = C and the intersection matrix (D · D ) is negative deﬁnite, so that D ⊂ Y can be contracted to a cusp singularity q ∈ Y . Then i j 1≤i,j≤n the dual cusp to q ∈ Y is smoothable. We continue to work with the setup at the beginning of Section 7.2,with L, σ ,P, m and J as given there. By Theorem 7.7, if I is a monomial ideal with I = J, we obtain a deformation X → S of X → S . For the remainder of the section, we shall write S I J I I J for S .Let f : X → S denote the formal deformation determined by the deformations J J J X N+1 → S N+1 for N ≥ 0. Thus, S is the formal complex analytic space obtained as the J J J completion of S along S , X is a formal complex analytic space, and X → S is an adic J J J J ﬂat morphism. We similarly have the family X → S of formal schemes already studied m m in Section 4. We refer to [G60]and [B78] for background on formal schemes and formal complex analytic spaces. We have a section s: S → X such that, for t ∈ S general, the point s(t) ∈ X J J J J,t o o o on the ﬁbre is the cusp. We write X := X \ s(S ) ⊂ X and X ⊂ X , X ⊂ X for the J J J I J J I J induced open embeddings. Let Z := Sing(f ) ⊂ X denote the singular locus of f : X → S , see Deﬁnition- I I I I I I Lemma 4.1.Thus Z ⊂ X is a closed embedding of schemes or complex analytic spaces. I I n n Since the singular locus is compatible with base-change, the singular loci Z ⊂ X deter- J J mine a closed embedding Z ⊂ X which we refer to as the singular locus of f : X → S . J J J J J Lemma 7.14. — In the above situation, there exists 0 = g ∈ k[P] such that Supp(g · O ) is contained in s(S ). In particular, f (g · O ) is a coherent sheaf on S . J J∗ Z J Proof. — The proof is essentially the same as that of Lemma 4.5.Let U be deﬁned i,J as in (4.1). Then X is a union of open subspaces V , i = 1,..., n,suchthat V is an i,J i,J analytic open subspace of U for each i.Wethentake g = a ··· a as in the proof of i,J 1 n Lemma 4.5, so that Supp(g · O ) is contained in s(S ). So the support of g · O is a Z J Z J J closed subset of s(S ), hence proper over S . It follows that f (g · O ) is coherent by J J J∗ Z [B78], 3.1. 164 MARK GROSS, PAUL HACKING, AND SEAN KEEL Let u(J) denote the natural map u(J): O → f (O ), S J∗ Z J J and u(m) similarly the natural map u(m): O → f (O ). S m∗ Z m m Lemma 7.15. —u(J) is injective if and only if u(m) is injective. Proof.—Let 0 = g ∈ k[P] be the element given by Lemma 7.14.Let K be the kernel of u(J) and K the kernel of g · u(J).Thus K , K are ideal sheaves in O and J S J J J g · K ⊂ K ⊂ K . The local rings of S are domains by Lemma 7.16,so K = 0if and J J J J J only if K = 0. The sheaf K is coherent because the image of g · u(J) is contained in the J J coherent subsheaf f (g · O ) ⊂ f O . J∗ Z J∗ Z J J We claim that the natural map K ⊗ O → K O S J S m m is an isomorphism. Let z ∈ S be the unique point, coinciding with the zero-dimensional torus orbit of S, and let O denote the completion of O at its maximal ideal. Note that S,z S,z O coincides with O and the completion of O at its maximal ideal. It sufﬁces to S,z S ,z S ,z m J show that the map K ⊗ O → K O S,z J,z S ,z m,z is an isomorphism. We have an exact sequence of coherent sheaves 0 → K → O → f (g · O ) S J∗ Z J J J and so an exact sequence of O -modules S,z ˆ ˆ ˆ 0 → K ⊗ O → O → f (g · O ) ⊗ O . S,z S,z J∗ Z z S,z J,z J Now ˆ ˆ ˆ f (g · O ) ⊗ O = (g · O ) ⊗ O = (g · O ) = g · O J∗ Z z S,z Z s(z) S,z Z Z ,s(z) J J J J s(z) where the hats denote completion with respect to the maximal ideal of O .Thus K ⊗ S ,z J J,z O is the kernel of the map S,z ˆ ˆ O → g · O . S,z Z ,s(z) By the base-change property for the singular locus, this map coincides with the corre- sponding map for m. This proves the claim. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 165 The support of the ideal sheaf K is either empty or S (because the local rings of S are domains and S is connected). So K = 0 if and only if K = 0by the claim. J J J m Lemma 7.16. — The local rings of S are integral domains. Proof. — The completion of the local ring of S at a point z ∈ S is identiﬁed J J with the completion of the local ring of the toric variety S at z. By Serre’s criterion for normality, the completion of a normal Noetherian ring at a maximal ideal is a local normal Noetherian ring; in particular, it is a domain. Since O is a local Noetherian S ,z ring, it is contained in its completion. We deduce that O is a domain. S ,z Proof of Theorem 7.13.—Let f : Y → Y be the contraction of D ⊂ Y. We may assume f is the minimal resolution of Y . We may further assume n ≥ 3. Indeed, the embedding dimension of the dual cusp equals max(n, 3) by [N80], Corollary 7.8, p. 232 and [KM98], Theorem 4.57, p. 143, see also [L81], p. 307. So in particular for n ≤ 3the dual cusp is a hypersurface and thus smoothable. Let L be a nef divisor on Y such that NE(Y) ∩ L =D ,..., D .Let σ ⊂ A (Y, R) be a strictly rational polyhedral R 1 n R P 1 ≥0 ≥0 cone containing NE(Y) such that σ ∩ L is a face of σ .Let P = σ ∩ A (Y, Z) and P P P 1 J = P \ P ∩ L . By Lemma 7.15 and Theorem 4.6, u(J) is not injective. Let x ∈ S be a point lying in the interior of the toric variety S and h ∈ O a J J S ,x nonzero element of the kernel of u(J) near x. By Lemma 7.17 thereisamorphism N+1 v: Spec C[t]/ t → S ∗ N+1 taking the unique point of the domain to x and 0 = v (h) ∈ C[t]/(t ).Let Y/ Spec(C[t] N+1 /(t )) be the pullback of X /S by v and Z ⊂ Y its singular locus. Then Y/ Spec(C[t]/ J J N+1 (t )) is a deformation of the dual cusp singularity. Furthermore, O is annihi- ∗ ∗ N lated by v (h), and the ideal generated by v (h) must contain t ,so O is annihi- lated by t .By[A76], Theorem 5.1, there is an algebraic ﬁnite type deformation N+1 Y / Spec C[[t]] whose restriction to Spec(C[t]/(t )) is locally analytically isomorphic to N+1 Y/ Spec(C[t]/(t )).Let Z ⊂ Y denote the singular locus of Y / Spec C[[t]].Then O is a ﬁnite C[[t]]-module because the ﬁbre Y has an isolated singularity (using [Ma89], N+1 N N N+1 Theorem8.4,p.58).Now O = O /t O and t O = 0, so t O = t O and Z Z Z Z Z Z thus t O = 0 by Nakayama’s lemma. Hence the general ﬁbre of Y / Spec C[[t]] is smooth, and Y / Spec C[[t]] is a smoothing of the dual cusp. Lemma 7.17. —Let A be the completion of a ﬁnitely generated normal Cohen-Macaulay C- algebra at a maximal ideal. Let 0 = a ∈ A. Then there exists N ≥ 0 and a C-algebra map f : A → N+1 C[t]/(t ) such that f (a) = 0. 166 MARK GROSS, PAUL HACKING, AND SEAN KEEL Proof.—Extend a to a regular sequence a, t ,..., t of length dim A. Then the 1 r normalization of A/(t ,..., t ) is a ﬁnite direct sum of copies of C[[t]]. Now the result is 1 r clear. Acknowledgements An initial (and ongoing) motivation for the project was to ﬁnd a geometric com- pactiﬁcation of moduli of polarized K3 surfaces. We received a good deal of initial inspi- ration in this direction from conversations with V. Alexeev. The project also owes a great deal to the ﬁrst author’s collaboration with B. Siebert. We learned a great many things from A. Neitzke, especially about the connections of our work with cluster algebras and moduli of local systems. Our thinking about Looijenga pairs was heavily inﬂuenced by conversations with R. Friedman and E. Looijenga. Many other people have helped us with the project, and discussions with D. Allcock, D. Ben-Zvi, V. Fock, D. Freed, A. Gon- charov, R. Heitmann, D. Huybrechts, M. Kontsevich, A. Oblomkov, T. Perutz, M. Reid, A. Ritter, and Y. Soibelman were particularly helpful. We would also like to thank IHÉS for hospitality during the summer of 2009 when part of this research was done. The ﬁrst author was partially supported by NSF grants DMS-0805328 and DMS-0854987. The second author was partially supported by NSF grant DMS-0968824 and DMS-1201439. The third author was partially supported by NSF grant DMS-0854747. REFERENCES [AC11] D. ABRAMOVICH and Q. CHEN, Stable logarithmic maps to Deligne-Faltings pairs II, arXiv:1102.4531. [A02] V. ALEXEEV, Complete moduli in the presence of semiabelian group action, Ann. Math. (2), 155 (2002), 611– [A76] M. ARTIN, Lectures on Deformations of Singularities, Lectures on Mathematics and Physics, vol. 54, Tata Inst. Fund. Res, Bombey, 1976. [A07] D. AUROUX, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. 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Ann., 252 (1980), 221–235. [Ob04] A. OBLOMKOV, Double afﬁne Hecke algebras of rank 1 and afﬁne cubic surfaces, Int. Math. Res. Not., 2004 (2004), 877–912. [R83] M. REID, Decomposition of toric morphisms, in Arithmetic and Geometry, Vol. II, Progr. Math., vol. 36, pp. 395– 418, Birkhäuser, Basel, 1983. [SYZ96] A. STROMINGER,S.-T. YAU,and E. ZASLOW, Mirror symmetry is T-duality, Nucl. Phys. B, 479 (1996), 243–259. [Ty99] A. TYURIN, Geometric quantization and mirror symmetry, arXiv:math/9902027. [W76] J. WAHL, Equisingular deformations of normal surface singularities I, Ann. Math. (2), 104 (1976), 325–356. 168 MARK GROSS, PAUL HACKING, AND SEAN KEEL M. G. DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK mgross@dpmms.cam.ac.uk P. H. Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts, Amherst, MA 01003-9305, USA hacking@math.umass.edu S. K. Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA keel@math.utexas.edu Manuscrit reçu le 12 avril 2013 Manuscrit accepté le 10 janvier 2015 publié en ligne le 25 mars 2015. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/mirror-symmetry-for-log-calabi-yau-surfaces-i-yJXwvJRmBr

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Publisher: Springer Journals
Copyright: Copyright © 2015 by IHES and Springer-Verlag Berlin Heidelberg
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-015-0073-1
Publisher site: See Article on Publisher Site

Abstract

by MARK GROSS, PAUL HACKING, and SEAN KEEL ABSTRACT We give a canonical synthetic construction of the mirror family to pairs (Y, D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule deﬁned in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y, D). CONTENTS Introduction ........................................................ 65 0.1. The main theorems ................................................ 65 0.2. The symplectic heuristic ............................................. 68 0.3. Outline of the proof ................................................ 70 0.4. Further directions ................................................. 73 1. Basics .......................................................... 75 1.1. Looijenga pairs . ................................................. 75 1.2. Tropical Looijenga pairs . ............................................ 77 1.3. The Mumford degeneration and Givental’s construction ............................ 81 2. Modiﬁed Mumford degenerations ........................................... 84 2.1. The uncorrected degeneration .......................................... 85 2.2. Scattering diagrams on B . ............................................ 90 2.3. Broken lines .................................................... 93 2.4. The algebra structure ............................................... 101 3. The canonical scattering diagram ........................................... 102 3.1. Deﬁnition ..................................................... 102 3.2. Consistency: overview of the proof ........................................ 107 3.3. Consistency: reduction to the Gross-Siebert locus ................................ 109 3.4. Step V: the proof of Theorem 3.25 and the connection with [GPS09] .................... 124 4. Smoothness: around the Gross-Siebert locus ..................................... 127 5. The relative torus .................................................... 133 6. Extending the family over boundary strata ...................................... 134 6.1. Theorem 0.2 in the case that (Y, D) has a toric model ............................. 135 6.2. Proof of Theorems 0.1 and 0.2 in general .................................... 139 6.3. The case that (Y, D) is positive .......................................... 141 7. Looijenga’s conjecture ................................................. 145 7.1. Duality of cusp singularities ............................................ 145 7.2. Cusp family .................................................... 147 7.3. Thickening of the cusp family .......................................... 152 7.4. Smoothness .................................................... 163 Acknowledgements ..................................................... 166 References ......................................................... 166 Introduction 0.1. The main theorems. — Throughout the paper (Y, D) with D = D +···+ D 1 n will denote a smooth rational projective surface over an algebraically closed ﬁeld k of characteristic zero, with D∈|−K | a singular nodal curve. The divisor D is necessarily either an irreducible rational nodal curve, or a cycle of n ≥ 2 smooth rational curves. We DOI 10.1007/s10240-015-0073-1 66 MARK GROSS, PAUL HACKING, AND SEAN KEEL call (Y, D) a Looijenga pair for, as far as we know, their rich geometry was ﬁrst investigated in [L81]. We cyclically order the components of D and take indices modulo n.Byas- sumption there is a holomorphic symplectic 2-form , unique up to scaling, on Y \ D, with simple poles along D, and thus U := Y \ D is a log Calabi-Yau surface. Our main result is a canonical synthetic construction of the mirror family to such a pair. The construction gives an embedded smoothing of the n-vertex V ⊂ A ,deﬁned as, for n ≥ 3, the n-cycle of coordinate planes in A : 2 2 2 n V := A ∪ A ∪···∪ A ⊂ A . x ,x x ,x x ,x x ,...,x 1 2 2 3 n 1 1 n (See (1.7)and (1.8) for the deﬁnition of V and V .) This family is in general parameter- 1 2 ized roughly by the formal completion of the afﬁne toric variety Spec k[NE(Y)] along the union of toric boundary strata corresponding to contractions f : Y → Y. Here NE(Y) de- notes the monoid NE(Y) ∩ A (Y, Z) where NE(Y) ⊂ A (Y, R) is the cone generated R 1 R 1 by effective curve classes. This is just an approximate statement of our result, as NE(Y) is not in general ﬁnitely generated. More precisely, ﬁx (Y, D),D = D + ··· + D as above. Let B (Z) be the set 1 n 0 of pairs (E, n) where E is a prime divisor on some blowup of Y along which has a pole and n is a positive integer. Set B(Z) := B (Z)∪{0}. Later we will describe this set as the set of integer points in a natural integral afﬁne manifold, the dual intersection complex, or tropicalization, of the pair (Y, D).Let v ∈ B(Z) be the pair (D , 1). Choose i i σ ⊂ A (Y, R) a strictly convex rational polyhedral cone containing NE(Y) ,let P := P 1 R σ ∩ A (Y, Z) be the associated monoid, and set R := k[P] to be the associated k-algebra. P 1 For each monomial ideal I ⊂ R, consider the free R := R/I-module (0.1)A := R · ϑ . I I q q∈B(Z) D n Let m ⊂ R denote the maximal monomial ideal. Let T := G be the torus with charac- ter group χ(T ) having basis e indexed by the components D ⊂ D. There is a homo- D i D gp D morphism T → Spec k[P ] induced by C → (C · D )e ,so T acts on Spec R . i D I Theorem 0.1. —Let I ⊂ R be a monomial ideal with I = m. In Sections 2 and 3,we construct a ﬁnitely generated R -algebra structure on A , determined by relative Gromov-Witten invariants I I of (Y, D) counting rational curves meeting D in a single point. In Section 5,weconstruct a T action on Spec A . This induces a ﬂat T -equivariant map f : X := Spec A → Spec R I I I with closed ﬁbre V . By taking the limit over all such I, this yields a formal ﬂat family f : X → S := Spf R, m m where R is the completion of R with respect to the ideal m. The generic ﬁbre of f is smooth in the sense of Deﬁnition 4.2,so f is a formal smoothing of V . n MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 67 We use the notation ϑ for generators of our algebra, as the construction ﬁts into a more general family of constructions which includes, as a special case, theta functions on Abelian varieties. The history of such functions is as follows. Tyurin conjectured the existence of canonical theta functions (i.e., a basis of global sections) for polarized K3 surfaces, see [Ty99]. In 2007, discussions of the ﬁrst author with Abouzaid and Siebert involving a tropicalization of the Fukaya category gave a stronger hint as to the existence of theta functions on arbitrary degenerations of Calabi-Yau manifolds in the context of the Gross-Siebert program. In particular, these discussions led to what is now understood to be a variant of the notion of broken line. The latter notion was introduced in [G09]. These were initially used to construct canonical perturbations of the Landau-Ginzburg potential for P . Broken lines were then used for constructing mirror Landau-Ginzburg potentials for varieties with effective anti- canonical divisor in the setting of the Gross-Siebert program by Carl, Pumperla and Siebert in [CPS]. The authors show that the mirrors to such varieties as constructed in [GS07] carry a canonical Landau-Ginzburg potential obtained by using broken lines to lift monomial functions on the central ﬁbre of a toric degeneration to the toric degenera- tion. Simultaneously, we used these same lifts to allow an extension of the construction developed by Gross and Siebert to prove the above main theorem. The main innovations we have introduced here are that we use theta functions to provide partial compactiﬁ- cations of certain canonically constructed deformations, and that these canonically con- structed deformations, along with the theta functions, can be constructed relying only on the Gromov-Witten theory of (Y, D). The key point is that it is easy to build deformations of the punctured n-vertex V := V \{0}, but it is difﬁcult to extend these to deformations of V . This is effectively done by using theta functions to embed a suitably chosen defor- mation of V in afﬁne space, where the closure may then be taken. This extension would be impossible without the existence of theta functions. This result can be viewed as log analogs of Tyurin’s conjecture. In work in progress we apply similar ideas to obtain Tyurin’s conjecture in the K3 case as well, and construct canonical bases for cluster algebras, to cite two other generalizations. These are large topics and will be expanded on elsewhere. See also [GSTheta] for more motivation from mirror symmetry, and upcoming papers [GHKS]and [K3]. Continuing with (Y, D), P and R as above, our second main theorem is: Theorem 0.2. — There is a unique smallest radical monomial ideal J ⊂ R with the following properties: (1) For every monomial ideal I with J ⊂ I there is a ﬁnitely generated R -algebra structure on N N A compatible with the R -algebra structure on A of Theorem 0.1 for all N > 0. I I+m I+m (2) If the intersection matrix (D · D ) is not negative semi-deﬁnite then J = 0. In general, the i j zero locus V(J) ⊂ Spec R contains the union of the closed toric strata corresponding to faces F of σ such that there exists an i such that [D] ∈ F. P i 68 MARK GROSS, PAUL HACKING, AND SEAN KEEL (3) Let R denote the J-adic completion of R and S := Spf R the associated formal scheme. The algebras A determine a canonical T -equivariant formal ﬂat family of afﬁne surfaces f : X → S J J max(n,3) with ﬁbre V over 0.The ϑ determine a canonical embedding X ⊂ A × S . n q J J Remark 0.3. — When NE(Y) ⊂ P ⊂ P ⊂ A (Y),then J ⊂ J and the formal family X for P comes from the family for P by base-change. In this sense the family is indepen- dent of the choice of P. Remark 0.4. — Note that in the case that the intersection matrix (D · D ) is not i j negative semi-deﬁnite (which includes the case that D supports an ample divisor), The- orem 0.2 tells us that our construction gives a family over Spec R, so in particular the construction is algebraic. In this paper, we will not address the question as to in what sense our construction can be proved to be a mirror family. We expect, however, that our families constructed by the above theorems are mirror to U = Y\ D in the sense of homological mirror symmetry in the case k = C. Further justiﬁcation for our construction yielding the mirror family comes from the heuristic description of the construction in terms of symplectic geometry as discussed below. The third main result of this paper is an application of our general construction, following from a more detailed analysis of the case where the matrix (D · D ) is negative i j deﬁnite: Theorem 0.5 (Looijenga’s conjecture). — A 2-dimensional cusp singularity is smoothable if and only if the exceptional cycle of the dual cusp occurs as an anti-canonical cycle on a smooth projective rational surface. This was conjectured by Looijenga in [L81], where he also proved the forward implication. Partial results were obtained in [FM83]and [FP84]. 0.2. The symplectic heuristic. — Much of what we do in this paper, following the philosophy of the Gross–Siebert program, is to tropicalize the SYZ picture [SYZ96]. Thus it is helpful to review informally this picture in the context of mirrors to Looijenga pairs (Y, D). The SYZ picture will be a heuristic philosophical guide, and hence we make no effort to be rigorous. Here we follow the exposition from [A07] concerning SYZ on the complement of an anti-canonical divisor, itself a generalization of ideas of Cho and Oh for interpreting the Landau-Ginzburg mirror of a toric variety in terms of counting Maslov index two holomorphic disks [CO06]. For the most part we follow Auroux’s notation, except that we use Y instead of his X, and our X is his M. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 69 We ﬁx a Kähler form ω on Y, and a nowhere vanishing holomorphic 2-form on U := Y\ D. Now suppose we have a ﬁbration f : U → B by special Lagrangian 2-tori (i.e., a ﬁbre L of f satisﬁes Im | = ω| = 0). Then the SYZ mirror X of (U,ω) is the L L dual torus ﬁbration f : X → B. This can be thought of as a moduli space of pairs (L,∇) consisting of a special Lagrangian ﬁbre L of f equipped with a unitary connection ∇ modulo gauge equivalence, or equivalently a holonomy map hol : H (L, Z) → U(1) ⊂ ∇ 1 C . The complex structure on X is subtle, speciﬁed by so-called instanton corrections. In this picture we can deﬁne local holomorphic functions on X associated to a basis of H (Y, L, Z) (in a neighbourhood of a ﬁbre of f corresponding to a non-singular ﬁbre Lof f ) as follows. For A ∈ H (Y, L, Z) deﬁne A ∗ (0.2) z := exp −2π ω hol (∂ A) : X → C . By choosing a splitting of H (Y, L, Z) H (L, Z) we can pick out local coordinates on 2 1 X which deﬁne a complex structure. See [A07], Lemma 2.7. Note that as the ﬁbre L varies, the relative homology group H (Y, L, Z) forms a local system over B ⊂ B, where 2 0 B is the subset of points with non-singular ﬁbres. This local system has monodromy, and as a consequence, the functions z are only well-deﬁned locally. However, there are also well-deﬁned global functions ϑ ,...,ϑ on X. These are 1 n deﬁned locally in neighbourhoods of ﬁbres of f corresponding to ﬁbres of f not bounding holomorphic disks contained in U, via a (rough) expression (0.3) ϑ = n z , i β β∈H (Y,L,Z) where n is a count of so-called Maslov index two disks with boundary on L representing the class β and intersecting D transversally in one point lying in D . (We note that in our setting the Maslov index μ of a holomorphic disk f : → Y with boundary lying on a special Lagrangian torus L ⊂ Yis given by μ = 2deg f D. See [A07], Lemma 3.1.) In the case that D is ample, there are, for generic L, only ﬁnitely many such disks; it is not known how to treat the general case in this symplectic setting. For ϑ to make sense the moduli space of Maslov index 2 disks with boundary on L must deform smoothly with the Lagrangian L. This fails for Lagrangians that bound holomorphic disks contained in U (Maslov index zero disks). This is a real codimension one condition on L, and thus deﬁnes canonical walls in the afﬁne manifold B. When we cross the wall the ϑ are discontinuous. But the discontinuity is corrected by a holomor- phic change of variable in the local coordinates z , according to [A07], Proposition 3.9: [∂β]·[∂α] β β α (0.4) z → z · h z where here α ∈ H (Y, L , Z) represents the class of the Maslov index zero disk with 2 0 boundary on L a Lagrangian ﬁbre over a point on the wall, and h(q) is a generating func- tion counting such holomorphic disks. Thus we can deﬁne a new complex manifold, with 70 MARK GROSS, PAUL HACKING, AND SEAN KEEL the same local coordinates, by composing the obvious gluing induced by identiﬁcations of ﬁbres of the local system on B with ﬁbres H (Y, L, Z) with the automorphism (0.4). 0 2 These regluings are the instanton corrections, and the modiﬁed manifold X should be the mirror. By construction it comes with canonical global holomorphic functions ϑ . In particular, the sum W = ϑ is a well-deﬁned global function, the Landau–Ginzburg potential. 0.3. Outline of the proof. — We now outline how we realise the symplectic SYZ heuristic in terms of algebraic geometry. There are three principal issues to consider: • What information about a putative SYZ ﬁbration can be seen inside algebraic geometry? • What is the analogue of a Maslov index two disk in algebraic geometry? • How do we obtain the mirror by gluing together varieties? The philosophy for dealing with the ﬁrst and third issues was developed by Gross and Siebert in [GS07]. For the ﬁrst item, while we cannot build an SYZ ﬁbration f : U → B in general, we can roughly describe B as a combinatorial object. Given the Looijenga pair (Y, D), we build a space B homeomorphic to R along with a decomposition of B into cones. We construct (B, ) as the dual intersection complex of (Y, D).For each double point of D, we take a copy of the ﬁrst quadrant in R , with the axes labelled by the two irreducible components of D (assuming D is not irreducible) passing through the double point. We then identify edges of these cones if they are labelled with the same irreducible component of D. We thus get a topological space abstractly homeomorphic to R subdivided into cones. This is (B, ).InSection 1.2, we show how we can put an additional structure on B, namely the structure of an afﬁne manifold with singularities. Indeed, we can give B := B\{0} a system of coordinate charts whose transition maps are integral afﬁne linear transformations. The afﬁne structure does not extend across the origin unless (Y, D) is in fact a toric pair, in which case we recover the fan deﬁning Y. The manifold B can be viewed as the base of the SYZ ﬁbration “seen from a great distance.” In general the base of an SYZ ﬁbration has the structure of an afﬁne manifold with singularities. Singular ﬁbres of the ﬁbration occur over the singular points. One would expect f : U → B to have a number of singular ﬁbres in general, hence B will have a number of singular points. So the above construction moves all these singular points to the origin. Next, let us consider the third item. Fixing (Y, D) with D = D +··· + D ,let 1 n P ⊂ A (Y, Z) be a ﬁnitely generated monoid containing the classes of all effective curves on Y, obtained by choosing a strictly convex rational polyhedral cone σ ⊂ A (Y, R) P 1 containing the Mori cone. Let m be the maximal monomial ideal in the ring k[P],Ia monomial ideal with radical m,and let R = k[P]/I. We will describe the basic pieces we will glue together to describe a scheme over S := Spec R whose special ﬁbre is V := V \{0}. Assume that the components D are I I n i n MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 71 numbered in cyclic order, with indices taken modulo n.Wecan deﬁneanopencover of V by taking sets, for 1 ≤ i ≤ n, U = V(X X ) ⊂ A × (G ) . i i−1 i+1 m X X ,X i i−1 i+1 Note as subsets of V , they are disjoint except for U := U ∩ U = (G ) . i,i+1 i i+1 m X ,X i i+1 In V they are glued in the obvious way, i.e., via the canonical inclusions U ={X = 0}⊂ U , U ={X = 0}⊂ U . i,i+1 i+1 i i,i+1 i i+1 A deformation of V over S is obtained by gluing thickenings of the U I i −D [D ] i 2 (0.5)U := V X X − z X ⊂ S × A × (G ) i,I i−1 i+1 I m X i X ,X i i−1 i+1 [D ] where z ∈ k[P] is the corresponding monomial. The overlaps are relative tori, U = S × G , and the gluings are the obvious ones. The details are given in Sec- i,i+1,I I tion 2.1. This gluing gives a ﬂat family X → S , which can be viewed as being analogous to the naive complex structure on the mirror described as the moduli of smooth special Lagrangian ﬁbres with U(1) connection. There is no reason in general to believe that X → S can be extended to a ﬂat deformation X → S of V . The reason is that such an X should be an afﬁne scheme, I I n I and hence have many functions, while X as constructed tends to have few functions. The only case where X extends to give a deformation of V is when (Y, D) is a toric pair. In this case, we recover an inﬁnitesimal version of Givental’s mirror family, which then easily extends to Givental’s mirror construction. We review this case in Section 1.3. To rectify this problem, we need to translate the instanton corrections of the sym- plectic heuristic. We do this using the notion of scattering diagram, here a variant of similar notions introduced in [KS06]and [GS07]. For us, a scattering diagram D will be a collection of pairs (d, f ) where d is a ray emanating from the origin of B with rational slope, and f is a kind of function attached to the ray. Any scattering diagram will dictate how to modify both the deﬁnition of the open sets U and the gluings of U with U . The precise details of this modiﬁcation i,I i,I i+1,I are given in Section 2.2. Brieﬂy, the rays deﬁne automorphisms of the open sets U i,i+1,I analogous to (0.4), and are used to modify the gluing. While any scattering diagram can be used to obtain a modiﬁed ﬂat deformation X , we need to choose D correctly to have a chance of extending this deformation I,D to V . The symplectic heuristic can be used to motivate the choice of the canonical scattering diagram. The functions f chosen are generating functions for certain Gromov-Witten invariants, intuitively counting ﬁnite maps A → U. Heuristically, each holomorphic disk contributing can be approximated by a proper rational curve meeting D in a single point. 72 MARK GROSS, PAUL HACKING, AND SEAN KEEL Thus the canonical scattering diagram encodes the chamber structure seen in the symplectic heuristic. But there still remains the question of extending X to a ﬂat de- I,D formation of V . To do so, we need to construct enough functions on X . This is where I,D the concept of theta function comes in. The symplectic heuristic suggests that there should be a canonical choice of holomorphic functions on X arising from a count of Maslov I,D index two holomorphic disks. Rather than trying to ﬁnd an algebro-geometric analogue of a Maslov index two holomorphic disk, one instead deﬁnes the counts using tropical ge- ometry. In particular, we use the notion of broken line, introduced in [G09] and developed further by [CPS] simultaneously with this work, to provide the count. A broken line is es- sentially a tropical analogue of a Maslov index two disk. They are piecewise linear paths which only bend when they cross rays of the scattering diagram D, in ways prescribed by the functions attached to the rays. For any point p ∈ B with integral coordinates, we can use a count of broken lines to deﬁne a function on U for any i. This procedure is described in Section 2.3. Since i,I this procedure is dependent on the scattering diagram D, we can then ask whether these functions on the various U glue. We say D is consistent if they always glue. If these i,I functions do glue, then we call the resulting global function on X a theta function, writing I,D it as ϑ . The bulk of the argument in this paper occurs in Section 3, where we prove that the canonical scattering diagram described above is in fact consistent. This argument is rather involved, so we leave it to Section 3.2 to give an overview of the full argument for consistency. Crucial to the argument is a reduction to methods of [CPS] using the main results of [GPS09]. Once consistency is proved, this gives global functions ϑ on X for each p ∈ I,D B with integral coordinates. Let v denote the ﬁrst integral point along the ray of corresponding to the divisor D , and write ϑ := ϑ . Then we can use the functions i i v ϑ ,...,ϑ to embed (in the case that n ≥ 3) X in A × S . Taking the closure of the 1 n I I,D image gives the desired deformation X → S of V . I I n This construction essentially proves the ﬁrst main theorem, Theorem 0.1.The statement about the scheme-theoretic singular locus of f is dealt with in Section 4.There we again make a connection with the techniques of [GS07]. The crucial point is to show the singularity 0 ∈ V is formally smoothed, and for this, we need to work in a family where we have a local model for the behaviour near 0, much as Gross and Siebert have in [GS07]. More work is required for Theorem 0.2. We need to show that the construction above, which really only produces a family over the completion of Spec k[P] at the zero- dimensional torus orbit of this scheme, extends across completions along larger strata. Since the coordinate rings of the families constructed above are generated by theta func- tions, we proceed by studying the products of theta functions. In general, one expects the product of two theta functions to be a formal series of theta functions. However, in many cases one can control the terms sufﬁciently in these products to obtain the desired exten- MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 73 sions. This relies on a tropical interpretation of the product of theta functions, given in Section 2.4, as well as the existence of a torus action on our families, given in Section 5. This torus action only exists because of the canonical nature of our scattering diagrams. Complete details for the arguments are given in the last section, Section 6. Turning to Theorem 0.5, the main point is that Looijenga’s conjecture is really a form of mirror symmetry. We start with a pair (Y, D) such that the intersection matrix (D · D ) is negative deﬁnite. Thus D can be contracted analytically to give a cusp sin- i j gularity p ∈ Y . (By deﬁnition, a cusp singularity is a surface singularity whose minimal resolution is a cycle of rational curves.) For the sake of exposition, assume this contraction is algebraic, so that there is a divisor L on Y which is the pull-back of an ample divisor gp on Y . We choose the monoid P so L ∩ Pis a face P of P, with P generated by the bdy bdy classes [D ],...,[D ]. The main goal is to extend our construction to a formal neigh- 1 n bourhood of Spec k[P ]⊂ Spec k[P]. The problem is that Theorem 0.2 explicitly does bdy not apply in this case. The main difﬁculty is that the charts (0.5)overlap toomuchwhen [D ] all the z are invertible (in fact the ﬁbres over such points in Spec k[P ] coincide under bdy the natural gluing maps). There is no way to glue these charts compatibly. However, this can be done after shrinking these charts to analytic open subsets and working over an analytic open neighborhood of the zero-dimensional stratum of Spec k[P ].Herewe bdy work of course with k = C only. In doing so, we ﬁnd over a general point of Spec k[P ] the dual cusp singular- bdy ity to p ∈ Y . Thus we see that our mirror symmetry construction naturally produces the dual cusp. We then would like to extend the family constructed over thickenings of Spec k[P ]. We use the same techniques as those used to prove Theorem 0.1.However, bdy the construction of theta functions is considerably more delicate. In general, theta func- tions are described as a sum of monomials associated to broken lines. In the situation of Theorem 0.1, these sums are always ﬁnite. However, in the current situation, they are al- ways inﬁnite. Thus there are serious convergence issues, and this makes the proof rather technical. A delicate analysis of the combinatorics of broken lines is necessary to prove convergence. Once convergence is shown, we then argue that the formal family produced actu- ally gives a smoothing of the cusp singularity. This follows from the fact proved in Theo- rem 0.1 that we already have a smoothing of the n-vertex in a formal neighbourhood of the zero-dimensional stratum, but again the argument is slightly delicate. All details are giveninSection 7. 0.4. Further directions. — Here we will brieﬂy indicate the results of further study of our mirror construction, to be given in sequal paper [GHKII], as well as connections with other recent work. There are three broad classes of behaviour for our construction, depending on the properties of the intersection matrix (D · D ): the matrix can be negative deﬁnite, i j negative semi-deﬁnite but not negative deﬁnite, or not negative semi-deﬁnite. The ﬁrst 74 MARK GROSS, PAUL HACKING, AND SEAN KEEL case is analyzed here in detail to prove Theorem 0.5. We will discuss the third case in the sequel paper. We call the case that the intersection matrix is not negative semi-deﬁnite the positive case. It holds if and only if U is the minimal resolution of an afﬁne surface, see Lemma 6.9. In this case, the cone NE(Y) is rational polyhedral, so we may take P = NE(Y). Further- more, the ideal J of Theorem 0.2 equals 0. Thus our construction deﬁnes an algebraic family over Spec k[NE(Y)], with smooth generic ﬁbre. We will show in Part II that the restriction of this family to the structure torus X → T := Pic(Y) ⊗ G = Spec k A (Y) ⊂ Spec k NE(Y) Y m 1 is close to a universal family of deformations of U = Y \ D. More precisely, we will show independently of the positivity of the intersection matrix that our formal family has a simple and canonical (ﬁbrewise) compactiﬁcation to aformalfamily (Z, D) of Looijenga pairs (with X = Z \ D), equivariant for the action D D of T ⊂ T , the subtorus generated by the components of D. The theta functions are T eigenfunctions, see Section 5. In the positive case this extends naturally over all of Spec k[NE(Y)], and its restric- tion (Z, D) → T comes with a trivialization of the boundary D = D × T realizing it Y ∗ Y as the universal family of Looijenga pairs (Z, D ) deformation equivalent to (Y, D) to- gether with an isomorphism D → D constructed in [GHK12]. In particular, choosing Z ∗ such an isomorphism D → D for our original pair (Y, D) canonically identiﬁes it with a ﬁbre of the family (Z, D)/T . More importantly, the restrictions of the theta functions ϑ to U ⊂ X endow the afﬁne surface U = Y \ D with canonical functions. We give a modular interpretation of the quotient of Z \ D → T by T as the universal deforma- tion of U (this shows in particular the quotient depends only on U, e.g., is independent of the choice of compactiﬁcation U ⊂ Y), and give a unique geometric characterisation of the theta function basis of H (U, O ). The fact that (Y, D) appears as a ﬁbre is perhaps a bit surprising as, after all, we set out to construct the mirror and have obtained the original surface back. Note however that dual Lagrangian torus ﬁbrations in dimension 2 are topologically equivalent by Poincaré duality, so this is consistent with the SYZ formulation of mirror symmetry. To illustrate, in Example 6.12 we explicitly compute the theta functions in the case (Y, D) is the del Pezzo of degree 5 together with a cycle of 5 (−1)-curves. In Ex- ample 6.13, we give the expression in the case of a triangle of lines on a cubic surface, deferring in this case the proof until Part II. In each of these cases there is a characterisa- tion of the ϑ in terms of classical geometry. In a different direction, in [GHKK], along with M. Kontsevich, we extend many of the methods introduced in this paper to prove a number of signiﬁcant conjectures about cluster varieties. In particular, the technology of theta functions leads to a proof of positivity of the Laurent phenomenon, and a proof of the Fock-Goncharov dual basis MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 75 conjecture for a broad class of cluster varieties. The latter can be viewed as a generaliza- tion of the construction of theta functions on Y \ D in the positive case, described above. In fact, in the case of cluster varieties associated to a skew-symmetric matrix of rank two, the Fock-Goncharov X variety ﬁbres over a torus with ﬁbres being interiors of Looijenga pairs. This is described in detail in [GHK13]. In this case, the general construction of theta functions in [GHKK] coincides with the ones constructed here. Let us end with some mild speculation in all dimensions suggested by the above discussion. By a Looijenga pair we mean a dlt pair (Y, D) (e.g., a simple normal crossings pair) with K + D trivial and (Y, D) having a zero-dimensional log canonical center. (In the simple normal crossing case, this means there is an intersection point of dim Y different components of D.) By a log Calabi-Yau with maximal boundary we mean a variety U which can be realized as the interior Y \ D of a Looijenga pair. See Section 1 of [GHK13] for background on these notions. We expect that many of the results in this paper will extend to Looijenga pairs of all dimensions. This generalization will require the further development of the technology of logarithmic Gromow-Witten invariants, [GS11],[AC11]. We should obtain in complete generality a mirror family X → Spf k[P] for suitable monoids P. Furthermore, one would expect in the case that U = Y \ Dis afﬁne that this family extends to X → Spec k[P]. Using the two-dimensional case as a guide, the general ﬁbre of X → Spec k[P] should itself be the interior of a Looijenga ¯ ¯ pair (X, E),with X \ E afﬁne by construction. Thus we can then repeat the process to obtain a family X → Spec k[P ], and it would be expected, as taking mirror twice should return to where we started, that X → Spec k[P ] contains a ﬁbre isomorphic to U. The family X carries our canonically deﬁned theta functions, indexed by tropical points of the mirror. This leads us to propose: Conjecture 0.6. —Let U be an afﬁne log Calabi-Yau variety with maximal boundary. Then H (U, O ) has a canonical basis of theta functions indexed by tropical points of the mirror. The structure constants for multiplication of theta functions can be described combinatorially in terms of broken lines. Versions of this conjecture have been proven for cluster varieties in many cases in [GHKK]. 1. Basics 1.1. Looijenga pairs. Deﬁnition 1.1. —A Looijenga pair (Y, D) is a smooth rational projective surface Y together with a reduced nodal curve D∈|−K | with at least one singular point. Note that for a Looijenga pair, p (D) = 1 by adjunction. Since H (Y, O ) = 0 a Y by rationality of Y, D is connected. Applying adjunction to each irreducible component 76 MARK GROSS, PAUL HACKING, AND SEAN KEEL of D, one sees easily that D is either an irreducible genus one curve with a single node, or a cycle of smooth rational curves. We will always write D = D +···+ D , with a cyclic 1 n ordering of the irreducible components, and take the indices modulo n. We will need a few basic facts about Looijenga pairs, which we collect here. Deﬁnition 1.2. —Let (Y, D) be a Looijenga pair. ˜ ˜ (1) A toric blow-up of (Y, D) is a birational morphism π : Y → Y such that if D is the −1 ˜ ˜ ˜ reduced scheme structure on π (D), then (Y, D) is a Looijenga pair. In particular, Y is smooth. ¯ ¯ (2) A toric model of (Y, D) is a birational morphism (Y, D) → (Y, D) to a smooth toric ¯ ¯ ¯ surface Y with its toric boundary D such that D → D is an isomorphism. Note that if π : Y → Y is the blow-up of a node of D, then π is a toric blow-up. ˜ ˜ Proposition 1.3. —Given (Y, D) there exists a toric blowup (Y, D) which has a toric model ˜ ˜ ¯ ¯ (Y, D) → (Y, D). Proof. — First observe: (1) Let p : Y → Y be the blowdown of a (−1)-curve not contained in D, and D := p (D) ⊂ Y . If the proposition holds for (Y , D ) then it holds for (Y, D). (2) Let Y → Y be the blowup at a node of D, and D ⊂ Y the reduced inverse image of D. The proposition holds for (Y , D ) if and only if it holds for (Y, D). By using (1) and (2) repeatedly we may assume Y is minimal, and thus is either a ruled surface or is P . In the latter case, by blowing up a node of D we reduce to the ruled case. So we have q : Y → P a ruling. We next consider the number of components of D which are ﬁbres of q. There cannot be more than two such components, for otherwise D cannot be a cycle. If there are precisely two such components, then D necessarily has precisely four components, and it is then easy to check that D is the toric boundary of Y, for a suitable choice of toric structure on Y. In this case the proposition obviously holds. Otherwise let D ⊂ D be the union of components not contained in ﬁbres. If D has a node, then we can blowup the node, blowdown the strict transform of the ﬁbre through the node, increasing the number of components of D contained in ﬁbres. After carrying out this procedure for each node of D ,weare then in oneoftwo cases. Case I. D has two components contained in ﬁbres, and then we are done. Case II. D consists of a ﬁbre f and a non-singular irreducible two-section D of q. Note that since D + f ∼−K and Y is isomorphic to the Hirzebruch surface F for Y e some e, we can write Pic Y = ZC ⊕ Zf ,with C =−e and −K = 2C + (e + 2)f .Thus 0 Y 0 D ∼ 2C + (e + 1)f and C · D =−e + 1. Since C is not contained in D , e = 0or1. 0 0 0 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 77 If e = 0, then there is a second ruling q : Y → P ,with D and f sections of this ruling. In this case, we follow the same procedure as above of blowing up nodes for this new ruling, arriving in Case I. If e = 1, then C is disjoint from D . Blowing down C ,weobtain P , and can then 0 0 blowup one of the nodes of the image of D ∪ f . Using this new ruled surface, we can again blowup a node and ﬁnd ourselves back in Case I. 1.2. Tropical Looijenga pairs. — We explain how to tropicalize a Looijenga pair, ﬁrst recalling the following basic deﬁnition. Fix a lattice M Z . In what follows, we will always use the notation M = M⊗ R,N = Hom (M, Z) and N = N⊗ R.Wedenote R Z Z R Z by Aff(M) the group of afﬁne linear transformations of the lattice M. Recall the following deﬁnitions from [GS06]. Deﬁnition 1.4. —An integral afﬁne manifold B is a (real) manifold B with an atlas of −1 charts {ψ : U → M } such that ψ ◦ ψ ∈ Aff(M) for all i, j. i i R i An integral afﬁne manifold with singularities B is a (real) manifold B with an open subset B ⊂ B which carries the structure of an integral afﬁne manifold, and such that := B \ B ,the 0 0 singular locus of B, is a locally ﬁnite union of locally closed submanifolds of codimension at least two. If B is an integral afﬁne manifold with singularities, there is a local system on B consisting B 0 of ﬂat integral vector ﬁelds: if y ,..., y are local integral afﬁne coordinates, then is locally given 1 n B by linear combinations of the vector ﬁelds ∂/∂ y ,...,∂/∂ y .If B is clear from context, we drop the 1 n subscript B. Similarly, is the dual local system, locally generated by dy ,..., dy . B 1 n We will be primarily interested in dim B = 2 in this paper, in which case will consist, in all our examples, of a ﬁnite number of points. All integral afﬁne manifolds we encounter will in fact be linear, in the sense that the coordinate transformations are in fact linear rather than just afﬁne linear. We associate to a Looijenga pair (Y, D) apair (B, ), where B is homeomorphic to R and has the structure of integral afﬁne manifold with one singularity at the origin, and is a decomposition of B into cones. We call (B, ) the tropicalization of (Y, D),and the fan of (Y, D). The idea is that we pretend that (Y, D) is toric and we try to build the associated fan. More precisely, the construction is as follows. For each node p := D ∩ D of D we take a rank two lattice M with basis i,i+1 i i+1 i,i+1 v ,v ,and thecone σ ⊂ M ⊗ R generated by v and v .Wethenglue σ to i i+1 i,i+1 i,i+1 Z i i+1 i,i+1 σ along the rays ρ := R v to obtain a piecewise-linear manifold B homeomorphic i−1,i i ≥0 i to R and a decomposition ={σ | 1 ≤ i ≤ n}∪{ρ | 1 ≤ i ≤ n}∪{0}. i,i+1 i We deﬁne an integral afﬁne structure on B\{0} by deﬁning charts ψ : U → M (where i i R M = Z ). Here (1.1)U = Int(σ ∪ σ ) i i−1,i i,i+1 78 MARK GROSS, PAUL HACKING, AND SEAN KEEL and ψ is deﬁned on the closure of U by i i ψ (v ) = (1, 0), ψ (v ) = (0, 1), ψ (v ) = −1,−D , i i−1 i i i i+1 with ψ linear on σ and σ . The reason for choosing these particular vectors is i i−1,i i,i+1 that they form the one-dimensional rays of a fan deﬁning a toric variety such that the divisor D corresponding to the ray generated by v has self-intersection D . i i We note this construction makes sense even when n = 1, i.e., the anti-canonical divisor D is an irreducible nodal curve. In this case there is one cone σ , and opposite 1,1 sides of the cone are identiﬁed. (Moreover, the integral afﬁne charts are deﬁned using 2 2 the integer D − 2 instead of D . This is the degree of the normal bundle of the map 1 1 from the normalization of D to Y.) However, this case will often complicate arguments in this paper, so we will usually replace Y with a surface obtained by blowing up the node of D, and replace D with the reduced inverse image of D under the blowup. This does not change the underlying integral afﬁne manifold with singularities, but reﬁnes the decomposition , exactly as in the toric case: ˜ ˜ Deﬁnition 1.5. —Given (B, ),a reﬁnement is a pair (B, ),where is a decomposition of B into rational polyhedral cones reﬁning ,eachconeof integral afﬁne isomorphic to the ﬁrst quadrant in R . Lemma 1.6. — There is a one-to-one correspondence between toric blow-ups of (Y, D) and ˜ ˜ ˜ ˜ reﬁnements of (B, ).Furthermore,if (Y, D) is a non-singular toric blow-up of (Y, D),and (B, ) ˜ ˜ ˜ is the afﬁne manifold with singularities constructed from (Y, D), then B and B are isomorphic as integral afﬁne manifolds with singularities in such a way that is the corresponding reﬁnement of . Proof.—Let π : Y → Y be a toric blow-up. It follows from the condition that −1 −1 π (D) is an anti-canonical divisor that π : Y \ π (Sing(D)) → Y \ Sing(D) is an red isomorphism. Indeed, if this restriction of π has an exceptional divisor, it must have discrepancy a(E, Y, D) =−1. But by [KM98], Cor. 2.31, (3), the smallest discrepancy occurring is 0. Thus necessarily π is a blow-up along a subscheme supported on Sing(D).Let x ∈ Sing(D) be a double point of D, corresponding to a cone σ ∈ .Note σ can be viewed as a rational polyhedral cone deﬁning a non-singular toric variety X A .Then étale locally near x, the pair (Y, D) is isomorphic to the pair (X ,∂ X ).One canthen σ σ check that in this local model, the only possible blow-ups satisfying the deﬁnition of toric blow-ups come from subdivisions of the cone σ , i.e., toric blow-ups of X .Indeed,the exceptional divisors of toric blowups are the only divisors with discrepancy −1. This gives the desired correspondence. The second statement is then easily checked. Example 1.7. — It is easy to see that if Y is a non-singular toric surface and D = ∂ Y is the toric boundary of D, then in fact the afﬁne structure on B extends across the MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 79 origin, identifying (B, ) with (M , ),where is the fan for Y. Indeed, if ρ ∈ R Y Y j Y corresponds to the divisor D and ρ = R v with v ∈ M primitive, then it is a standard j j ≥0 j j fact that v + (D ) v + v = 0. i−1 i i i+1 Since Y is non-singular, there is always a linear identiﬁcation ϕ : M → Z taking v to i i−1 (1, 0), v to (0, 1),and thus v must map to (−1,−D ).Soon U ,achartfor theafﬁne i i+1 i −1 structure on B is ψ = ϕ ◦ ψ : U → M .The maps ψ glue to give an integral afﬁne i i R i i i isomorphism B → M . In fact, the converse is also true: Lemma 1.8. — If the afﬁne structure on B = B\{0} extends across the origin, then Y is toric and D = ∂ Y. Proof. — We ﬁrst note that by Lemma 1.6, we can replace (Y, D) with a non- singular toric blow-up without affecting the afﬁne manifold B. By Proposition 1.3,we ¯ ¯ ¯ can thus assume the existence of a toric model π : (Y, D) → (Y, D).If D is the image of 2 2 D under this map, then D ≥ D . i i ¯ ¯ We ﬁrst claim that (Y, D) is isomorphic to (Y, D) if and only if equality holds for every i. Indeed, if equality holds for a given i,then π can’t contract any curves which intersect D . On the other hand, π can’t contract any curves contained in Y \ D since then D would not be an anti-canonical cycle. Now assume that (Y, D) is not toric, so that π is not an isomorphism. Let (M , ) ¯ ¯ be the fan for the toric pair (Y, D),withrays ρ¯ ,..., ρ¯ corresponding to ρ ,...,ρ .In 1 n 1 n general, B \ ρ has a coordinate chart ψ : B \ ρ → M , constructed by gluing together 1 1 R coordinate charts for U ,..., U . This can be done so that σ is mapped to the cone of 2 n 1,2 generated by ρ¯ and ρ¯ . It is now enough to show the following: 1 2 Claim. — For suitable choice of ρ , in fact ψ is injective and ψ(B \ ρ ) is strictly contained in 1 1 M \¯ ρ . R 1 To show this, ﬁrst let us analyze the effects of one blow-up on these charts. Let (Y, D) → (Y , D ) be given by a blow-up of a single point p ∈ D for some i,where ¯ ¯ (Y , D ) is obtained from (Y, D) by a sequence of blow-ups with centers at smooth points of the boundary. Let (B , ) be the tropicalization of (Y , D ). Let us examine the dif- ference between the charts ψ : B \ ρ → M and ψ : B \ ρ → M deﬁned as above. 1 R R If i = 1, then B \ ρ and B \ ρ are afﬁne isomorphic and ψ , ψ agree. Otherwise, let σ = σ ⊂ B, with σ ⊂ B deﬁned similarly. Then σ and σ are afﬁne 1,i j−1,j 1,i j=2 1,i 1,i isomorphic and ψ| = ψ | . On the other hand, ψ| = T ◦ ψ | where σ \ρ σ \ρ B\σ i B \σ 1 1 1,i 1,i 1,i 1,i T : M → M is the shear T (m) = m +m, n v ,where v is a primitive generator of i R R i i i i ψ (ρ ) and n ∈ N is primitive, annihilates v , and is positive on ψ (σ ). i i i i,i+1 80 MARK GROSS, PAUL HACKING, AND SEAN KEEL FIG.1.— (B, ) for Example 1.9 2 2 Now note that D > D for at least one i, and by choosing ρ appropriately, we can i i assume that this is the case for some i = 1. Furthermore, we can also assume that if there 2 2 ¯ ¯ is a ρ¯ ∈ with ρ¯ =−ρ¯ , then there is an i with D > D with i = 1, j . Applying the j j 1 i i above description of the change of the coordinate charts under one blow-up repeatedly then shows the claim. Now if the afﬁne structure on B extended across the origin, then ψ would extend to an isomorphism ψ : B → M , which contradicts the claim. Example 1.9. — Let Y be a del Pezzo surface of degree 5. Thus Y is isomorphic to the blowup of P in 4 points in general position. The surface Y contains exactly 10 (−1)-curves. It is easy to ﬁnd an anti-canonical cycle D of length 5 among these 10 curves. In this case, consider B \ ρ .Eachchart ψ : U → M can be composed with an 1 i i R integral linear function on M in such a way that the charts ψ ,ψ ,ψ and ψ glue to R 2 3 4 5 give a chart ψ : B \ ρ → M . This can be done, for example, with 1 R ψ(v ) = (1, 0), ψ (v ) = (0, 1), ψ (v ) = (−1, 1), 1 2 3 ψ(v ) = (−1, 0), ψ (v ) = (0,−1). 4 5 We can then take a chart ψ : U ∪ U → M which agrees with ψ on σ , and hence 5 1 R 5,1 takes the values ψ (v ) = (0,−1), ψ (v ) = (1,−1), ψ (v ) = (1, 0), 5 1 2 see Figure 1. Thus B, as an afﬁne manifold, can be constructed by cutting M along the posi- tive real axis, and then identifying the two copies of the cone σ via an integral linear 1,2 transformation. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 81 Example 1.10. — Suppose given a Looijenga pair (Y, D) with D ≤−2for all i and D is negative deﬁnite (which is equivalent to D ≤−3for some i). Then we have an ¯ ¯ analytic contraction p : Y → Ywith Y having a single cusp singularity. This case will lead to our proof of Looijenga’s conjecture. We can describe (B, ) as follows. Let M = Z and take v ,v to be a basis for M, and deﬁne v for i ∈ Z by the relation 0 1 i (1.2) v + D v + v = 0. i−1 i i+1 i mod n We deﬁne an inﬁnite fan in M whose two-dimensional cones are the cones generated by v and v , i ∈ Z. It is easy to check that these cones do indeed form a fan and that i i+1 the support of the fan | | is a strictly convex cone. If we deﬁne T ∈ SL(M) by T(v ) = v 0 n and T(v ) = v ,then T(v ) = v for each i. Necessarily T takes | | to itself, so the 1 n+1 i i+n boundary rays of the closure of | | are real eigenspaces for T. Hence T is hyperbolic, i.e., Tr T > 2. We now obtain (B, ) by dividing out | | by the action of T. 1.3. The Mumford degeneration and Givental’s construction. — The toric case of Theo- rem 0.1 yields Givental’s construction for mirrors of toric varieties in the surface case, and can also be seen as a special case of a construction due to Mumford [Mum]. Mumford’s construction in general produces degenerations of arbitrary toric varieties; the construc- tion as we review it here only gives degenerations of the algebraic torus. This should be regarded as a warmup for our general construction. gp A toric monoid P is a (commutative) monoid whose Grothendieck group P is a gp gp ﬁnitely generated free Abelian group and P = P ∩ σ ,where σ ⊆ P ⊗ R is a convex P P Z rational polyhedral cone. Let M = Z be a lattice, for some arbitrary rank n.Fix afan in M = M ⊗ R, whose support, | |, is convex. In what follows, we view B=| | as an R Z afﬁne manifold with boundary. We denote by the set of maximal cones in . max We now generalize the usual notion of a convex piecewise linear function on a fan. If one is interested in R-valued convex functions, then one can take P = N, σ = R . P ≥0 Then the following deﬁnition yields the notion of a piecewise linear R-valued function with integral slopes, and convexity here means upper convexity, i.e., the function is the supremum of a collection of linear functions. gp Deﬁnition 1.11. —A -piecewise linear function ϕ :| |→ P is a continuous function gp gp such that for each σ ∈ , ϕ| is given by an element ϕ ∈ Hom (M, P ) = N ⊗ P . max σ σ Z Z For each codimension one cone ρ ∈ contained in two maximal cones σ ,σ ∈ ,wecan + − max write ϕ − ϕ = n ⊗ κ σ σ ρ ρ,ϕ + − gp where n ∈ N is the unique primitive element annihilating ρ and positive on σ ,and κ ∈ P .We ρ + ρ,ϕ call κ the bending parameter. Note (as the notation suggests) it depends only on the codimension one ρ,ϕ cone ρ (not on the ordering of σ ,σ ). + − 82 MARK GROSS, PAUL HACKING, AND SEAN KEEL gp We say a -piecewise linear function ϕ :| |→ P is P-convex (strictly P-convex) if × × for every codimension one cone ρ ∈ , κ ∈ P(κ ∈ P \ P ,where P is the group of invertible ρ,ϕ ρ,ϕ elements of P). Example 1.12. —Takeacomplete fan in M . This deﬁnes a toric variety Y = gp Y , which we assume is non-singular. We let P ⊂ P be given by the cone of effective curves, NE(Y) ⊂ A (Y, Z). Each codimension one cone ρ ∈ corresponds to a one-dimensional toric stratum D ⊂ ∂ Y, hence a class [D ]∈ NE(Y) = P. If ω ∈ (1), the set of rays of , we also write D ρ ω for the corresponding toric divisor. (1) Lemma 1.13. —Deﬁne s : T := Z → M to send the basis element t , ω ∈ (1) to the ﬁrst lattice point m on ω. Then A (Y, Z) β → (D · β)t 1 ω ω ω∈ (1) identiﬁes A (Y, Z) with Ker(s), giving rise to an exact sequence (1.3)0 → A (Y, Z) → T → M → 0. Then there is a unique -piecewise linear section ϕ˜ : M → T satisfying ϕ( ˜ m ) = t .Let π : T → ω ω gp A (Y, Z) be any splitting, and set ϕ := π ◦˜ ϕ. Then ϕ : M → A (Y, Z) = P is -piecewise 1 1 linear and strictly P-convex, with (1.4) κ =[D ] ρ,ϕ ρ for each codimension one cone ρ ∈ . Up to a linear function, ϕ is the unique -piecewise linear map with these bending parameters. Proof. — The exact sequence is standard. Since is a complete non-singular fan, it is clear that there exists such a unique ϕ˜ . To calculate the kink along a codimension one ρ ∈ , suppose ρ is generated by basis vectors e ,..., e and ρ is contained in 1 n−1 n−1 two maximal cones, generated by e ,..., e and e ,..., e , e := −e + a e .Let 1 n 1 n−1 n i i n i=1 t ,..., t , t be the generators of T mapping to e ,..., e , e respectively. Then the kink 1 n 1 n n n n−1 is κ = t + t − a t . On the other hand, if D ,..., D , D are the divisors corre- ρ,ϕ˜ n i i 1 n n i=1 n sponding to the rays generated by e ,..., e , e respectively, then D · D = D · D = 1 1 n n ρ ρ n n and using the rational function z ,D is linearly equivalent to −a D plus a sum of toric i i divisors disjoint from D .Thus D · D =−a and we see that κ is the image of [D ] ρ i ρ i ρ,ϕ˜ ρ under the inclusion A (Y, Z) → T .Thus κ = π(κ )=[D ] as required. 1 ρ,π◦˜ ϕ ρ,ϕ˜ ρ MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 83 gp Given a -piecewise linear and P-convex function ϕ :| |→ P we can deﬁne a gp monoid P ⊂ M × P by (1.5)P := m,ϕ(m) + p | m∈| |, p ∈ P . This is the set of integral points lying above the graph of ϕ, in the sense given by the gp partial order on P deﬁned by p ≥ p if p − p ∈ P. The convexity of ϕ is equivalent 1 2 1 2 to P being closed under addition. Furthermore, we have a natural inclusion P → P ϕ ϕ given by p → (0, p), which gives us a morphism f : Spec k[P ]→ Spec k[P]. This morphism is ﬂat as k[P ] is freely generated as a k[P]-module by all elements of (m,ϕ(m)) the form z , m ∈| |. It is easy to see that a general ﬁbre of f is isomorphic to the gp algebraic torus Spec k[M]: in fact, if we consider the big torus orbit U = Spec k[P ]⊂ −1 Spec k[P], f (U) = U × Spec k[M]. We now describe the ﬁbres over other toric strata of Spec k[P].Let x ∈ Spec k[P] be a point in the torus orbit corresponding to a face Q ⊂ P. Then by replacing P with the localized monoid P − Q obtained by inverting all elements of Q, we may assume that x is contained in the smallest toric stratum of Spec k[P]. Consider the composed map gp gp × ϕ¯ :| |−→P → P /P . Note ϕ¯ is also piecewise linear. Let be the fan (of convex but not necessarily strictly convex cones) whose maximal cones are the maximal domains of linearity of ϕ¯.Then −1 f (x) can be written as −1 f (x) = Spec k[ ]. Here, k[ ]= kz m∈M∩| | with multiplication given by m+m z if m, m lieinacommonconeof , m m (1.6) z · z = 0 otherwise. −1 In particular, the irreducible components of f (x) are the toric varieties Spec k[σ ∩ M] for σ ∈ . max In the particular case that rank M = 2and deﬁnes a non-singular complete sur- face with n toric divisors, suppose ϕ is strictly convex. If x is a point of the smallest toric 84 MARK GROSS, PAUL HACKING, AND SEAN KEEL −1 n stratum of Spec k[P],then f (x) is just V ⊂ A , the reduced cyclic union of coordi- nate A ’s: 2 2 2 n V = A ∪ A ∪···∪ A ⊂ A . x ,x x ,x x ,x x ,...,x 1 2 2 3 n 1 1 n We call V the vertex, or more speciﬁcally, the n-vertex. We will need in the sequel the degenerate case of the n-vertex for n = 2. This is a union of two afﬁne planes and can be described as the double cover 2 2 2 2 2 (1.7) V = Spec k[x , x , y]/ y − x x = A ∪ A . 2 1 2 1 2 x ,x x ,x 1 2 2 1 Of course, this does not appear as a central ﬁbre of a Mumford degeneration. Analo- gously, one can deﬁne 2 3 (1.8) V = Spec k[x, y, z]/ xyz − x − z , the afﬁne cone over a nodal curve embedded in weighted projective space WP (3, 1, 2). Example 1.14. — In Example 1.12, with the choice of ϕ given by Lemma 1.13,the family Spec k[P ]→ Spec k NE(Y) in fact gives the family of mirror manifolds to the toric variety Y, as constructed by Given- tal [Giv]. In fact, the mirror of a toric variety also includes the data of a Landau-Ginzburg potential, which is a regular function. If Y is Fano, the potential is (m ,ϕ(m )) ρ ρ W = z where we sum over all rays ρ ∈ ,and m ∈ M denotes the primitive generator of ρ . If Y is not Fano, the potential receives corrections which can be viewed as coming from degenerate holomorphic disks on Y with irreducible components mapping into D. 2. Modiﬁed Mumford degenerations In this section, we ﬁx (Y, D) a Looijenga pair, and let (B, ) be the tropicalisation of (Y, D) deﬁned in Section 1.2.The fan contains rays ρ ,...,ρ corresponding to 1 n divisors D ,..., D , ordered cyclically. As usual, we write the two-dimensional cones of 1 n as σ being the cone with edges ρ and ρ , with indices taken modulo n. i,i+1 i i+1 We explain how to generalize Mumford’s degeneration, to give a canonical formal deformation of V = V \{0} associated to (Y, D) if n ≥ 3. Locally on B the picture is n 0 n MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 85 toric and we have Mumford’s degenerations described in Section 1.3.AsMumford’s con- struction is functorial, the deformations built locally patch together canonically: this is a minor variation on the ideas of [GS07]. In particular, Sections 2.1 and 2.2 are variations of ideas in [GS07]. However, it differs crucially in several respects which prevent us from just referring to [GS07]. First, we work with piecewise linear functions with values in a gp vector space P rather than just R. This allows us to construct higher dimensional for- mal families, namely over the completion of Spec k[P] at the zero-dimensional stratum. Second, by avoiding a description of local models in codimension at least two, we avoid some of the technical complexities of [GS07]. Here are the details. 2.1. The uncorrected degeneration. — We ﬁx some notation. For any locally constant sheaf F on B , and any simply connected subset τ ⊂ B we write F for the stalk of this 0 0 τ local system at any point of τ (as any two such stalks are canonically identiﬁed by parallel transport). In particular, we apply this for the sheaf of integral constant vector ﬁelds, as well as for the sheaf := ⊗ R. R Z −1 For each cone τ ∈ with dim τ = 1 or 2, we write τ for the localized fan of convex (but not strictly convex) cones in described as follows. If dim τ = 2, then τ,R −1 −1 τ just consists of the single cone .If dim τ = 1, then τ consists of three τ,R cones: the tangent line to τ and the two half-spaces bounded by the tangent line to τ . gp Let P ⊆ P be a toric monoid as in Section 1.3. gp Deﬁnition 2.1. —A (P -valued) -piecewise linear multivalued function on B is a gp collection ϕ ={ϕ } with ϕ a -piecewise linear function on U with values in P . i i i gp −1 Note this is equivalent to giving a ρ -piecewise linear function ϕ : → P for each i R,ρ i i R ray ρ ∈ . Two such functions ϕ, ϕ are said to be equivalent if ϕ − ϕ is linear for each i. Note i i gp the equivalence class of ϕ is determined by the collection of bending parameters κ ∈ P .Wesay the ρ,ϕ function is convex (strictly convex) if κ ∈ P (κ ∈ P \ P )for each ρ . ρ,ϕ ρ,ϕ We drop the modiﬁers and P when they are clear from context. Construction 2.2. — The collection {ϕ } determines a local system P on B as fol- i 0 lows. First, we can construct an afﬁne manifold P which comes along with the structure gp of P -principal bundle π : P → B and a piecewise linear section ϕ : B → P as follows: 0 0 0 0 gp gp gp we glue U × P to U × P along (U ∩ U ) × P by i i+1 i i+1 R R R (x, p) → x, p + ϕ (x) − ϕ (x) . i+1 i By construction we have local sections x → (x,ϕ (x)) which patch to give a piecewise gp linear section ϕ. One checks immediately the isomorphism class (of the P -principal bundle together with the section) depends only on the equivalence class of {ϕ }.The bundle P → B can be viewed as a tropical analogue of a sum of line bundles, and {ϕ } 0 0 i 86 MARK GROSS, PAUL HACKING, AND SEAN KEEL yield a section of this vector bundle. Convexity is analogous to holomorphicity of the section. We then deﬁne −1 P := π ϕ ∗ P P 0 0 on B .Wehaveanexact sequence gp (2.1)0 → P → P−→ → 0 of local systems on B ,where r is the derivative of π . gp gp Note over U , the description of P as U × P gives a splitting P| | × P . i 0 i U U R i i Example 2.3. — Our standard example, fundamental to this paper, will be as fol- lows. Suppose P is a monoid which comes with a homomorphism η : NE(Y) → Pof monoids. Choose ϕ by specifying ϕ on U by the formula i i κ = η [D ] . ρ ,ϕ i i i Such a ϕ is well-deﬁned up to linear functions, and always exists. This is always convex, and is strictly convex provided η([D ]) is not invertible for any i. Now suppose given a piecewise linear multivalued P-convex function ϕ on B. We explain how Mumford’s construction determines a canonical formal deformation of V , restricting to the case n ≥ 3 for ease of exposition. −1 For each τ ∈ with dimτ> 0, ϕ determines a canonically deﬁned τ - piecewise linear section ϕ : → P of the projection P → .If U ∩ τ = ∅, τ R,τ R,τ R,τ R,τ i −1 we use the representative ϕ on U and extend it linearly on each cone in the fan τ i i −1 to obtain a P-convex piecewise linear function on τ , which we also write as ϕ .Then the section ϕ is deﬁned as in Construction 2.2 by x → (x,ϕ (x)), using the splitting τ i gp P = × P . We note a different choice of representative of ϕ leads to a different R,τ R,τ i choice of splitting and the same section ϕ , so this section is well-deﬁned. Now deﬁne the toric monoid P ⊂ P by ϕ τ (2.2)P := q ∈ P | q = p + ϕ (m) for some p ∈ P, m ∈ . ϕ τ τ τ By the deﬁnition of convexity of ϕ, we have canonical inclusions (2.3)P ⊂ P ⊂ P ϕ ϕ ρ ρ σ whenever ρ ⊂ σ ∈ .If ρ ∈ is a ray with ρ ⊂ σ ∈ we have the equality ± max (2.4)P ∩ P = P . ϕ ϕ ϕ σ σ ρ + − MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 87 Deﬁnition 2.4 (Monomial ideals). — A (monoid) ideal of a monoid P is a subset I ⊂ P such that p ∈ I, q ∈ P implies p + q ∈ I. An ideal determines a monomial ideal in the monoid ring k[P], generated by monomials z for p ∈ I. We also denote this ideal by I, hopefully with no confusion. As a consequence, we shall sometimes write certain ideal operations either additively or multiplicatively, i.e., for J ⊂ P, kJ={p +···+ p | p ∈ J, 1 ≤ i ≤ k}, 1 k i and the corresponding monomial ideal is J . Let m = P \ P . This is the unique maximal ideal of P, deﬁning a monomial ideal m ⊂ k[P]. Note k[P]/m k[P ]. We say an ideal I ⊂ P is m-primary if m = I := {p ∈ P| there exists a positive integer k such that kp ∈ I}, in which case the same holds for the associated monomial ideal I ⊂ k[P]. Recall from Section 1.3 that we are only considering toric monoids P, i.e., monoids which are the intersection of rational polyhedral cones σ with lattices. Such monoids are always ﬁnitely generated, so that k[P] is Noetherian. If σ is strictly convex, then m is the maximal ideal corresponding to the unique torus ﬁxed point of Spec k[P]. Fix an ideal I ⊂ P, and recalling that we write R = k[P], set R := k[P]/I. We deﬁne for τ ∈ ,dimτ> 0, the ring R := k[P ]⊗ R , τ,I ϕ R I noting that P acts naturally on P by addition. So Spec R is a base-change of the ϕ τ,I −1 Mumford degeneration induced by ϕ on the localized fan τ . One observes Proposition 2.5. —Let v denote the primitive generator of the tangent ray to ρ , for each i. Then i i ρ,ϕ viewing z ∈ k[P] as determining an element in R ,wehave R [X , X , X ] I i−1 i+1 (2.5) = R ρ ,I −D κ i ρ ,ϕ (X X − z X ) i−1 i+1 via the map ϕ (v ) ρ j X → z , j ∈{i − 1, i, i + 1}. Furthermore, there are natural maps ψ : R → R ,ψ : R → R ρ ,− ρ ,I σ ,I ρ ,+ ρ ,I σ ,I i i i−1,i i i i,i+1 88 MARK GROSS, PAUL HACKING, AND SEAN KEEL induced by the inclusions P ⊆ P which induce isomorphisms ϕ ϕ ρ σ ∼ ∼ (R ) R ,(R ) R . = = ρ ,I X σ ,I ρ ,I X σ ,I i i−1 i−1,i i i+1 i,i+1 Proof. — We need to check that the ideal on the left-hand side is mapped to zero, as the rest is obvious. Note by construction of B, v + D v + v = 0 aselementsof , i−1 i i+1 ρ ρ i so one sees in fact that ϕ (v ) + ϕ (v ) = κ − D ϕ (v ). The result then follows ρ i−1 ρ i+1 ρ ,ϕ ρ i i i i ρ i easily. Remark 2.6. — Since Spec R → Spec R is a base-change of the Mumford de- ρ ,I I generation, we can in fact say what a ﬁbre of this morphism is over a closed point x in the smallest toric stratum of Spec k[P], i.e., a point in Spec R . This depends on whether −1 κ ∈ P is invertible or not. If it is not invertible, then the ﬁbre is Spec k[ρ ] ρ ,ϕ i i ±1 Spec k[X , X , X ]/(X X ).If κ is invertible, then the ﬁbre is Spec k[Z ].In i−1 i+1 i−1 i+1 ρ ,ϕ i i this latter case, if ρ ⊂ σ ,infactthe map R → R induced by the inclusion P ⊆ P i ρ ,I σ,I ϕ ϕ i ρ σ is an isomorphism. Somewhat more generally, if J ⊂ P is a radical ideal with κ ∈ J, then in fact ρ ,ϕ −1 R = R [ρ ]. ρ ,J J i i For τ ∈ ,dim τ ≥ 1, set U := Spec R . τ,I τ,I The maps ψ induce open immersions U → U and U → U . Denoting ρ ,± σ ,I ρ ,I σ ρ ,I i i−1,i i i,i+1 i the image of each of these immersions as U and U respectively, we note ρ ,σ ,I ρ ,σ ,I i i−1,i i i,i+1 that ∼ ∼ ρ ,ϕ (2.6)U ∩ U = Spec(R ) = (G ) × Spec(R ) ρ ,σ ,I ρ ,σ ,I ρ ,I X X m I z i i−1,i i i,i+1 i i−1 i+1 κρ ,ϕ Note that if κ ∈ I then the localization (k[P]/I) is zero, and the intersection is ρ ,ϕ z empty. We can now deﬁne our analogue of the Mumford degeneration. Construction 2.7. — Suppose that the number of irreducible components n of D satisﬁes n ≥ 3, that ϕ is a PL multivalued function, and I ⊂ Pan ideal such that κ ∈ I ρ,ϕ for all rays ρ ∈ . Then there are canonical identiﬁcations of open subsets ∼ ∼ U ⊃ U U U ⊂ U = = ρ ,I ρ ,σ ,I σ ,I ρ ,σ ,I ρ ,I i i i,i+1 i,i+1 i+1 i,i+1 i+1 which generate an equivalence relation on U , and the quotient by this equivalence ρ ,I relation deﬁnes a scheme X over Spec R . One checks easily that the canonical isomorphisms of U ⊆ U and U ⊆ U ρ ,σ ,I ρ ,I ρ ,σ ,I ρ ,I i i,i+1 i i+1 i,i+1 i+1 satisfy the requirements for gluing data for schemes along open subsets, see e.g., [H77], Ex. II 2.12. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 89 Remark 2.8. —X only depends on the equivalence class of ϕ, since the monoids P are canonically deﬁned, independently of the choice of representative for ϕ. We ﬁrst analyze this construction in the purely toric case: Lemma 2.9. —For (Y, D) toric and ϕ a single-valued convex function on B = M , X is an open subscheme of the Mumford degeneration Spec k[P ]/Ik[P ],and ϕ ϕ 0 o H X , O = k[P ]/Ik[P ]. X ϕ ϕ Proof. — Note that for τ ∈ , τ = {0}, the monoid P is isomorphic to the local- ization of P along the face {(m,ϕ(m))| m ∈ τ ∩ M}.Thus Spec k[P ] is an open subset of ϕ ϕ Spec k[P ] and Spec k[P ]⊗ k[P]/I is an open subset of Spec k[P ]/Ik[P ]. Further- ϕ ϕ k[P] ϕ ϕ more, the gluing procedure constructing X is clearly compatible with these inclusions, so X is an open subscheme of Spec k[P ]/Ik[P ]. Next, looking at the ﬁbre over a closed ϕ ϕ point, one sees easily that the underlying topological space of these ﬁbres is obtained just by removing the zero-dimensional torus orbit from the corresponding ﬁbre of the Mumford degeneration. The closed ﬁbres of the Mumford degeneration are S by [A02], 2.3.19. Thus by Lemma 2.10, the result follows. Lemma 2.10. —Let π : X → S be a ﬂat family of surfaces such that the ﬁbre X satisﬁes Serre’s condition S for each s ∈ S.Let i: X ⊂ X be the inclusion of an open subset such that the complement has ﬁnite ﬁbres. Then i O = O . Similarly, if F is a coherent sheaf on S then ∗ X X ∗ ∗ i (O ⊗ π F ) = O ⊗ π F . ∗ X Proof. — For the ﬁrst statement see, e.g., [H04], Lemma A.3, (the assumption that the ﬁbres are semi log canonical is not used). The second statement follows from the ﬁrst by dévissage. Deﬁnition 2.11. —Let B (Z) denote the set of points of B with integral coordinates in an 0 0 integral afﬁne chart. We also write B(Z) = B (Z)∪{0}. Given the description of Remark 2.6, the following lemma is obvious. Lemma 2.12. — Suppose n ≥ 3 and we are given a convex multivalued piecewise linear function ϕ and a radical monomial ideal J ⊂ P such that κ ∈ J for all rays ρ ∈ .Thenif x ∈ Spec R ρ,ϕ J is a closed point, the ﬁbre of X → Spec R over x is (Spec k[ ])\{0}.Here, k[ ] denotes the k- algebra with a k-basis {z | m ∈ B(Z)} with multiplication given exactly as in (1.6), and 0 is the closed m o point whose ideal is generated by {z | m = 0}. In particular, the ﬁbre is isomorphic to V .Furthermore, with R [ ]:= R ⊗ k[ ], J J k X Spec R [ ] \ (Spec R )×{0}. J J J 90 MARK GROSS, PAUL HACKING, AND SEAN KEEL 2.2. Scattering diagrams on B.— Next we translate into algebraic geometry the in- stanton corrections. To construct our mirror family we will use the canonical scattering can diagram D deﬁned in Section 3.1, (which is the translation of the instanton corrections associated to Maslov index zero disks), but as the regluing process works for any scattering diagram (and we will make use of this greater generality in [K3]), we carry it out for an arbitrary scattering diagram. We continue with the notation of the previous sections, with (Y, D), (B, ),Pan arbitrary toric monoid, and ϕ given. We also ﬁx a monomial ideal J ⊂ Psuch thatJ = J. Denote by R the completion of k[P] with respect to the ideal J, and for any τ ∈ , τ = 0, denote by k[P ] the completion of the ring k[P ] with respect to the ideal Jk[P ]. ϕ ϕ τ τ We will now deﬁne a scattering diagram, which encodes a modiﬁcation of the con- struction of X . Unlike the previous subsection, where we assumed n ≥ 3 for ease of exposition throughout, in this subsection we can allow any number of irreducible com- ponents of D except where noted. Deﬁnition 2.13. —A scattering diagram for the data (B, ), P,ϕ, and J is a set D = (d, f ) where (1) d ⊂ B is a ray in B with endpoint the origin with rational slope. d may coincide with a ray of , or lie in the interior of a two-dimensional cone of . (2) Let τ ∈ be the smallest cone containing d.Then f is a formal sum d d f = 1 + c z ∈ k[P ] d p ϕ for c ∈ k and p running over elements of P such that r(p) = 0 and r(p) is tangent p ϕ to d. Here r is deﬁned by (2.1). We further require that d satisfy one of the following two properties: (a) For those p with c = 0,r(p), viewed as a tangent vector at an interior point of d, points towards the origin, in which case we say that d is an outgoing ray. (b) For those p with c = 0,r(p) points away from the origin, in which case we say that d is an incoming ray. (3) If dim τ = 2 or if dim τ = 1 and κ ∈ J,then f ≡ 1mod J. d d τ ,ϕ d (4) For any ideal I ⊂ P with I = J, there are only a ﬁnite number of (d, f ) ∈ D such that ≡ 1mod Ik[P ]. d ϕ d MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 91 Construction 2.14. — We now explain how a scattering diagram D is used to modify the construction of X , as given in Construction 2.7. Suppose we are given a scattering diagram D for the data (B, ),P, ϕ and J, and an ideal I with I = J. We assume that κ ∈ Jfor all rays ρ ∈ and that n ≥ 3 as in Construction 2.7. ρ,ϕ We will use the scattering diagram D to modify both the deﬁnition of the rings R as well as the gluings of the schemes deﬁned by these rings. First, we modify the ρ ,I deﬁnition of R , setting ρ ,I ±1 R [X , X , X ] I i−1 i+1 (2.7)R := ρ ,I i 2 −D κ i ρ ,ϕ (X X − z X f ) i−1 i+1 ρ i i ±1 Here f is an element of R [X ] deﬁned by ρ I i i f = f mod Ik[P ], ρ d ϕ i ρ (d,f )∈D d=ρ ϕ (v ) ρ i identifying X with z as in Proposition 2.5. Note this is a generalization of the old deﬁnition of R ,whichweobtainif f = 1. Thus we continue to use the same notation. ρ ,I ρ i i Retaining the deﬁnition R = k[P ]⊗ R for dim σ = 2 from the previous sub- σ,I ϕ R I section,wenotethatthere aremaps ψ : R → R ,ψ : R → R , ρ ,− ρ ,I σ ,I ρ ,+ ρ ,I σ ,I i i i−1,i i i i,i+1 given by ϕ (v ) ϕ (v ) ϕ (v ) ρ i ρ i−1 ρ i+1 i i i ψ (X ) = z ,ψ (X ) = z ,ψ (X ) = f z , ρ ,− i ρ ,− i−1 ρ ,− i+1 ρ i i i i (2.8) ϕ (v ) ϕ (v ) ϕ (v ) ρ ρ ρ i i−1 i+1 i i i ψ (X ) = z ,ψ (X ) = f z ,ψ (X ) = z . ρ ,+ i ρ ,+ i−1 ρ ρ ,+ i+1 i i i i Furthermore, ψ induce isomorphisms ρ ,± ψ : (R ) → R ,ψ : (R ) → R . ρ ,+ ρ ,I X σ ,I ρ ,− ρ ,I X σ ,I i i i+1 i,i+1 i i i−1 i−1,i Set for τ ∈ \{0} U := Spec R . τ,I τ,I One checks easily that the natural map U → Spec R is ﬂat. The maps ψ induces ρ,I I ρ ,± canonical embeddings U , U → U , and we denote their image by U σ ,I σ ,I ρ ,I ρ ,σ ,I i−1,i i,i+1 i i i−1,i and U respectively. Note that (2.6) continues to hold. ρ ,σ ,I i i,i+1 Next, consider (d, f ) ∈ D with τ = σ ∈ .Let γ be a path in B which crosses d d max 0 d transversally at time t . Then deﬁne θ : R → R γ,d σ,I σ,I 92 MARK GROSS, PAUL HACKING, AND SEAN KEEL FIG.2.—Thepath γ .The solid lines indicate the fan, the dotted lines are additional rays in D.The solid lines may also support rays in D by n ,r(p) p p d θ z = z f γ,d where n ∈ is primitive and satisﬁes, with m a non-zero tangent vector of d, n , m= 0, n ,γ (t ) < 0. d d 0 If γ is not differentiable at t , which might occur for broken lines, see Deﬁnition 2.16, this inequality is interpreted to mean that n is positive at γ(t − ) and negative at γ(t + ) d 0 0 for > 0 small. Note that f is invertible in R since f ≡ 1mod Jk[P ],so f − 1is d σ,I d ϕ d nilpotent in R . σ,I Let D ⊂ D be the ﬁnite set of rays (d, f ) with f ≡ 1mod Ik[P ].For apath γ I d d ϕ wholly contained in the interior of σ ∈ and crossing elements of D transversally, max I we deﬁne θ := θ ◦···◦ θ , γ,D γ,d γ,d n 1 where γ crosses precisely the elements (d , f ), ...,(d , f ) of D ,inthe givenorder. 1 d n d I 1 n Note that if two rays d , d in fact coincide as subsets of B, then θ and θ com- i i+1 γ,d γ,d i i+1 mute, so the ordering is not important for overlapping rays. To construct X , we modify the gluings of the sets U along the open subsets ρ,I I,D U .For each i, we have canonical identiﬁcations of open subsets ρ,σ,I ∼ ∼ U ⊃ U U U ⊂ U = = ρ ,I ρ ,σ ,I σ ,I ρ ,σ ,I ρ ,I i i i,i+1 i,i+1 i+1 i,i+1 i+1 We can modify this identiﬁcation via any automorphism of U . We do this by choosing i,i+1 apath γ :[0, 1]→ B whose image is contained in the interior of σ ,with γ (0) a point i i,i+1 i in σ close to ρ and γ (1) ∈ σ close to ρ , chosen so that γ crosses every ray i,i+1 i i i,i+1 i+1 i (d, f ) of D with τ = σ exactly once, see Figure 2. d I d i,i+1 We then obtain an automorphism θ : R → R , γ ,D σ ,I σ ,I i i,i+1 i,i+1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 93 hence, after taking Spec, an isomorphism θ : U → U . γ ,D ρ ,σ ,I ρ ,σ ,I i i+1 i,i+1 i i,i+1 We now deﬁne X by dividing out U by the equivalence relation given by identi- ρ ,I I,D i i fying x ∈ U ⊆ U with θ (x) ∈ U ⊆ U . ρ ,σ ,I ρ ,I γ ,D ρ ,σ ,I ρ ,I i+1 i,i+1 i+1 i i i,i+1 i 2.3. Broken lines. — We continue to ﬁx a rational surface with anti-canonical cycle (Y, D) as usual, D having an arbitrary number of irreducible components, giving (B, ), as well as a monoid P, a multivalued P-convex function ϕ on B, J ⊂ P an ideal with J = J, and a scattering diagram D for this data. Broken lines were introduced in [G09] and their theory was further developed in [CPS]. Deﬁnition 2.15. —Let B be an integral afﬁne manifold. An integral afﬁne map γ : (t , t ) → 1 2 B from an open interval (t , t ) is a continuous map such that for any integral afﬁne coordinate chart 1 2 n −1 n ψ : U → R of B, ψ ◦ γ : γ (U) → R is integral afﬁne, i.e., is given by t → tv + bfor some n n v ∈ Z and b ∈ R . Note that for an integral afﬁne map, γ (t) ∈ . B,γ (t) Deﬁnition 2.16. —A broken line γ in (B, ) for q ∈ B (Z) with endpoint Q ∈ B is 0 0 a proper continuous piecewise integral afﬁne map γ : (−∞, 0]→ B with only a ﬁnite number of domains of linearity, together with, for each L ⊂ (−∞, 0] a maximal connected domain of linearity q × −1 of γ , a choice of monomial m = c z where c ∈ k and q ∈ (L,γ (P )| ), satisfying the L L L L L following properties. (1) For the unique unbounded domain of linearity L, γ| goes off to inﬁnity in a cone σ ∈ L max as t →−∞,and q ∈ σ . Furthermore, using the identiﬁcation of the stalk P for x ∈ σ ϕ (q) with P ,m = z . σ L (2) For each L and t ∈ L, −r(q ) = γ (t),where r isdeﬁnedin(2.1). Also γ(0) = Q ∈ B . L 0 (3) Let t ∈ (−∞, 0) be apointatwhich γ is not linear, passing from domain of linearity L to L .If γ(t) ∈ τ ∈ , then P = P , so that we can view q ∈ P and r(q ) ∈ . γ(t) τ L τ L τ Let d ,..., d ∈ D be the rays of D that contain γ(t), with attached functions f . Then 1 p d we require that γ passes from one side of these rays to the other at time t, so that θ is γ,d deﬁned. Let n = n be the primitive element of used to deﬁne θ . Expand d γ,d j τ j n,r(q ) (2.9) f j=1 as a formal power series in k[P ]. Then there is a term cz in this sum with m = m · cz . L L 94 MARK GROSS, PAUL HACKING, AND SEAN KEEL Remark 2.17. — Using the notation of item (3) above, by item (2) of the deﬁnition, (2.10) n, r(q ) > 0. This is vital to interpret (2.9). Indeed, if τ is a ray, f need not be invertible in k[P ],so d ϕ i τ (2.10) tells us that (2.9) makes sense in this ring. Example 2.18. — We give a ﬁrst example of broken lines, in the case where B is as given in Example 1.10 and D=∅, so that there is no possibility of bending. Nevertheless, there is quite non-trivial behaviour. For an example including bending, see Example 3.7 after the introduction of the canonical scattering diagram. Given q ∈ B (Z),Q ∈ B general, we can choose lifts q˜, Q to the universal cover 0 0 ˜ ˜ ˜ ˜ B =| |\{0} of B .Let π : B → B be the covering map. Fixing the lift Q, for any lift 0 0 0 0 q˜ we obtain a broken line γ : (−∞, 0]→ B given by γ(t) = π(Q − tq˜). As this has one ϕ (q) domain of linearity L, we decorate L with the monomial z ,where q ∈ σ ∈ .Note there are an inﬁnite number of such broken lines, one for each lift of q. Dealing with this non-ﬁniteness is a key part of the proof of Looijenga’s conjecture in Section 7. The next lemma and corollary are crucial for interpreting the monomials m : Lemma 2.19. —Let σ ,σ ∈ be the two maximal cones containing the ray ρ ∈ .If − + max −1 q ∈ P with −r(q) ∈ Int(ρ σ ) ⊂ ⊗ R, then ϕ + ρ Z q ∈ P = P ∩ P . ϕ ϕ ϕ ρ σ σ − + Proof. — By the deﬁnitions there exist p,κ ∈ Pand n ∈ annihilating the ρ,ϕ ρ tangent space to ρ and positive on σ such that q = ϕ r(q) + p ϕ −r(q) = ϕ −r(q) + n ,−r(q) κ . σ σ ρ ρ,ϕ + − Since n ,−r(q) > 0, q = ϕ r(q) + p + n ,−r(q) κ ∈ P . σ ρ ρ,ϕ ϕ + σ An immediate consequence of this lemma is Corollary 2.20. (1) Let γ :[t , t ]→ B be integral afﬁne. Suppose that γ(t ) ∈ τ , γ(t ) ∈ τ . Suppose 1 2 0 1 1 2 2 −1 also we are given a section q ∈ (γ P ) such that −r(q) = γ (t) for each t. If q(t ) ∈ P ⊂ P = P , 1 ϕ τ γ(t ) τ 1 1 1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 95 then q(t ) ∈ P ⊂ P = P . 2 ϕ τ γ(t ) τ 2 2 (2) If γ is a broken line, t ∈ L a maximal domain of linearity with γ(t) ∈ τ , then q ∈ P ⊂ P = P . L ϕ τ γ(t) Proof. — The ﬁrst item follows immediately from the lemma. The second item fol- lows from the fact that if t 0 lies in the unbounded domain of linearity with γ(t) ∈ σ , ϕ (q) then m = z ∈ P by construction. Then this holds for all t by item (1) and Deﬁni- L ϕ tion 2.16,(3). The convexity of ϕ puts further restrictions on the monomial decorations of a broken line. Deﬁnition 2.21. —Let J ⊂ P be a proper monoid ideal. For p ∈ J there exists a maximal k ≥ 1 such that p = p +···+ p with p ∈ J.Wedeﬁne ord (p) to be this maximum, and deﬁne 1 k i J ord (p) = 0 if p ∈ P \ J. For x ∈ τ,q ∈ P ,deﬁne ord (q) := ord (q − ϕ (r(q))). This measures how high q is ϕ J,x J τ above the graph of ϕ .If γ is a broken line and t ∈ L a maximal domain of linearity, deﬁne ord (t) = ord (q ), J,γ J,γ (t) L using γ(t) ∈ τ and q ∈ P ⊂ P . L ϕ γ(t) Lemma 2.22. —Let γ be a broken line. Then if t < t , ord (t) ≤ ord t , J,γ J,γ with strict inequality if either t and t lie in different domains of linearity or for some t with t < t < t , γ(t ) lies in a ray ρ ∈ with bending parameter κ ∈ J. ρ,ϕ Proof. — This is immediate from the deﬁnitions and the proof of Lemma 2.19. Deﬁnition 2.23. —For I an ideal in P with I = J,let Supp (D) := d where the union is over all (d, f ) ∈ D such that f ≡ 1mod Ik[P ]. By Deﬁnition 2.13, (4), this d d ϕ is a ﬁnite union. 96 MARK GROSS, PAUL HACKING, AND SEAN KEEL Deﬁnition 2.24. —Let I be an ideal of P with I = J,and let Q ∈ B \ Supp (D), Q ∈ τ ∈ .For q ∈ B (Z),deﬁne (2.11)Lift (q) := Mono(γ ) ∈ k[P ]/I · k[P ], Q ϕ ϕ τ τ where the sum is over all broken lines γ for q with endpoint Q,and Mono(γ ) denotes the monomial attached to the last domain of linearity of γ . The word “Lift” is used to indicate that this is, as we shall q o o show, a lifting of a monomial z on X to X . The fact that Lift (q) lies in the stated ring follows J,D I,D from: Lemma 2.25. —Let Q ∈ σ ∈ ,q ∈ B (Z).Let I be an ideal with I = J. Assume max 0 that κ ∈ J for at least one ray ρ ∈ . Then the following hold: ρ,ϕ (1) The collection of γ in Deﬁnition 2.24 with Mono(γ ) ∈ I · k[P ] is ﬁnite. (2) If one boundary ray of the connected component of B \ Supp (D) containing Q is a ray ρ ∈ , then Mono(γ ) ∈ k[P ], and the collection of γ with Mono(γ ) ∈ I · k[P ] is ﬁnite. Proof. — Note there is some k such that J ⊂ I because k[P] is Noetherian. If γ is a broken line with Mono(γ ) ∈ I · k[P ],then γ crosses the rays of in a cyclic order. Indeed, this follows from condition (3) of Deﬁnition 2.16, as a broken line must cross from one side to the other of each ray of D it intersects. From this, the hypotheses on the κ ρ,ϕ imply that in any set of at least n consecutive rays of that it crosses, there is at least one ray ρ with κ ∈ J. By Lemma 2.22,ord increases every time γ crosses such a ray, ρ,ϕ J,γ and also every time γ bends at a ray d not contained in a ray of .Onceord ≥ k, J,γ Mono(γ ) ∈ I · k[P ]. Hence there is an absolute bound on the number of rays of that γ can cross, and the number of times γ can bend. When γ crosses a ray, there are a ﬁnite number of terms in (2.9) modulo I · k[P ], as the exponent is always positive, see Remark 2.17. Thus there are a ﬁnite number of possible choices of bend, and hence only a ﬁnite number of possible choices for the exponent of Mono(γ ) modulo I · k[P ] once the initial monomial of γ is ﬁxed. Given any prescribed sequence of bends and initial direction, one sees that there is only one possibility for the underlying map γ with endpoint a ﬁxed point Q by tracing the broken line back from Q. Each such underlying map γ supports only a ﬁnite number of broken lines modulo I · k[P ]. This yields the ﬁniteness of (1). MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 97 The argument for the ﬁniteness statement in (2), once the ﬁrst part of (2) is es- tablished, is the same. For the ﬁrst part of (2), consider a broken line γ contributing to Lift (q).Wetake Q ∈ σ ,inthe notation of Lemma 2.19.Write Mono(γ ) = c z .If Q + L −1 r(q ) ∈ ρ σ then the statement follows from Lemma 2.19. Otherwise (by the deﬁnition L + of broken line) γ crosses ρ , which is the last ray of and the last ray of Supp (D) it crosses before reaching Q. Now the result follows from Lemma 2.19 and the deﬁnition of broken line. Deﬁnition 2.26. — Assume that κ ∈ J for at least one ray ρ ∈ . We say a scattering ρ,ϕ diagram D is consistent if for all ideals I ⊂ P with I = J and for all q ∈ B (Z), the following holds. Let Q ∈ B be chosen so that the line joining the origin and Q has irrational slope, and Q ∈ B 0 0 similarly. Then: (1) If Q, Q ∈ σ ∈ , then we can view Lift (q) and Lift (q) as elements of R ,and max Q Q σ,I as such, we have Lift (q) = θ Lift (q) Q γ,D Q for γ a path contained in the interior of σ connecting Q to Q . (2) If Q ∈ σ and Q ∈ σ with σ ∈ and ρ = σ ∩ σ a ray, and furthermore − − + + ± max + − Q and Q are contained in connected components of B\ Supp (D) whose closures contain − + ρ , then Lift (q) ∈ R are both images under ψ of a single element Q σ ,I ρ,± ± ± Lift (q) ∈ R . ρ ρ,I Of course the deﬁnition is introduced so that the following construction works: Construction 2.27 (Construction of ϑ ). — Suppose D ⊂ Yhas n ≥ 3 irreducible com- ponents, and that D is a consistent scattering diagram for data (B, ),P, ϕ and J. Assume further that κ ∈ Jfor all ρ ∈ , so that we may apply Construction 2.14.Wenow con- ρ,ϕ struct for any I with I = Ja function ϑ ∈ (X , O ) for q ∈ B(Z) = B (Z)∪{0}. q X 0 I,D I,D We deﬁne ϑ = 1. Next, let q ∈ B (Z).For each ray ρ ∈ contained in σ ∈ , 0 0 ± max choose two points Q ∈ B, one each in the two connected components of B\ (Supp (D)∪ ρ I + − ρ) which are adjacent to ρ,with Q ∈ σ and Q ∈ σ . + − ρ ρ We ﬁrst note that Lift (q) is a well-deﬁned element of R , independent of the Q σ ,I ± + particular choice of Q : given a choice say of Q = Q and another choice Q ,we ρ ρ take a path γ connecting Q and Q wholly contained in the connected component of B \ (Supp (D) ∪ ρ) containing Q and Q . By Deﬁnition 2.26, (1), it then follows that Lift (q) = Lift (q). Q Q By Deﬁnition 2.26,(2),wehaveanelement Lift (q) ∈ R whose image under ρ ρ,I ψ is Lift ± (q). It then follows via another application of Deﬁnition 2.26, (1), applied ρ,± to the path of Figure 2,thatif ρ , ρ are adjacent rays in , then Lift (q) and Lift (q) ρ ρ glue under the identiﬁcation of open subsets of U and U given by θ .Thusall ρ,I ρ ,I γ,D 98 MARK GROSS, PAUL HACKING, AND SEAN KEEL these elements of the rings R for ρ ∈ glue to give a regular function on X ,by ρ,I I,D construction of this latter space. This regular function is what we call ϑ . Theorem 2.28. —Suppose D has n ≥ 3 irreducible components, and let ϕ be a multivalued piecewise linear function on B such that κ ∈ J for all rays ρ ∈ .Let D be a consistent scattering ρ,ϕ diagram and I ⊂ P an ideal with I = J.Set X := Spec X , O . I X I,D I,D Since X has the structure of a scheme over Spec R ,sodoes X , which we write as I I I,D f : X → Spec R . I I I Then (1) X contains X as an open subset and f is ﬂat with ﬁbre over a closed point x of Spec R I I I I,D isomorphic to the n-vertex V . (2) For each q ∈ B(Z), there is a section ϑ ∈ (X , O ),and theset q I X ϑ | q ∈ B(Z) is a free R -module basis for (X , O ). I I X Proof. — Construction 2.27 constructs regular functions ϑ on X ,hence by def- I,D inition of X ,weobtain ϑ ∈ (X , O ). I q I X o o Now note that X = X as deﬁned in Section 2.1.Indeed,for any (d, f ) ∈ D with J,D J dim τ = 2we have f ≡ 1 mod J, so the open sets U ,U are glued trivially. Similarly, d d ρ ,J ρ ,J i i+1 if dim τ = 1 then since κ ∈ J, the rings R as given in (2.7)and (2.5) coincide and are d ρ,ϕ ρ,I glued trivially. Thus with I = J, we see the gluing constructions 2.7 and 2.14 coincide. Note that with the assumption that κ ∈ Jfor all rays ρ , ρ,ϕ X = Spec R [ ] \ (Spec R )×{0} J J by Lemma 2.12. We see that the canonical map R · ϑ → X , O J q X q∈B(Z) is an isomorphism. Indeed, by Lemma 2.10, (X , O ) R [ ]. Furthermore, under X J this isomorphism, ϑ is clearly taken to z ∈ R [ ]. This is because the only broken lines q J contributing to Lift (q) modulo J for any Q is the straight line with endpoint Q, and this provides a contribution only if Q lies in the same maximal cone as q. It also follows that X := Spec (X , O ) = Spec R [ ] is ﬂat over Spec R and J X J J the ﬁber over a closed point x is given by Spec k[ ]. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 99 Now let I be an ideal with I = J. Let i: X ⊂ X be the inclusion. Deﬁne a ringed space X with underlying topological space X by O := i O . Then the natural map J X ∗ X I,D O → O is surjective by the existence of the lifts ϑ .Thus X / Spec R is a ﬂat defor- X X q I J I mation of X / Spec R by Lemma 2.29 below. Now since X is afﬁne it follows that X is J J J also afﬁne, so X = X := Spec (X , O ). I X I I,D I,D We showed above that the ϑ form an R -module basis of (X , O ). Now since q J J X X / Spec R is a ﬂat inﬁnitesimal deformation of X / Spec R it follows that the ϑ form a I I J J q R -module basis of (X , O ), see Lemma 2.30 below. I I X Lemma 2.29. —Let X /S be a ﬂat family of surfaces such that the ﬁbres satisfy Serre’s 0 0 condition S .Let i: X ⊂ X be the inclusion of an open subset such that the complement has ﬁnite 2 0 ﬁbres. Note that i O = O by Lemma 2.10. ∗ X X o o Let S ⊂ S be an inﬁnitesimal thickening of S and let X → S be a ﬂat deformation of X /S 0 0 0 over S. Deﬁne a family of ringed spaces X → S by O := i O . X ∗ X Then X/S is a ﬂat deformation of X /S (that is, X/S is ﬂat and X = X × S )ifand 0 0 0 S 0 only if the map (2.12) O := i O o → i O = O X ∗ X ∗ X X is surjective. Proof. — The condition is clearly necessary. Conversely, suppose (2.12) is surjective. Let I ⊂ O be the nilpotent ideal deﬁning S ⊂ S. Let X /S denote the nth order inﬁnitesimal thickening of X /S determined by 0 n 0 0 o n+1 n+1 X /S, that is, O o = O o /I · O o and O = O /I .Deﬁne X /S by O := i O o . X X X S S n n X ∗ X n n n n Note that O → O is surjective because O → O is surjective by assumption. We X X X X n 0 0 show by induction on n that X /S is a ﬂat deformation of X /S .For n = 0 there is n n 0 0 nothing to prove. Suppose the induction hypothesis is true for n. Since X /S is ﬂat n+1 n+1 (being the restriction of the ﬂat family X /StoS )wehaveashortexact sequence n+1 n+1 n+2 o o 0 → I /I ⊗ O → O → O o → 0. X X X 0 n+1 Applying i we obtain an exact sequence n+1 n+2 0 → i I /I ⊗ O → O → O . ∗ X X X n+1 n n+1 n+2 By Lemma 2.10 the ﬁrst term is equal to I /I ⊗ O .Moreover, thelastarrow is surjective because O → O is surjective, O /I · O = O by the induction X X X X X n+1 0 n n 0 hypothesis, and I is nilpotent, as in Theorem 8.4 of [Ma89] (where the module M need not be ﬁnitely generated for the argument given there to work). So we have an exact sequence n+1 n+2 (2.13) 0 → I /I ⊗ O → O → O → 0. X X X 0 n+1 n 100 MARK GROSS, PAUL HACKING, AND SEAN KEEL n+1 It follows that O /I · O = O . (Indeed, consider the map X X X n+1 n+1 n n+1 α: I ⊗ O → O ,α(f ⊗ g) = fg. X X n+1 n+1 We claim that α is equal to the composition of the map n+1 n+1 n+2 β : I ⊗ O → I /I ⊗ O X X n+1 0 given by the natural maps on the factors and the ﬁrst map γ of the exact sequence (2.13). Since O = i O by deﬁnition, it sufﬁces to check the equality after restriction to X ∗ X n+1 n+1 X , where it is obvious. The map β is surjective because O → O is surjective. So X X n+1 0 n+1 the image of γ is equal to the image of α,namely I · O .) Now by [Ma89], Theo- n+1 rem 22.3, p. 174, the exact sequence (2.13)shows that X /S is a ﬂat deformation of n+1 n+1 X /S . 0 0 Lemma 2.30. —Let A → B be a ﬂat homomorphism of Noetherian rings and I ⊂ A a nilpotent ideal. Suppose given a set S of elements of B such that the reductions of the elements of S form an A/I-module basis of B/IB.Then S is an A-module basis of B. Proof. — Since I is nilpotent and S generates B/IB it is clear that S spans B by Theorem 8.4 of [Ma89]. So we have an exact sequence 0 → K → A → B → 0. Tensoring with A/I we obtain an exact sequence 0 → K/IK → (A/I) → B/IB → 0 using ﬂatness of B over A. We deduce that K/IK = 0 by our assumption, hence K = 0 because I is nilpotent. Proposition 2.31. —Let X /S := Spec R be the family of Theorem 2.28. Then the relative I I I dualizing sheaf ω is trivial. It is generated by the global section givenonlocal patches U by X /S ρ ,I I I i dlog X ∧ dlog X = dlog X ∧ dlog X . Here we take the rays ρ in counter-clockwise order, i−1 i i i+1 j after choosing an orientation on B, to obtain a consistent choice of signs. Proof. — By the adjunction formula for the closed embedding U ⊂ A × G × S , ρ m,X I i X ,X i i−1 i+1 the dualizing sheaf ω is freely generated over U by the local section in the statement. X /S ρ I I i These sections patch to give a generator of ω because the scattering automor- X /S I,D phisms preserve the torus invariant two-forms. Both ω and O satisfy the relative X /S X I I I ∗ o S property i i F = F where i: X ⊂ X is the inclusion ([H04], Appendix, where the 2 ∗ I hypothesis of slc is not needed), hence ω is freely generated by . X /S I I MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 101 2.4. The algebra structure. — In the previous section, we saw that the R -algebra A = X , O I X I,D I,D deﬁning the ﬂat deformation X has an R -module basis of theta functions {ϑ | m ∈ I I m B(Z)}. Here we derive a description of the multiplication rule on R using the geometry of the integral afﬁne manifold B. Besides being an attractive combinatorial description of the multiplication rule, we will use this (in the case of the canonical scattering diagram can D )inSection 6 to prove that our deformation extends over completions of larger strata of Spec k[P], as well as for the case that D has 1 or 2 irreducible components. Deﬁnition 2.32. — For a broken line γ with endpoint Q ∈ τ ∈ ,deﬁne s(γ ) ∈ ,c(γ ) ∈ k[P] by demanding that ϕ (s(γ )) Mono(γ ) = c(γ ) · z . Write Limits(γ ) = (q, Q) if γ is a broken line for q and has endpoint Q ∈ B. Remark 2.33. —Recallthat B in fact has the structure of an integral linear mani- fold. One feature of such manifolds is that for any simply connected set U ⊂ B , there is a canonical linear immersion U → , compatible with parallel transport inside U. R,U In particular, if q is a point of B with q ∈ σ ∈ ,and τ ⊂ σ , then the canonical embedding of a neighbourhood of τ \{0} in identiﬁes q with a point of . τ,R τ,R Theorem 2.34. —Let q , q ∈ B(Z). In the canonical expansion 1 2 ϑ · ϑ = α ϑ , q q q q 1 2 q∈B(Z) where α ∈ R for each q, we have q I α = c(γ )c(γ ) q 1 2 (γ ,γ ) 1 2 Limits(γ )=(q ,z) i i s(γ )+s(γ )=q 1 2 Here z ∈ B is a point very close to q contained in a cell τ , and we identify q with a point of using 0 τ Remark 2.33. Proof. — To identify the coefﬁcient of ϑ , choose a point z ∈ B very close to q,and q q 1 2 describe the product using the lifts of z , z at z: Lift (q ) Lift (q ) = α Lift q . z 1 z 2 q z q 102 MARK GROSS, PAUL HACKING, AND SEAN KEEL Now observe ﬁrst that there is only one broken line γ with endpoint z and s(γ ) = q ∈ : this is the broken line whose image is z + R q. Indeed, the ﬁnal segment of such a γ is ≥0 on this ray, and this ray meets no scattering rays, so the broken line cannot bend. Thus the coefﬁcient of Lift (q) on the right-hand side of the above equation can be read off by ϕ (q) looking at the coefﬁcient (in R )of z . This gives the desired description. 3. The canonical scattering diagram can 3.1. Deﬁnition. — Here we give the precise deﬁnition of D . As explained in the introduction, it is, roughly speaking, deﬁned in terms of maps A → Y \ D, which are algebro-geometric analogues of the holomorphic disks used for instanton corrections in the symplectic heuristic. We begin by recalling necessary facts about relative Gromov- Witten invariants used to count these curves. ˜ ˜ ˜ Deﬁnition 3.1. —Let (Y, D) be a non-singular rational surface with D an anti-canonical ˜ ˜ cycle of rational curves, and let C be an irreducible component of D. Consider a class β ∈ A (Y, Z) such that k D = C β i (3.1) β · D = 0 D = C for some k > 0.Let F be the closure of D \ C,and let o o ˜ ˜ Y := Y \ F, C := C \ F. o o Let M(Y /C ,β) be the moduli space of stable relative maps of genus zero curves representing the class β with tangency of order k at an unspeciﬁed point of C .(See[Li00], [Li02] for the algebraic deﬁnition for these relative Gromov-Witten invariants, and [LR01], [IP03] for the original symplectic deﬁnitions.) We refer to β informally as an A -class. The virtual dimension of this moduli space is −K · β + (dim Y − 3) − (k − 1) = 0. ˜ β Here the ﬁrst two terms give the standard dimension formula for the moduli space of stable rational curves in Y representing the class β,and theterm k − 1 is the change in dimension given by imposing the k -fold tangency condition. The moduli space carries a virtual fundamental class. Furthermore, we have o o Lemma 3.2. — M(Y /C ,β) is proper over k. Proof. — This follows as in the proof of [GPS09], Theorem 4.2. In brief, let R be a valuation ring with quotient ﬁeld K, with S = Spec R, T = Spec K. We would like to o o ˜ ˜ extend a morphism T → M(Y /C ,β) to S. We know that the moduli space M(Y/C,β) is proper, so we obtain a family of relative stable maps C → Sto Y. We just need to show MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 103 that in fact the image of the closed ﬁbre C lies in Y . However, the argument in the proof of [GPS09], Theorem 4.2 shows that if the image of C intersects F, then C must be of 0 0 genus at least 1, which is not the case. Given this, we deﬁne N := 1. o o vir [M(Y /C ,β)] ˜ ˜ Morally, one should view N as counting maps from afﬁne lines to Y \ D whose closures represent the class β . In what follows, we ﬁx as usual the pair (Y, D), with tropicalisation (B, ),and ϕ the function given by Example 2.3 for some choice of η : NE(Y) → P. We can assume here that D has an arbitrary number of irreducible components. Deﬁnition 3.3. —Fix aray d ⊂ B with endpoint the origin, with rational slope. If d coincides with a ray of ,set := ; otherwise, let be a reﬁnement of obtained by adding the ray d and a number of other rays chosen so that each cone of is integral afﬁne isomorphic to the ﬁrst quadrant of R . −1 This gives a toric blow-up π : Y → Y (the identity in the ﬁrst case) by Lemma 1.6.Let C ⊂ π (D) be the irreducible component corresponding to d. Let τ ∈ be the smallest cone containing d.Let m ∈ be a primitive generator of the d d τ tangent space to d, pointing away from the origin. Deﬁne η(π (β))−ϕ (k m ) ∗ τ β d f := exp k N z . d β β Here the sum is over all classes β ∈ A (Y, Z) satisfying (3.1). Note that if N = 0, then necessarily 1 β o o ˜ ˜ M(Y /C ,β) is non-empty, and thus β ∈ NE(Y),so π (β) ∈ NE(Y). We note the numbers N ∗ β do not depend on the particular choice of reﬁnement . Indeed, further reﬁning does not change the o o pair Y /C , and hence does not change the numbers N . We deﬁne can D := (d, f )| d ⊂ B a ray of rational slope . We call a class β ∈ A (Y, Z) an A -class if N = 0. 1 β Note that all rays of the canonical scattering diagram are outgoing. Remark 3.4. — In theory, one should be able to use logarithmic Gromov-Witten invariants ([GS11]or[AC11]) to deﬁne N without the technical trick of blowing up and working on an open variety. This would be done by working relative to D, and counting rational curves of class β with one point mapping to the boundary with speciﬁed orders of tangency with each boundary divisor, with non-zero order of tangency with either 104 MARK GROSS, PAUL HACKING, AND SEAN KEEL one divisor D or two adjacent divisors D ,D . However, some additional arguments i i i+1 are required to compare logarithmic invariants with the invariants described above as developed in [GPS09], and we do not wish to do this here. Lemma 3.5. —Let J ⊂ P be an ideal with J = J.Suppose themap η: NE(Y) → P satisﬁes the following conditions: (1) For any ray d ⊂ B of rational slope, let π : Y → Y be the corresponding blow-up. We re- quire that if dim τ = 2 or dim τ = 1 and κ ∈ J then for any A -class β contributing d d τ ,ϕ to f , we have η(π (β)) ∈ J. d ∗ (2) For any ideal I with I = J, there are only a ﬁnite number of d and A -classes β such that η(π (β)) ∈ I. can Then D is a scattering diagram for the data (B, ), P,ϕ, and J. Proof. — Note that η(π (β))−ϕ (k m ) ∗ τ β d z ∈ Ik[P ] if and only if η(π (β)) ∈ I. So the hypotheses of the lemma imply conditions (2)–(4) of Deﬁnition 2.13. Example 3.6. —Let σ ⊂ A (Y) ⊗ R be a strictly convex rational polyhedral cone 1 Z containing NE(Y). (This can be obtained as the dual of a strictly convex rational polyhe- dral cone in Pic(Y) ⊗ R which spans this latter space and is contained in the nef cone.) Let P = σ ∩ A (Y). Since σ is strictly convex, P = 0. For any m-primary ideal I, P \ I is a ﬁnite set. Let η : NE(Y) → P be the inclusion. Then the ﬁniteness hypotheses of the above Lemma hold for J = m (note that the conditions (3.1) determine β ∈ A (Y) given π (β)). Example 3.7. — We return to the example (Y, D) of a del Pezzo surface together with a cycle of 5 (−1)-curves studied in Example 1.9.Let P = NE(Y) and η be the can identity. Let J = m ⊂ Pand I ⊂ P be an ideal with I = J. Then D consists of ﬁve rays: can [E ]−ϕ (v ) i ρ i D = ρ , 1 + z | 1 ≤ i ≤ 5 . Here E is the unique (−1)-curve in Y which is not contained in D and meets D transver- i i sally, and v is the primitive generator of the ray ρ corresponding to D . To derive this i i i formula from the above deﬁnition of the canonical scattering diagram one needs to can show that the only possible stable relative maps contributing to D are multiple cov- ers of the E ’s, and that a k-fold multiple cover contributes a Gromov-Witten invariant of k−1 2 (−1) /k . It is easier to compute this using the main result of [GPS09], which is done by way of Theorem 3.25. See Example 3.26. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 105 FIG. 3. — The different types of broken lines in Example 3.7 can If we accept this description of D , then we can describe all broken lines and the multiplication law given by this diagram. We ﬁrst note that no broken line can wrap around 0 ∈ B, i.e., if a broken line leaves acone σ ∈ , it will never return to that cone. It is enough to check this for a straight max line (as the bending in any broken line for D is always away from the origin), and this can is easily veriﬁed, using e.g., Figure 1. Next, since the only scattering rays are the rays ρ ∈ ,if q, Q ∈ σ ∈ , then the max obvious straight line is the unique broken line for q with endpoint Q. Thus if we describe o ϕ (q) ϑ in the open subset of X corresponding to σ , ϑ is just the monomial z .It can q q I,D follows that a b ϑ ϑ = ϑ av +bv v v i i+1 i i+1 for a, b ≥ 0. In particular, the ϑ ’s generate the k[P]/I-algebra (X , O ),and thealge- v I X i I bra structure is determined once we compute ϑ · ϑ . v v i i+2 We consider a broken line for v . One checks the following, using Figure 1 and the can above description of D : The broken line can cross at most two rays of , and it bends at most once, at the last ray of that it crosses. See Figure 3. From this one deduces using Theorem 2.34: [D ] [E ] i i (3.2) ϑ ϑ = z ϑ + z . v v v i−1 i+1 i [D ] The term z · ϑ corresponds to two straight broken lines for v ,v , with endpoint v i−1 i+1 [D ] [E ] i i the point v of ρ .The term z · z is the coefﬁcient of 1 = ϑ . To compute this we use i i 0 the invariance of broken lines, and so choose a generic point Q near 0 and compute the coefﬁcient α of ϑ using pairs γ as in Theorem 2.34 whose ﬁnal directions are opposite, 0 0 i i.e., s(γ ) + s(γ ) = 0. If we take Q ∈ σ , then there is exactly one term contributing 1 2 i,i+1 to α : γ will bend once where it crosses ρ ,and γ is straight. Alternatively, one can 0 1 i 2 106 MARK GROSS, PAUL HACKING, AND SEAN KEEL use the explicit expressions for Lift (v ), j = i − 1, i and i + 1, and see they satisfy the Q j relation (3.2). Onecan checkthatthe ﬁveequations(3.2)deﬁne X . These equations are alge- braic, and in fact deﬁne a ﬂat family over Spec k[NE(Y)]. (This is always the case in the non-negative semi-deﬁnite case, see Corollary 6.11). can Our goal now is to prove consistency of D , as stated in the following (the ﬁnal step in the construction of our mirror family): Theorem 3.8. — Suppose that we are given a map η : NE(Y) → P such that ϕ is deﬁned as in Example 2.3 by κ = η([D ]). Suppose furthermore the following conditions hold: ρ,ϕ ρ (I) For any A -class β , η(π (β)) ∈ J; (II) For any ideal I with I = J, there are only a ﬁnite number of A -classes β such that η(π (β)) ∈ I. (III) η([D ]) ∈ J for at least one boundary component D ⊂ D. i i can Then D is a consistent scattering diagram. We include here an observation we will need later showing that the canonical scat- tering diagram only depends on the deformation class of (Y, D). Lemma 3.9. —Let (Y , D) → S be a ﬂat family of pairs over a connected base S, with each ﬁbre (Y , D ) being a non-singular rational surface with anti-canonical cycle. Suppose further that there s s is a trivialization D = D × S and the restriction map Pic(Y ) → Pic(Y ) is an isomorphism for any s ∈ S. This in particular gives a canonical identiﬁcation A (Y , Z) with A (Y , Z) for any s, s ∈ S. 1 s 1 s Then for any s, s ∈ S, (Y , D ) and (Y , D ) induce the same canonical scattering diagram. s s s s Proof. — It is enough to show that the numbers N are deformation invariants ˜ ˜ in the above sense, i.e., if we are given a family π : (Y , D) → Swith each ﬁbre as in Deﬁnition 3.1, with an irreducible component C ⊂ D, then the number N := 1 β,s ˜ o o vir [M(Y /C ,β)] s s is independent of s. Indeed, once this is shown, then if N = 0, necessarily β deﬁnes a β,s class in NE(Y ),aswellasin NE(Y ), under the chosen identiﬁcation. This invariance s s follows from the standard argument that (relative) Gromov-Witten invariants are defor- mation invariants, with a little care because our target spaces are open. For this, one o o o o ˜ ˜ considers the moduli space M(Y /C ,β) of stable maps to Y relative to C and whose o o composition with π is constant. Then one has a map ψ : M(Y /C ,β) → Swhose ﬁbre o o o o ˜ ˜ over s is M(Y /C ,β). Letting ξ be the inclusion of this ﬁbre in M(Y /C ,β),deforma- s s tion invariance will follow if we know that vir vir ! o o o o ˜ ˜ ξ M Y /C ,β = M Y /C ,β s s MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 107 and ψ is proper. The ﬁrst statement is standard in Gromov-Witten theory, see e.g. the argument of Theorem 4.2 of [LT98] in the non-relative case, which carries over to the relative case. The second point, the properness of ψ , follows exactly as in the proof of Lemma 3.2. 3.2. Consistency: overview of the proof. — We will describe in detail the intuition be- hind each step of the proof of consistency. In the next subsection, we will work somewhat more generally with a more general scattering diagram D for certain steps, as this will be needed in [K3] for the K3 case. However, for the discussion here let us assume we are only studying the consistency of the canonical scattering diagram. Step I. We can replace (Y, D) with a toric blow-up of (Y, D). This is straightforward— toric blowups just correspond to reﬁnements of , but do not change broken lines or scattering diagrams. Step II. We can assume that (Y, D) has a toric model and P is a ﬁnitely generated submonoid of A (Y, Z) containing NE(Y), with η the inclusion.ByStepIandProposition 1.3,wecan η¯ ψ assume (Y, D) has a toric model. We can then always factor η as NE(Y)−→P−→P where P is a ﬁnitely generated submonoid of A (Y, Z) containing NE(Y) with η¯ the inclusion. In this case there are two canonical scattering diagrams, D and D deﬁned ¯ ¯ using η¯ : NE(Y) → Pand η : NE(Y) → P respectively. Then D can be obtained from D essentially just by applying ψ to each exponent appearing in each function f . In this case we show that if consistency holds for D then it holds for D. The idea is that given a broken line γ for D, we can get something like a broken line for D by applying ψ to the exponents of monomials attached to γ . However, this isn’t necessarily a broken line for D. Indeed, there might be two different broken lines for D,say γ and γ , which after we apply ψ give broken lines with the same sequence of attached exponents. These should not arise as distinct broken lines for D, and we have to combine the monomials attached to these broken lines. This requires a certain amount of book- keeping. Step III. Reduction to the Gross-Siebert locus. By Step II we can assume we have a toric ¯ ¯ model p : Y → Y. Let H be an ample divisor on Y. Shrinking P if necessary, we can ∗ ⊥ assume that P has a face of the form P ∩ (p H) . Let G be the monomial ideal which is gp the complement of this face, E the subgroup of P generated by P \ G. The main work in this step is to show that we can replace P by P+ E. This requires a bit of analysis of the can rays (ρ , f ) of D . In particular, we need to understand the contribution to f coming i ρ ρ i i from the exceptional curves of p meeting D . × o After doing this, we have P = E, so now X lives over the thickening of a torus I,D gs T we call the Gross-Siebert locus. Step IV. Pushing the singularities to inﬁnity. This is the crucial step, and we explain carefully the intuition here. In [GS07], Gross and Siebert considered a smoothing con- struction associated to an integral afﬁne manifold with singularities where (in the two- dimensional case) the singularities occurred only in the interior of edges of a polyhedral 108 MARK GROSS, PAUL HACKING, AND SEAN KEEL decomposition of B, rather than at the vertices. The case at hand, with one singularity at the origin, does not ﬁt into that framework. In particular, in the Gross-Siebert world, the 1 k singularities must have monodromy of the form for some k > 0, with the tangent line to the edge containing the singularity being the invariant direction; in analogy with the Kodaira classiﬁcation, we call this an I singularity. Indeed, one expects a cycle of k two-spheres as ﬁbre over such a point in the SYZ picture. Here, we can view such a surface as being obtained by factoring the complicated sin- gularity 0 ∈ B into I singularities along the edges of . We should have an I singularity k k on the ray ρ where k is the number of exceptional divisors of p : Y → Y intersecting D . i i i ¯ ¯ ¯ ¯ This process can be described as follows. Let (B, ) be the fan associated to (Y, D). There is a piecewise linear isomorphism ν : B → B which identiﬁes each cone in with the corresponding cone in . This is an isomor- phism of integral afﬁne manifolds outside of ρ , but it is not afﬁne along ρ . There is a i i natural one-parameter family of integral afﬁne manifolds interpolating between the two structures by a process Kontsevich and Soibelman [KS06]call moving worms. Precisely, choose points y ∈ ρ \{0}.Let := {y | 1 ≤ i ≤ n},B := B \ . Put a new afﬁne struc- i i i ture on B compatible with the afﬁne structures on the interior of each maximal cell by deﬁning a -piecewise linear function to be linear if its restriction to a small neighbour- hood of (y ,+∞) ⊂ ρ in B is B-linear, and its restriction to a small neighbourhood of i i [0, y ) ⊂ ρ in B is B-linear. Call the resulting integral afﬁne manifold with singulari- i i ties B .The map ν : B → B is a linear isomorphism near 0. This new manifold can be seen to have an I singularity at y , with invariant direction ρ . k i i Now if we were to apply the algorithm of Gross and Siebert [GS07]to B ,one would ﬁnd roughly that one obtains a scattering diagram which initially has two rays em- anating from each singularity. The rays emanating from ρ are initially contained in ρ ; i i one of these goes out to inﬁnity and the other passes through the origin and then to in- ﬁnity. Where all these rays meet at the origin, one must follow a procedure of Kontsevich and Soibelman [KS06] and add some additional rays to ensure that the composition of automorphisms associated to the rays about a loop centered at the origin is the identity. We then obtain a scattering diagram which can be shown to be very close to the canoni- cal scattering diagram, the only difference being the segments of the rays between the y and the origin. We do not actually work with this afﬁne manifold with singularities. Rather, we instead push the singularities y to inﬁnity. In doing so, we replace B with B. We transfer the canonical scattering diagram D to a scattering diagram D on B, differing from D essentially only by changing the rays supported on the ρ ’s in a simple way motivated by the above description. Once this is done, we show consistency of D is equivalent to consistency of D. Now we no longer have to deal with any singularities. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 109 It is much easier to determine consistency when there are no singularities. In par- ticular, we appeal to a result in [CPS], which shows that D is consistent provided that the composition of automorphisms associated to the rays about a loop centered at the origin is the identity. We say such a scattering diagram is compatible. The important point is that we can now make sense of such a statement: when we had a singularity at the origin, there was no common ring which the automorphisms associated to rays could act on. However, without a singularity at the origin, there are such rings, as appeared in [GS07]. Step V. D satisﬁes the required compatibility condition. This step is really the punch-line, explaining why the particular choice of the canonical scattering diagram D gives a dia- gram D which is compatible. We make use of [GPS09] to link the enumerative deﬁnition of D to the notion of compatibility. Indeed, the deﬁnition of the canonical scattering dia- gram was originally obtained by working backwards from the enumerative description of [GPS09]. This connection is worked out in Section 3.4. 3.3. Consistency: reduction to the Gross-Siebert locus. — We now begin the proof of The- orem 3.8, following the outline given in Section 3.2. We will, however, prove a number of lemmas in a slightly more general context, as we will need some more general consis- tency results in [K3]. We assume we are given (Y, D), η : NE(Y) → Pand ϕ deﬁned as in Example 2.3, and a radical ideal J ⊆ P. Suppose we are given a scattering diagram D can for this data; the application in this paper will be D = D . In particular, the hypotheses can of Theorem 3.8 imply D is a scattering diagram for this data. Step I. Replacing (Y, D) with a toric blowup. ˜ ˜ Proposition 3.10. —Let p : (Y, D) → (Y, D) be a toric blowup. Then if we take η˜ := η ◦ p : NE(Y) → P, then D can also be viewed as a scattering diagram for B , P.Furthermore, ˜ ˜ (Y,D) if D is consistent for this latter data, it is consistent for the data B , P. (Y,D) ˜ ˜ Proof. — Decorate notation, writing for example B, for the singular afﬁne mani- ˜ ˜ fold with subdivision into cones associated to (Y, D). By Lemma 1.6, we have a canonical ˜ ˜ identiﬁcation of the underlying singular afﬁne manifolds B = B, and is the reﬁnement of obtained by adding one ray for each p-exceptional divisor. We have multivalued piecewise linear functions ϕ on B and ϕ˜ on B. We can in fact choose representatives so ˜ ˜ ˜ that ϕ˜ = ϕ. Indeed, κ = η(p ([D ])) where D is the irreducible component of Dcor- ρ,ϕ˜ ∗ ρ ρ ˜ ˜ responding to ρ.But p ([D ]) = 0if ρ ∈ ,and p ([D ]) =[D ] if ρ ∈ .Thus ϕ˜ in ∗ ρ ∗ ρ ρ fact has the same domains of linearity as ϕ, and the same bending parameters, so we can choose representatives which agree. As a consequence, we note that the sheaves P and P on B deﬁned using ϕ and ϕ˜ coincide. Furthermore, if τ˜ ⊂˜ σ are cones in ,with τ ∈ the smallest cone containing τ˜ and σ ∈ the smallest cone containing σ˜ , there is a canonical identiﬁcation of P with P and a canonical isomorphism ϕ˜ τ˜ (3.3) R R ; τ, ˜ I τ,I 110 MARK GROSS, PAUL HACKING, AND SEAN KEEL note the slightly non-trivial case when dim τ˜ = 1 but dim τ = 2, in which case we use the fact that κ = 0. τ, ˜ ϕ˜ Using these identiﬁcations, we can view D as living on B, and as such, one sees from the deﬁnition that D is a scattering diagram for the data B, P, ϕ˜ . Now suppose I = J. One observes that the set of broken lines contributing to ˜ ˜ Lift (q) are the same whether we are working in B or B. Thus if Q∈˜ σ ∈ ,Lift (q) ∈ Q max Q ˜ ˜ R , deﬁned using B, coincides under the isomorphism (3.3) with Lift (q) ∈ R .From σ, ˜ I Q σ,I this one sees easily that if D is consistent for Y, it is consistent for Y. Corollary 3.11. —Given Y, P,η, J satisfying the hypotheses of Theorem 3.8, then Theo- rem 3.8 holds for this data if it holds for the data Y, P, η, ˜ J. Proof. — By the proposition, one just needs to check that the canonical scattering diagrams deﬁned using Y or Y are identical. Indeed, given a ray d ⊂ B, we can choose ˜ ˜ a reﬁnement of which is also a reﬁnement of , giving maps π˜ : Y → Yand π : Y → Y. Then for an A -class β ∈ A (Y , Z), η(π (β))=˜ η(π˜ (β)),and so f is the 1 ∗ ∗ d same for Yand Y. Step II. Changing the monoid P. We would like to change the monoid P, which was fairly arbitrary, to one with better properties. For this step, assume we are given monoid homomorphisms η¯ ψ NE(Y)−→P−→P with η = ψ ◦¯ η.Then η and η¯ induce multivalued piecewise linear functions ϕ and ϕ¯ respectively, via Example 2.3,with ϕ = ψ◦¯ ϕ. The monoids P,P and functions ϕ, ϕ¯ yield ¯ ¯ ¯ sheaves P and P over B .The map ψ : P → P induces a map of sheaves ψ : P → P using ϕ = ψ ◦¯ ϕ, and hence it also induces monoid homomorphisms ψ : P → P for ϕ¯ ϕ τ τ any τ ∈ \{0}. ¯ ¯ ¯ Suppose D is a scattering diagram for the data B, P, m¯ = P \ P .For each ray (d, f ), f ∈ k[P ].Now we cantry to deﬁne ψ(f ) by applying ψ to each exponent of d d ϕ¯ d f , but in general, this need not make sense even formally since ψ may take an inﬁnite number of exponents occurring in f to a single element of P. However, we shall write ψ(f ) for such an expression if it does make sense as an element of k[P ].If ψ(f ) makes d ϕ d sense for each (d, f ) ∈ D, we write ¯ ¯ ψ(D) = d,ψ(f ) | (d, f ) ∈ D . d d Proposition 3.12. — In the above situation, suppose D is a scattering diagram for the data ¯ ¯ ¯ ¯ B, P, m¯ = P\ P , such that D = ψ(D) makes sense and is a scattering diagram for the data B, P, J, where J is a radical ideal in P. Assume that κ ∈ J for at least one ray ρ ∈ .If D is consistent for ρ,ϕ P, η, ¯ m¯ , then D is consistent for P,η, J. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 111 Proof.—Let q ∈ B (Z).Thenif γ¯ is a broken line for q with endpoint Q with respect to the barred data, i.e., P, P etc., we can construct what we shall call ψ(γ) ¯ . This will be the data required for deﬁning a broken line for the unbarred data. The underlying map of ψ(γ) ¯ coincides with that of γ¯ . For the attached monomials, we simply apply ψ to the monomial m (γ) ¯ attached to a domain of linearity L of γ¯ to get the attached monomial for ψ(γ) ¯ . This is not a broken line for the unbarred data, as condition (3) of Deﬁnition 2.16 need not hold. Indeed, when a broken line bends at a ray, the attached monomial will be replaced by a term in (2.9). However, there might be several different ¯ s terms c z appearing in (2.9)with ¯ s ∈ P such that ψ(¯ s ) all coincide with some s ∈ P . i i ϕ i ϕ τ τ ¯ s Each choice c z leads to a different broken line γ¯ ,but ψ(γ¯ ) is not a broken line because i i i ψ(¯ s ) s c z = c z is not a term in the formula (2.9) for the monoid P. Rather, one needs to i i replace the collection of broken lines γ¯ with a single one which has monomial c z i i attached after the bend. To deal with this, we need to do a certain amount of book- keeping. Fix an ideal I ⊂ Pwith I = J, Q ∈ σ ∈ ,and let B be the set of broken lines γ¯ for the barred data with endpoint Q such that ψ(Mono(γ) ¯ ) ∈ I · k[P ].The same ﬁniteness argument of Lemma 2.25 shows that B is a ﬁnite set. Note this uses the facts (1) at least one κ ∈ J and (2) all but a ﬁnite number of monomials appearing in D lie ρ,ϕ in I. We deﬁne an equivalence relation on B by saying γ¯ ∼¯ γ provided ψ(γ¯ ) and 1 2 1 ψ(γ¯ ) coincide except possibly for the k-valued coefﬁcients of the monomials attached to the domains of linearity. Given an equivalence class ξ ⊂ B with respect to this equiv- alence relation, we will show there is at most one broken line γ for the unbarred data such that (3.4) ψ Mono(γ) ¯ = Mono(γ ), γ¯∈ξ with there being no such broken line precisely if the above quantity is zero. Furthermore, every broken line γ for the unbarred data with Mono(γ ) ∈ I · k[P ] arises in this way. Deﬁne γ to be the broken line with underlying piecewise linear map given by any element of ξ , with the following attached monomials. For any domain of linearity L=[s, t] for γ , choose a maximal subset ξ ⊂ ξ of broken lines such that the attached ξ L monomials for γ¯ and γ¯ on (−∞, t] do not coincide for any γ¯ , γ¯ ∈ ξ . Then deﬁne 1 2 1 2 L m (γ ) = m ψ(γ) ¯ . L ξ L γ¯∈ξ Assuming that the ﬁnal monomial attached to γ is not zero, one checks easily that γ is a ξ ξ broken line, now satisfying (3) of Deﬁnition 2.16,and (3.4) is satisﬁed since for L the last domain of linearity of γ , one takes ξ = ξ . Furthermore, it is easy to see that any broken ξ L line for the unbarred data with the same underlying map and attached monomials at 112 MARK GROSS, PAUL HACKING, AND SEAN KEEL most differing by their coefﬁcients from γ must in fact coincide with γ . This shows the ξ ξ claim. ¯ ¯ ¯ Since B is ﬁnite, there is some k > 0such that for any γ¯ ∈ B,Mono(γ) ¯ ∈ k[P ] k −1 ¯ ¯ does not lie in m¯ · k[P ].Ifwetake I = ψ (I),thenitisclear from (3.4)that (3.5) ψ Lift (q) = Lift (q), Q Q where Lift (q) is the lift deﬁned with respect to the ideal I and the other barred data, and Lift (q) is deﬁned with respect to the unbarred data and the ideal I. Now D is consistent for m¯ , which implies (1) and (2) of Deﬁnition 2.26 hold for the ideal m¯ + I. Since any ¯ ¯ ¯ ¯ monomial in P \ I appearing in Lift (q) is in P \ (m + I),wecan use(3.5) to deduce consistency of D from consistency of D. Step III. Reduction to the Gross-Siebert locus. As a consequence of Proposition 1.3 and can Corollary 3.11, in order to prove Theorem 3.8 (i.e., with D = D ), we may assume we ¯ ¯ ¯ ¯ ¯ have a toric model p : (Y, D) → (Y, D) with D = D +···+ D . Furthermore, by replac- 1 n ing (Y, D) with a deformation equivalent pair and using Lemma 3.9, we can assume that p is the blowup at distinct points x ,1 ≤ j ≤ ,along D , with exceptional divisors E . ij i i ij Assume D is the proper transform of D , corresponding to the ray ρ ∈ . i i i By Proposition 3.12, we can replace P with a better suited choice of monoid. We can shall do this as follows in the case that D = D .AsinExample 3.6, the nef cone K(Y) ⊂ 1 ∨ A (Y, R) contains a strictly convex rational polyhedral cone σ,so σ ⊂ A (Y, R) is a strictly convex rational polyhedral cone containing NE(Y).The map η : NE(Y) → P gp induces a map η : A (Y, R) → P . Since P is toric, there is some rational polyhedral gp gp cone σ ⊂ P such that P = σ ∩ P . In addition, let H be an ample divisor on Y, so that P P ∗ ⊥ NE(Y) ∩ (p H) is a face of NE(Y), generated by the classes [E ].Now take ij −1 ∨ ∗ σ = η (σ ) ∩ σ ∩ q ∈ A (Y, R)| p H · q ≥ 0 , P 1 and take P = σ ∩ A (Y, Z). ¯ ¯ ¯ As σ is strictly convex, (P) ={0}, m¯ = P\{0}, and if Iis an m¯ -primary ideal, ¯ ¯ ¯ P \ I is ﬁnite. Thus the hypotheses of Theorem 3.8 trivially hold for η¯ : NE(Y) → P. By Proposition 3.12, we can replace P with Pto prove Theorem 3.8. The above discussion shows that in order to complete a proof of consistency of can D , (i.e., Theorem 3.8), we can operate under the following assumptions: Assumptions 3.13. • There is a toric model ¯ ¯ p : (Y, D) → (Y, D) which blows up distinct points x on D , with exceptional divisors E . ij i ij MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 113 • η : NE(Y) → P is an inclusion, and P ={0}. Via Example 2.3, this gives the function ϕ. ∗ ⊥ • There is a face of P whose intersection with NE(Y) is NE(Y) ∩ (p H) .Let G be the prime monomial ideal given by the complement of this face. Note that G = m unless p is an isomorphism. • J = m = P\{0}. • D is a scattering diagram for the data P,ϕ and J. gs Deﬁnition 3.14. —The Gross-Siebert locus is the open torus orbit T of Spec k[P]/G. We now want to work not with the maximal ideal m but with the ideal G, effec- tively extending the families X with I = m to inﬁnitesimal neighbourhoods of the I,D toric boundary stratum of Spec k[P] associated to G. We will then ﬁnd it easier to check the explicit equalities of Deﬁnition 2.26 after restricting to these neighbourhoods of the Gross-Siebert locus. To do so requires showing that the diagram D we are working with can can (D in this paper) is also a scattering diagram for the data P,η, G. In the case of D , this requires analyzing elements of this scattering diagram supported on the ρ . can We ﬁrst perform this analysis for D ; we will then continue our proof assuming that the elements of D supported on the rays ρ take the same form as the corresponding can elements of D modulo G. can For each ray ρ in , we have a unique ray (ρ , f ) ∈ D with support ρ .The i i ρ i following describes f mod G. can Lemma 3.15. — Given Assumption 3.13 with D = D ,viewing f as an element of k[P ]⊗ R with I = m,wehave ϕ I −1 f = g 1 + b X ρ ρ ij i i i j=1 η([E ]) 1 ij where b = z and g ≡ 1mod G. The jth term of the product is the contribution from A -classes ij ρ coming from multiple covers of the p-exceptional divisor E ,and g is the product of contributions from ij ρ all other A -classes. Proof. — Note that in deﬁning f using the deﬁnition of the canonical scattering diagram, we take Y = Y. Now the only terms that contribute to f mod G will involve classes β ∈ NE(Y) ⊂ A (Y) with η(β) ∈ G, so in particular, such a β must be a linear combination c [E ],with k = c . Furthermore, if f : C → Y contributes to N , j ij β j β j=1 f (C) must be contained in E .Indeed,if f (C) has an irreducible component C ij i,j not contained in this set, then η([C ]) ∈ G, so η(f ([C])) ∈ G, as G is an ideal. But η(f ([C])) = η(β), which we have assumed is not an element of G. Since f (C) is connected and intersects D , we now see that the image of f is E for i ij some j , and in particular, f is a degree k cover of E .ThenTheorem 6.1of[GPS09] β ij 114 MARK GROSS, PAUL HACKING, AND SEAN KEEL k −1 2 tells us that the contribution from k -fold multiple covers of E is (−1) /k . From this β ij we conclude that k−1 (−1) −1 f = exp h + k b X ρ ij k=1 j=1 −1 = exp(h) 1 + b X ij j=1 where h ≡ 0mod G. We take g = exp(h). can Corollary 3.16. — D is a scattering diagram for the data (B, ), P, ϕ and G. Proof. — Fixing an I ⊂ Pwith I = G, there exists a bound n such that q ∈ P \ I ∗ 1 implies q · p H < n, where H is a ﬁxed ample divisor on Y. Thus if β is an A -class with η(π (β)) ∈ P\ I, there are only a ﬁnite number of choices for p π β . We need to examine ∗ ∗ ∗ the possible choices for π β . Given a choice for α = p π β,wehave π β = p α + a E ∗ ∗ ∗ ∗ ij ij for some collection of a ∈ Z. Clearly the a are bounded below by the requirement that ij ij (π β) · D ≥ 0for each i. On the other hand, if a > 0for some i, j,then (π β) · E < 0, ∗ i ij ∗ ij so if f : C → Yis an A -curve with f [C]= π β then its reduced image C must contain ∗ ∗ E . (Technically a relative stable map in Y is a map to an expanded degeneration of ij ◦ ◦ ◦ ˜ ˜ ˜ Y , but we compose with the projection to Y and then the natural map Y → Y.) Write C = C ∪ E ,with C a reduced divisor distinct from E . Suppose C is non-empty. ij ij Necessarily either C ∩ Dis empty or C intersects D only at E ∩ D; otherwise C cannot ij be the image of a relative stable map with one point of tangency with D. In either case there is an integer k such that O (C + kE )| is the trivial sheaf. However, by [GHK12], Y ij D Proposition 4.1, for a general deformation (Y , D) of (Y, D), the kernel of the restriction map Pic Y → Pic D is trivial. Thus by Lemma 3.9,N = 0. We conclude that there are only a ﬁnite number of choices of π β,exceptwhen π β is a multiple of some E . This ∗ ∗ ij shows condition (4) in Deﬁnition 2.13 of scattering diagrams, as well as condition (2). Note that κ =[D]∈ Gfor each i so condition (3) is vacuous for dim τ = 1. If dim τ = 2, ρ ,ϕ i d d any contributing A -class β satisﬁes π β ∈ G, so (3) holds. can Theorem 3.17. — We follow the above notation. If D is consistent as a scattering diagram for (B, ), P, ϕ,and G, then Theorem 3.8 is true. Proof. — This just follows from the series of reductions of Theorem 3.8 already made and the observation that if I is an m-primary ideal, then since G ⊂ m one can ﬁnd some k such that kG ⊂ I . To show consistency holds for the ideal I , we use the assumed consistency to observe consistency holds for the ideal I = kG, and this gives the desired result. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 115 Remark 3.18. — Given a consistent scattering diagram D for (B, ),P, ϕ,and G, and κ ∈ Gfor all rays ρ ∈ ,Theorem 2.28 shows that with I = G, ρ,ϕ X := Spec X , O I X I,D I,D is ﬂat over Spec k[P]/I, and X = V × Spec k[P]/G. G n gs gs Let T ⊂ Spec k[P]/G be the Gross-Siebert locus, Deﬁnition 3.14.Note T de- termines open subschemes of the thickenings Spec k[P]/I, which we will shall denote gs by T . gs gp We can describe the subscheme T of Spec k[P]/I as follows. Let E ⊂ P be the gs lattice generated by the face P \ G. Then as a subset of Spec k[P]/G, T Spec k[E]. gs Furthermore, if we take the localization P + EofPalong the face P \ G, then T as a subscheme of Spec k[P]/IisSpec k[P + E]/(I + E). Note that m = (P + E) \ E, and G = P ∩ m , so we can write k[E]= k[P + P+E P+E E]/m . P+E We can now view ϕ as a multivalued strictly (P + E)-convex function. Then we have the following obvious Lemma 3.19. — Suppose D is a consistent scattering diagram for the data (B, ), P + E, ϕ, m , and a scattering diagram for the data (B, ), P, ϕ,and G. Then D is also consistent as P+E can a scattering diagram for the latter data. In particular, by Theorem 3.17,Theorem 3.8 holds if D is consistent as a scattering diagram for P + E, m . P+E Proof. — Since P ∩ m = G, the equalities in Deﬁnition 2.26 can be tested for P+E √ √ an ideal I of P with I = G by choosing some ideal I ⊆ P + Ewith I = m and P+E I ∩ P ⊂ I. Then the equalities of Deﬁnition 2.26 hold for the data P + Eand I by the assumed consistency, and hence also for P and I. ¯ ¯ ¯ Let I ⊂ m be an ideal with I = m .Set I = I ∩ P. Then X is ﬂat over P+E P+E I,D o o Spec k[P]/I. Restricting X to the open set Spec k[P + E]/Igives the ﬂat family X . I,D ¯ I,D We now replace P by P + Eand Jby m in what follows. We now summarize P+E our current situation with the following assumptions: Assumptions 3.20. • There is a toric model ¯ ¯ p : (Y, D) → (Y, D) which blows up distinct points x on D , 1 ≤ j ≤ , with exceptional divisors E . ij i i ij • η : NE(Y) → P is an inclusion. Via Example 2.3, this gives the function ϕ. E = P = ∗ ⊥ P ∩ (p H) is generated by the classes of exceptional curves of p. Let G = P \ E = m . P 116 MARK GROSS, PAUL HACKING, AND SEAN KEEL • D is a scattering diagram for the data P,ϕ and G. Furthermore, for each ray ρ ∈ ,the unique outgoing ray (ρ , f ) ∈ D satisﬁes i ρ −1 f = g 1 + b X ρ ρ ij i i i j=1 [E ] ij with g ≡ 1mod G and b = z . ρ ij can We note we have shown that D = D achieves these assumptions. Step IV. Pushing the singularities to inﬁnity. We work with Assumptions 3.20.Consider ¯ ¯ ¯ ¯ ¯ the tropicalisation (B, ) of (Y, D). By Example 1.7, B in fact has no singularity at the 2 2 origin, and is afﬁne isomorphic to M = R (with M = Z ), while is precisely the fan ¯ ¯ ¯ for Y. In order to distinguish between constructions on (Y, D) and (Y, D),wedecorate all existing notation with bars. For example, if τ ∈ , denote the corresponding cone of gp ¯ ¯ by τ¯.Let ϕ¯ be the multivalued P -valued function on Bsuch that (3.6) κ = p [D ]. ρ, ¯ ϕ¯ ρ¯ Note that by Lemma 1.13, we can assume ϕ¯ is in fact a single-valued function on M . This single-valuedness will be important to be able to apply the method of Kontsevich and Soibelman, Theorem 3.23. We now have sheaves P on B and P on B , induced by the two functions ϕ¯ and 0 0 ϕ respectively. Note that since ϕ¯ is single-valued and B has no singularities, P is the constant sheaf gp with ﬁbre P ⊕ M. There is a canonical piecewise linear map ν : B → B which restricts to an integral afﬁne isomorphism ν| : σ →¯ σ ,where σ ∈ and σ¯ ∈ σ max ¯ ¯ ¯ is the corresponding cell of . Note this map identiﬁes B(Z) with B(Z). max For each maximal cone σ ∈ , the derivative ν of ν induces a canonical iden- max ∗ tiﬁcation of with ¯ . This then gives an induced isomorphism of monoids: B,σ B,σ (3.7) ν˜ : P → P σ ϕ ϕ¯ σ σ¯ given by ϕ (m) + p → ϕ¯ ν (m) + p, σ σ¯ ∗ for p ∈ Pand m ∈ . This identiﬁes the k[P]-algebras k[P ] and k[P ],and thecom- σ ϕ ϕ¯ σ σ¯ pletions k[P ] and k[P ]. ϕ ϕ¯ σ σ¯ MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 117 Because the map ν is only piecewise linear around rays ρ ∈ , there is only a piecewise linear identiﬁcation of P with P and hence no identiﬁcation of the corre- ϕ ϕ¯ ρ ρ¯ sponding rings. However, ν is still deﬁned on the tangent space to ρ,and thereisan identiﬁcation ν˜ : ϕ (m)+ p| m is tangent to ρ, p ∈ P → ϕ¯ (m)+ p| m is tangent to ρ, ¯ p ∈ P ρ ρ ρ¯ given by ϕ (m) + p → ϕ¯ ν (m) + p. ρ ρ¯ ∗ We now explain the Kontsevich-Soibelman lemma. This has to do with scattering diagrams on the smooth afﬁne surface M = R (such as B = B ). For this general ¯ ¯ R (Y,D) discussion, we ﬁx the data of a monoid Q which comes along with a map r : Q → M. Let m = Q\ Q ,and let k[Q] denote the completion of k[Q] with respect to the monomial ideal m . (In our application we take Q = P as deﬁned in (1.5).) Q ϕ¯ We can then consider a variant of the notion of scattering diagram: Deﬁnition 3.21. —Wedeﬁne a scattering diagram for the pair Q, r : Q → M.Thisis aset D = (d, f ) where • d ⊂ M is given by d=−R m ≥0 0 if d is an outgoing ray and d = R m ≥0 0 if d is an incoming ray,for some m ∈ M\{0}. • f ∈ k[Q]. • f ≡ 1mod m . d Q • f = 1 + c z for c ∈ k,r(p) = 0 a positive multiple of m . d p p 0 • For any k > 0, there are only a ﬁnite number of rays (d, f ) ∈ D with f ≡ 1mod m . d d Deﬁnition 3.22. — Given a loop γ in M around the origin, we deﬁne the path ordered product θ : k[Q]→ k[Q] γ,D 118 MARK GROSS, PAUL HACKING, AND SEAN KEEL as follows. For each k > 0,let D[k]⊂ D be the subset of rays (d, f ) ∈ D with f ≡ 1mod m . d d This set is ﬁnite. For d ∈ D[k] with γ(t ) ∈ d,deﬁne k k k θ : k[Q]/m → k[Q]/m γ,d Q Q by n ,r(q) k q q d θ z = z f γ,d for n ∈ M primitive satisfying, with m a non-zero tangent vector of d, n , m= 0, n ,γ (t ) < 0. d d 0 Then, if γ crosses the rays d ,..., d in order with D[k]={d ,..., d }, we can deﬁne 1 n 1 n k k k θ = θ ◦···◦ θ . γ,D γ,d γ,d n 1 We then deﬁne θ by taking the limit as k →∞. γ,D The following is a slight generalisation of a result of Kontsevich and Soibelman which appeared in [KS06]. Theorem 3.23. —Let D be a scattering diagram in the sense of Deﬁnition 3.21. Then there is another scattering diagram Scatter(D) containing D such that Scatter(D) \ D consists only of outgoing rays and θ is the identity for γ a loop around the origin. γ,Scatter(D) For a proof of this theorem essentially as stated here, see [GPS09], Theorem 1.4. The result is unique if Scatter(D) \ D has at most one ray in each possible direction; we shall assume Scatter(D) has been chosen to have this property. This can always be done. We apply this in the following situation. We take Q to be the monoid P which ϕ¯ ¯ ¯ ¯ yields the Mumford degeneration associated to the data (B, ), ϕ¯ (recalling B = M ), deﬁned by gp P = m, ϕ( ¯ m) + p | m ∈ M, p ∈ P ⊂ M × P . ϕ¯ This comes with a canonical map r : P → M by projection. ϕ¯ Deﬁnition 3.24. — Suppose we are in the situation of Assumptions 3.20. We deﬁne a scattering diagram ν(D) on B as follows. For every ray (d, f ) ∈ D not equal to (ρ , f ) for some i, ν(D) d i ρ contains the ray (ν(d), ν˜ (f )), and for each ray (ρ , f ), ν(D) contains two rays, (ρ¯ , ν˜ (g )) and τ d i ρ i τ ρ d i d i i −1 (ρ¯ , (1 + b X )). i i j=1 ij We note that ν(D) may not actually be a scattering diagram in the sense of Deﬁnition 3.21,as it is possible that f ∈ k[P ]:if p ∈ P , then ν˜ (p) ∈ P but need not lie in P . d ϕ¯ ϕ τ ϕ ϕ τ MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 119 can In the case of D = D , we can use the Kontsevich-Soibelman lemma to describe can can ν(D ). This will both show that ν(D ) is a scattering diagram in the sense of Deﬁni- tion 3.21 and that it satisﬁes an important additional property, namely the condition that θ can is equal to the identity. This will allow us to prove consistency. Let γ,ν(D ) −1 ¯ ¯ (3.8) D = ρ¯ , 1 + b X 1 ≤ i ≤ n . 0 i i ij j=1 ¯ ¯ Let m = P \ P as usual. Then by the strict convexity of ϕ¯ , X ∈ m so that D is a ϕ¯ ϕ¯ i ϕ¯ 0 ϕ¯ scattering diagram for the pair P , r in the sense of Deﬁnition 3.21.Now deﬁne ϕ¯ ¯ ¯ D := Scatter(D ) ¯ ¯ where we require D\ D to have only one outgoing ray in each direction (and no incom- ing rays). The following will be Step V, which we defer until Section 3.4. can can Theorem 3.25. — D = ν(D ). In particular, ν(D ) is a scattering diagram in the sense of Deﬁnition 3.21 and θ ≡ 1 for a loop γ around the origin. γ,D Example 3.26. — Continuing with Example 3.7, note that the pair (Y, D) can be ¯ ¯ ¯ obtained from the toric pair (Y, D) deﬁned by the fan with rays generated by (1, 0), ¯ ¯ (1, 1), (0, 1), (−1, 0) and (0,−1), corresponding to D ,..., D , by blowing up one point 1 5 ¯ ¯ ¯ ¯ on each of D and D . This description determines D and hence D. One can check this 4 5 0 can description agrees with that given in Example 3.7 for D , see e.g. [GPS09], Example 1.6 for a similar computation. Returning to the situation of Assumptions 3.20, suppose in addition that ν(D) is a scattering diagram in the sense of Deﬁnition 3.21.(Forexample,byTheorem 3.25, can D = D satisﬁes these assumptions.) For I ⊂ P an ideal with I = J, we now have o o deformations X and X . The latter scheme is glued from open sets I,D I,ν(D) U = Spec R ρ, ¯ I ρ, ¯ I along open sets identiﬁed with Spec R . Here we are decorating the rings coming from σ, ¯ I the data on B with bars as before, while we maintain the notation R , etc., for those ρ,I rings coming from the data on B. Lemma 3.27. — Given Assumptions 3.20, assume also that ν(D) is a scattering diagram in the sense of Deﬁnition 3.21. Then there are isomorphisms p : R → R i ρ ,I ρ¯ ,I i i 120 MARK GROSS, PAUL HACKING, AND SEAN KEEL and p : R → R i−1,i σ ,I σ¯ ,I i−1,i i−1,i for all i such that the diagrams ψ ψ ρ ,− ρ ,+ i i R R R R ρ ,I σ ,I ρ ,I σ ,I i i−1,i i i,i+1 p p p p i i−1,i i i,i+1 R R R R ρ¯ ,I σ¯ ,I ρ¯ ,I σ¯ ,I i i−1,i i i,i+1 ψ ψ ρ¯ ,− ρ¯ ,+ i i and γ,D R R σ ,I σ ,I i−1,i i−1,i p p i−1,i i−1,i R R σ¯ ,I σ¯ ,I i−1,i i−1,i γ, ¯ ν(D) are commutative, where γ is any path in σ for which θ is deﬁned, and γ¯ = ν ◦ γ . i−1,i γ,D Consequently, the maps p and p induce an isomorphism i i−1,i o o p : X → X I,D I,ν(D) over Spec k[P]/I. Proof.—Recall that R [X , X , X ] I i−1 i+1 (3.9) R = , ρ,I −D i −1 η([D ]) (X X − z X g (1 + b X )) i−1 i+1 ρ ij i i j=1 i ¯ ¯ ¯ R [X , X , X ] I i−1 i+1 (3.10) R = . ρ,I −D ∗ ¯ −1 i i η(p [D ]) ¯ ¯ i ¯ ¯ (X X − z X g¯ (1 + b X )) i−1 i+1 ρ i i i ij j=1 We simply deﬁne p to be the identity on R and p (X ) = X . This makes sense i I i j j 2 2 ∗ i ¯ ¯ since D = D − and [D]= p [D]− E ,sothat i i i ij i i j=1 i i i 2 2 −D ∗ −D η([D ]) i −1 η(p [D ]) i −1 −1 i i ¯ ¯ ¯ p z X 1 + b X = z X b X 1 + b X i ij i ij i i i ij i j=1 j=1 j=1 ∗ −D η(p [D ]) i −1 ¯ ¯ = z X 1 + b X . ij j=1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 121 The map p is induced by ν˜ deﬁned in (3.7). It is then straightforward to check the i−1,i σ i−1,i commutativity of the three diagrams. Lemma 3.28. — Given Assumptions 3.20,suppose ν(D) is a scattering diagram in the sense of Deﬁnition 3.21.For Q ∈ σ , we distinguish between i−1,i Lift (q) ∈ R Q σ ,I i−1,i for the lift of q ∈ B (Z) and Lift ν(q) ∈ R ν(Q) σ¯ ,I i−1,i the lift of ν(q). Then (1) p (Lift (q)) = Lift (ν(q)). i−1,i Q ν(Q) gp gp (2) Under the natural identiﬁcations (P ) = (P ) ,for τ ∈ \{0}, P ⊂ P ,and for ϕ¯ ϕ¯ ϕ¯ ϕ¯ τ τ any broken line γ for q, Mono(γ ) ∈ k[P ]. ϕ¯ (3) ν induces a bijection between broken lines: If γ : (−∞, 0]→ B is a broken line in B , 0 0 ¯ ¯ then ν ◦ γ is a broken line in B , and conversely, if γ¯ : (−∞, 0]→ P is a broken line in −1 B , then ν ◦¯ γ is a broken line in B . 0 0 Proof. — (3) implies (1). For (3), clearly it is enough to compare bending and at- tached monomials of broken lines near a ray ρ . Consider a broken line γ in B passing from σ to σ ,and let cz be the 0 i−1,i i,i+1 monomial attached to the broken line before it crosses over ρ ,sothat q ∈ P .Let θ , i ϕ ρ σ i i−1,i θ be deﬁned by ρ¯ p p n,r(p) θ z := z f i ρ ¯ n,r¯(p) p p −1 ¯ ¯ θ z := z g¯ 1 + b X ρ¯ ρ i i i ij j=1 where (ρ¯ , g¯ ) ∈ ν(D) is the outgoing ray with support ρ¯ .Here n, n¯ are primitive cotan- i ρ i gent vectors vanishing on tangent vectors to ρ , ρ¯ and positive on σ , σ¯ respectively. i i i−1,i i−1,i Then we need to show that q q (3.11) p θ cz = θ p cz i,i+1 ρ ρ¯ i−1,i i i to get the correspondence between broken lines. Note that ¯ ¯ ¯ p (X ) = X , p (X ) = X , p (X ) = X , i−1,i i−1 i−1 i,i−1 i i i,i+1 i i but to compute p (X ), we need to use the relation (see Proposition 2.5) i,i+1 i−1 −D η([D ]) X X = z X i−1 i+1 i 122 MARK GROSS, PAUL HACKING, AND SEAN KEEL in k[P ] to write −D η([D ]) i −1 X = z X X . i−1 i i+1 On the other hand, one has the relation −D η(p [D ]) i i ¯ ¯ ¯ X X = z X i−1 i+1 in k[P ],so ϕ¯ ρ¯ 2 2 −D +D η([D ]−p [D ]) i i i i ¯ ¯ p (X ) = z X X i,i+1 i−1 i−1 i −1 ¯ ¯ = X X b . i−1 ij j=1 Thus using Assumptions 3.20 for the form of f ,wehave p θ (X ) = p (X f ) i,i+1 ρ i−1 i,i+1 i−1 ρ i i i i i −1 −1 ¯ ¯ ¯ = X X b 1 + b X g¯ i−1 ij ρ ij i i j=1 j=1 −1 ¯ ¯ = X g¯ 1 + b X i−1 ρ i ij j=1 = θ p (X ) ρ¯ i−1,i i−1 as desired. Also, ¯ ¯ ¯ p θ (X ) = X = θ p (X ) . i,i+1 ρ i i ρ¯ i−1,i i i i Thus (3.11) holds. This shows (3). For (2), the statement that P ⊂ P is obvious. For q ∈ σ ∈ , by deﬁnition the ϕ¯ ϕ¯ ϕ (q) monomial attached to the ﬁrst domain of linearity of a broken line for q is z , which (ν(q),ϕ( ¯ ν(q))) is identiﬁed under ν˜ with z ∈ k[P ].For any (d, f ) ∈ ν(D), f ∈ k[P ] by σ ϕ¯ d d ϕ¯ assumption, and hence all monomials associated to broken lines in B lie in k[P ], 0 ϕ¯ hence (2). Deﬁnition 3.29. —Let D be a scattering diagram in the sense of Deﬁnition 3.21 for the pair P, r : P → M for some toric monoid P.Let I ⊂ P be an ideal with I = m .Wedeﬁne for q ∈ B (Z) P 0 and Q ∈ B , Lift (q) = Mono(γ ) ∈ k[P]/I Q MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 123 where the sum is over all broken lines γ for q with endpoint Q in B with respect to the scattering diagram D. One sees easily as in Lemma 2.25 that this is a ﬁnite sum. The last crucial result we need for consistency is the following result of [CPS]. Theorem 3.30. — With the assumptions of Deﬁnition 3.29, suppose furthermore that θ ≡ 1 γ,D for a loop γ around the origin. Fix an ideal I ⊂ P with I = m and q ∈ B (Z).If Q, Q ∈ P 0 M \ Supp(D ) are general, and γ is a path connecting Q and Q for which θ is deﬁned, then R I γ,D Lift (q) = θ Lift (q) Q Q γ,D as elements of k[P]/I. Proof. — This is shown in [CPS] in a rather more general setup. For a version of the argument closer to the current setup, see the proof of Theorem 5.35 of [G11]. Proof of Theorem 3.8. — By Lemmas 3.15 and 3.19, we can assume we are in the can can situation of Assumptions 3.20 with D = D . In checking (1) of Deﬁnition 2.26 for D in this situation, we want to check equalities can Lift (q) = θ Lift (q) Q γ,D Q for Q, Q ∈ σ . By Lemmas 3.27 and 3.28, it is sufﬁcient to show that i−1,i (3.12)Lift ν(q) = θ Lift ν(q) . ν(Q ) γ, ¯ ν(D) ν(Q) To check this equality we can compare coefﬁcients of monomials, and given any mono- mial z appearing on the left- or right-hand sides, we can apply Theorems 3.25 and 3.30,where we take P = P ,I = m for sufﬁciently large k so that p ∈ I. The hypothesis ϕ¯ ϕ¯ θ ≡ 1of Theorem 3.30 holds by Theorem 3.25. γ,ν(D) To show (2) of Deﬁnition 2.26,wecan take Q = Q and Q = Q on opposite − + sides of a ray ρ .If γ is a short path joining Q and Q ,westill have (3.12) after inverting f .Wehaveamap ψ := (ψ ,ψ ) : R → R × R . ρ¯ ,− ρ¯ ,+ ρ ,I σ ,I σ ,I i i i−1,i i,i+1 i −1 If f =¯ g (1 + b X ) then ψ is given by i ρ i i j=1 ij ϕ (v ) ϕ (v ) i−1 i−1 ρ ρ i i X → z , f z , i−1 i ϕ (v ) ϕ (v ) ρ i ρ i i i X → z , z , ϕ (v ) ϕ (v ) i+1 i+1 ρ ρ ¯ i i X → f z , z . i+1 i 124 MARK GROSS, PAUL HACKING, AND SEAN KEEL One checks easily that this map is injective. Furthermore, the image is described as fol- lows. Let I ⊂ P be the monoid ideal ρ¯ ϕ¯ i ρ¯ I = q ∈ P | q − ϕ r(q) ∈ Ior q − ϕ r(q) ∈ I . ρ¯ ρ σ σ i i−1,i i,i+1 Then the image consists of those elements (g , g ) such that every monomial of g and − + − g has exponent in P ⊂ P , P , and the images g¯ of g in (k[P ]/I ) satisfy + ϕ¯ ϕ¯ ϕ¯ ± ± ρ ρ¯ f ρ¯ σ σ i i i i i−1,i i,i+1 θ (g¯ )=¯ g , where this makes sense as we have localized at f . (See e.g., the proof of γ,ν(D) − + i Lemma 2.34 in [GS07] for a similar statement.) Thus by (3.12) and Lemma 2.25,(2), thereisan α ∈ R such that ρ ,I ψ (α) = Lift ν(q),ψ (α) = Lift ν(q) . ρ ,− ν(Q) ρ ,+ ν(Q ) i i −1 Thus we may take Lift (q) = p (α), and by Lemma 3.28, ψ (Lift (q)) = Lift (q), ρ ρ ,± ρ Q i i i i ± giving consistency. 3.4. Step V: the proof of Theorem 3.25 and the connection with [GPS09]. — Here we derive Theorem 3.25 from the main result of [GPS09]. We will need to review one form of this result, which gives an enumerative interpretation for the output of the Kontsevich- Soibelman lemma. Fix M = Z as usual. Suppose we are given positive integers ,..., and prim- 1 n itive vectors m ,..., m ∈ M. Let = and Q = M ⊕ N ,with r : Q → Mthe 1 n i i=1 projection. Denote the variables in k[Q] corresponding to the generators of N as t ,for ij 1 ≤ i ≤ n and 1 ≤ j ≤ . Consider the scattering diagram for the data r : Q → M(in the sense of Deﬁnition 3.21) D = R m , 1 + t z | 1 ≤ i ≤ n . ≥0 i ij j=1 We wish to interpret (d, f ) ∈ Scatter(D) \ D. Choose a complete fan in M which d d R contains the rays R m ,..., R m as well as the ray d (which may coincide with one of ≥0 1 ≥0 n the other rays). Let X be the corresponding toric surface, and let D ,..., D , D be the d 1 n out divisors corresponding to the above rays. Choose general points x ,..., x ∈ D ,and let i1 i i ν : X → X d d ˜ ˜ ˜ be the blow-up of all the points {x }.Let D ,..., D , D be the proper transforms of the ij 1 n out divisors D ,..., D , D and E the exceptional curve over x . 1 n out ij ij Now introduce the additional data of P = (P ,..., P ),where P denotes a se- 1 n i quence p ,..., p of non-negative numbers. We will use the notation P = p +···+ i1 i i i i1 p and call P an ordered partition.Deﬁne i i |P|= p . i ij j=1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 125 We shall restrict attention to those P such that (3.13) − |P |m = k m i i P d i=1 where m ∈ M is a primitive generator of d and k is a positive integer. d P Given this data, consider the class β ∈ A (X , Z) speciﬁed by the requirement 1 d that, if D is a toric divisor of X with D ∈ {D ,..., D , D },then D · β = 0; if D d 1 n out out {D ,..., D }, 1 n D · β =|P |, D · β = k ; i i out P while if D = D for some j,then out j |P | i = j, D · β = |P |+ k i = j. i P That such a class exists follows easily from (3.13) and Lemma 1.13. It is also unique. We can then deﬁne β = ν (β) − p [E ]∈ A (X , Z). P ij ij 1 d i=1 j=1 ˜ ˜ ˜ ˜ ˜ We deﬁne N := N as in Deﬁnition 3.1, using (Y, D) = (X , D),where Dis the proper P β d transform of the toric boundary of X , and using C = D . Then one of the main theo- d out rems of [GPS09] (see Section 5.7 of that paper) states Theorem 3.31. P −k m P d (3.14)log f = k N t z , d P P ij where the sum is over all P satisfying (3.13)and t denotes the monomial t . ij ij We can adapt this theorem for our purposes as follows. Fix a fan in M deﬁning ¯ ¯ a complete non-singular toric surface Y, with D = D +··· + D the toric boundary. 1 n Choose points x ,..., x ∈ D , and deﬁne a new surface Y as the blow-up ν : Y → Yat i1 i i the points {x }.Let E be the exceptional curve over x . ij ij ij ¯ ¯ Let P = NE(Y); because Y is toric, this is a ﬁnitely generated monoid with P = gp {0}.Let ϕ¯ : M → P be the -piecewise linear strictly P-convex function given by Lemma 1.13. We will need the following, an immediate corollary of Lemma 1.13, using the no- tation of that lemma applied to the fan for Y: 126 MARK GROSS, PAUL HACKING, AND SEAN KEEL Lemma 3.32. —If a t ∈ ker s, then the corresponding element of ker s = A (Y, Z) is ρ ρ 1 gp a ϕ( ¯ m ) ∈ P . ρ ρ Let E ⊂ A (Y, Z) be the lattice spanned by the classes of the exceptional curves of ν,sothat A (Y, Z) = ν A (Y, Z) ⊕ E. We then obtain a map 1 1 ∗ ∗ gp ϕ = ν ◦¯ ϕ : M → ν P ⊕ E. Let Q = (m, p) ∈ M ⊕ A (Y, Z)|∃p ∈ ν P ⊕ Esuchthat p = p + ϕ(m) . Thereisanobvious projection r : Q → M, and by strict convexity of ϕ¯,Q = E. We consider the scattering diagram, D ,over k[Q] given by (m ,ϕ(m )−E ) i i ij D = R m , 1 + z 1 ≤ i ≤ n . 0 ≥0 i j=1 Then we have ¯ ¯ Theorem 3.33. —Let (d, f ) ∈ Scatter(D ) \ D ,assuming that thereisatmostone ray d 0 0 ¯ ¯ ¯ of Scatter(D ) \ D in each possible outgoing direction. (Note by deﬁnition of Scatter(D ), (d, f ) 0 0 0 d cannot be incoming.) Then, following the notation of Deﬁnition 3.1 and 3.3, (−k m ,π (β)−ϕ(k m )) β d ∗ β d (3.15)log f = k N z . d β β ˜ ˜ Here π : Y → Y is the toric blow-up of Y determined by d and C ⊂ Y is the component of the boundary determined by d.If d is not one of the rays R m , then we sum over all A -classes β ∈ A (Y, Z) ≥0 i 1 satisfying (3.1), and if d = R m we sum over all such classes except for classes given by multiple ≥0 i covers of one of the exceptional divisors E . ij Proof.—Let Q be the submonoid of M ⊕ N generated by elements of the form (m , d ),where d is the (i, j)-th generator of N .Notethat Q itself is freely generated by i ij ij these elements. Thus we can deﬁne a map α : Q → Q by (m , d ) → (m ,ϕ(m ) − E ). The scattering diagram i ij i i ij (m ,d ) i ij D := R m , 1 + z | 1 ≤ i ≤ n ≥0 i j=1 MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 127 then has image under the map α (applying α to each f ) the scattering diagram D .Thus d 0 if we apply α to each element of Scatter(D ), we must get Scatter(D ),as θ 0 γ,Scatter(D ) being the identity on k[Q ] implies that θ is the identity on k[Q]. γ,α(Scatter(D )) To obtain the result, we now note that the set of possible A -classes in Y occurring in the expression (3.15) are precisely the classes {β } where P runs over all partitions satisfying (3.13). Now applying α to a term appearing in (3.14)ofthe form ij P −k m |P |m d i i i=1 k N t z = k N t z , P P β β P P ij we get (−k m , |P |ϕ(m )− p E ) P d i i ij ij i,j i=1 k N z . β β P P But by Lemma 3.32 and (3.13), |P |ϕ(m ) − p E = π (β ) − ϕ(k m ), i i ij ij ∗ P P d i=1 i,j hence the result. A direct comparison of the formula of the above theorem and the formula in the deﬁnition of the canonical scattering diagram then yields Theorem 3.25. 4. Smoothness: around the Gross-Siebert locus Next we prove that our deformation of V is indeed a smoothing. The main the- orem of this section (Theorem 4.6) will show this in the situation of Theorem 0.1 when (Y, D) has a toric model. The full smoothness statement of Theorem 0.1 will require some more work,whichwillbecarried outinSection 6. We prove smoothness by working over the Gross-Siebert locus (Deﬁnition 3.14). Here our deformation (when restricted to one-parameter subgroups associated to p A, A an ample divisor on Y) agrees with the construction of [GS07]. This is important here because the deformations of [GS07] come with explicit charts that cover all of V ,from which it is clear that they give a smoothing. So conceptually, the smoothing claim is clear. Because we work with formal families the actual argument is a bit more delicate. First we make rigorous the notion of a smooth generic ﬁbre for a formal family: Deﬁnition-Lemma 4.1. —Let f : Z → W be a ﬂat ﬁnite type morphism of schemes of relative dimension d . Then Sing(f ) ⊂ Z is the closed embedding deﬁned by the d th Fitting ideal of . Z/W Sing(f ) is empty if and only if f is smooth. Formation of Sing(f ) commutes with all base extensions of W. 128 MARK GROSS, PAUL HACKING, AND SEAN KEEL Proof. — For the deﬁnition of the Fitting ideal, see e.g., [E95], 20.4. The fact that it commutes with base-change follows from the fact that commutes with base-change Z/W and [E95], Cor. 20.5. That Sing(f ) is empty if and only if f is smooth follows from [E95], Prop. 20.6 and the deﬁnition of smoothness. Now for a formal family, smoothness of the generic ﬁbre is measured by the fact that Sing(f ) does not surject scheme-theoretically onto the base. More precisely: Deﬁnition 4.2. —Let S be a normal variety, V ⊂ S a connected closed subset, and S the formal completion of S along V.Let f: X → S be an adic ﬂat morphism of formal schemes of pure relative dimension and Z ⊂ X the scheme theoretic singular locus of f. Then we say the generic ﬁber of f is smooth if the map O → f O is not injective. S ∗ Z For the statement of Proposition 4.3, we ﬁx our usual setting of a surface (Y, D), and assume given Assumptions 3.20 and that ν(D) is a scattering diagram in the sense of Deﬁnition 3.21. Suppose furthermore that θ ≡ 1 for a loop γ around the origin. γ,ν(D) Thus by Theorem 3.30, D is consistent. These hypotheses on D apply in particular when can D = D . gs gs Let T be the Gross-Siebert locus; we have T = Spec k[P]/G. Consistency of D gives a ﬂat family f : X → Spec k[P]/I I I gs over a thickening of T whenever I = G. ¯ ¯ ¯ On the other hand, letting be the fan for Yin B = M , we have the piecewise gp ¯ ¯ linear function ϕ¯ : B → P with κ = p [D ],asin(3.6). This now determines the ρ, ¯ ϕ¯ ρ¯ Mumford family ¯ ¯ f : X → Spec k[P]/I. I I gs gs Our goal is to compare these two families. Note that both X → T and X → T G G gs gs gs are the trivial family V × T → T . Thus either family contains a canonical copy of T , gs i.e., {0}× T , where 0 is the vertex of V . Proposition 4.3. — In the above situation, ﬁx an ideal I with I = G. There are open afﬁne gs ¯ ¯ sets U ⊂ X , U ⊂ X , both sets containing the canonical copy of T , and an isomorphism I I I I μ : U → U I I I of families over Spec k[P]/I. Moreover, there is a non-zero monomial y ∈ k[P] whose pullback to X is in the stalk at any gs gs point x∈{0}× T ⊂ V × T of the ideal of Sing for all I. I MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 129 Proof. — The generic ﬁbre of the Mumford family over Spec k[P] is smooth: indeed the family is trivial over the open torus orbit of Spec k[P], with ﬁbre an algebraic torus. It follows that there is a non-zero monomial y ∈ k[P] in the ideal of Sing for the global Mumford family f : Spec k[P ]→ Spec k[P]. ϕ¯ Of course its restriction then lies in the stalk at any point x of the ideal sheaf of Sing for all I. Thus once we establish the claimed isomorphisms, the ﬁnal statement follows. ¯ ¯ Recall from Section 3.3 the construction of D := ν(D) and the scheme X from I,D o o ¯ ¯ D. By Lemma 3.27,X X ,soweinfacthaveanisomorphism I,D I,D ¯ ¯ X = Spec X , O ¯ =: X ¯ . ¯ X I,D I,D I,D So we can work with X instead of X . On the other hand, the Mumford family I,D Spec k[P ]/Ik[P ] over Spec k[P]/I can be described similarly. Using the empty scat- ϕ¯ ϕ¯ tering diagram instead of the scattering diagram D, one has by Lemma 2.9 ¯ ¯ X = Spec X , O o . I,∅ I,∅ X I,∅ Now deﬁne an ideal I ⊂ P as follows. For σ ∈ ,let ϕ¯ denote the linear 0 ϕ¯ max σ extension of ϕ¯| . We set I := (m, p) ∈ P | p−¯ ϕ (m) ∈ Ifor some σ ∈ . 0 ϕ¯ σ max Note that I = m . By assumption, D is a scattering diagram for P , and hence there 0 P ϕ¯ ϕ¯ are only a ﬁnite number of (d, f ) ∈ D for which f ≡ 1mod I . Furthermore, modulo I , d d 0 0 each f is a polynomial. ¯ ¯ Let D be the scattering diagram obtained from D by, for each outgoing ray (d, f ), I d truncating each f by throwing out all terms which lie in I . The incoming rays remain d 0 unchanged. Thus D can be viewed as a ﬁnite scattering diagram. Let h := f . d∈D This is an element of k[P ]. Note that necessarily h ≡ 1mod m .Thus h = 0deﬁnes ϕ¯ P ϕ¯ gs ¯ ¯ ¯ ¯ an open subset U ⊂ Spec k[P ]/Gk[P ]= X ¯ = X = V × T . Furthermore, U ϕ¯ ϕ¯ G,∅ n G,D gs ¯ ¯ contains the canonical copy of T . Since X ¯ and X both have underlying topological I,∅ I,D ¯ ¯ ¯ ¯ ¯ space X , this deﬁnes open sets U of X and U of X . We shall show these two ¯ ¯ G I,∅ I,∅ I,D I,D open subschemes are isomorphic. ¯ ¯ ¯ First the following claim shows that X X ,asfor any τ ∈ \{0},the auto- ¯ = ¯ I,D I,D morphisms involved will have the same effect modulo I . As a consequence, we can work with the scattering diagram D . I 130 MARK GROSS, PAUL HACKING, AND SEAN KEEL Claim 4.4. —Let τ ∈ , and suppose (m, p) ∈ P satisﬁes −m ∈ τ . Then (m, p) ∈ I if ϕ¯ 0 and only if (m, p) ∈ I ,where I := (m, p) ∈ P | p−¯ ϕ (m) ∈ I for some σ ∈ with τ ⊂ σ . τ ϕ¯ σ max Proof of claim. — Clearly I ∩ P ⊂ I , so one implication is clear. Conversely, sup- τ ϕ 0 pose that (m, p) ∈ I ,sothat p − ϕ (m) ∈ Ifor some σ ∈ .If τ ⊂ σ ∈ ,let 0 max max ρ ,...,ρ be the sequence of rays traversed in passing from σ to σ , chosen so that all 1 n ρ ,...,ρ lie in a half-plane bounded by the line Rm.Then 1 n ϕ (m) = ϕ (m) + n , mκ , σ σ ρ ρ ,ϕ i i i=1 with n primitive, vanishing on ρ ,and positive on ρ . Note that since −m ∈ τ ,wemust ρ i i+1 have n , m≤ 0for each i, and hence p− ϕ (m) = p− ϕ (m)+ p for some p ∈ P. Hence i σ σ (m, p) ∈ I . ¯ ¯ To show that U and U are isomorphic, let us describe these open subschemes I,D I,∅ explicitly away from the origin. Recall that X is obtained by gluing together schemes I,D which are spectra of rings R for τ ∈ . However in the case that dim τ = 1, this ring τ,I depends on the scattering diagram, so we write R for D = D or ∅. τ,I,D If dim τ = 2, then R = k[P ]/Ik[P ]. Since h ∈ k[P ]⊂ k[P ], h deﬁnes an τ,I,D ϕ¯ ϕ¯ ϕ¯ ϕ¯ τ τ τ element of R in this case. τ,I,D If dim τ = 1, then τ = ρ for some i, and we have a surjection ±1 ¯ ¯ ¯ ¯ ¯ R → R = R X , X , X /(X X ) = k[P ]/Gk[P ], ρ ,I,D ρ ,G,D G i−1 i+1 i−1 i+1 ϕ¯ ϕ¯ i i ρ ρ i i so that h ∈ k[P ]⊂ k[P ] deﬁnes an element of R . Choosing any lift of h to R , ϕ¯ ϕ¯ ρ ,G,D ρ ,I,D i i we note the localization (R ) is independent of the lift since the kernel of the above ρ ,I,D h surjection is nilpotent. We can then deﬁne regardless of dim τ , S := (R ) . τ,I,D τ,I,D h Note there is an isomorphism ψ : S → S i ρ ,I,D ρ ,I,∅ given by −1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ X → X g¯ 1 + b X , X → X , X → X . i−1 i−1 ρ i i i i+1 i+1 i ij j=1 This has an inverse because of the localization at h. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 131 Given a path γ in M \{0}, note that by construction of h, θ makes sense as γ,D an automorphism of the localization k[P ] , since to deﬁne the automorphism associated ϕ¯ h with crossing a ray (d, f ), we only need f to be invertible. However since by construction d d h is divisible by f , f is invertible. In particular, θ also makes sense as an automorphism d d γ,D of (k[P ]/Ik[P ]) = S for any τ ∈ \{0}. Thus using the equality S = S ¯ ϕ¯ ϕ¯ h τ,I,∅ σ,I,∅ τ τ σ,I,D for dim σ = 2 we see that θ ¯ also makes sense as an automorphism of S ¯ . γ,D σ,I,D Choose an orientation on M , labelling the rays ρ ,...,ρ of in a counterclock- R 1 n wise order, with σ as usual the maximal cone containing ρ and ρ . For two distinct i−1,i i−1 i points p, q on the unit circle in M not contained in Supp(D ),let γ be a counterclockwise R I p,q path from p to q, and write θ for θ acting on any of the rings S . p,q τ,I,∅ γ ,D p,q I For each ρ ,let p be a point on this unit circle contained in the connected com- i i,+ ponent of σ \ Supp(D ) adjacent to ρ ,and p a point in this unit circle contained in i,i+1 I i i,− the connected component of σ \ Supp(D ) adjacent to ρ . i−1,i I i Choose a base-point q on the unit circle not in Supp(D ). Recall in the construction of X , the open sets Spec R and Spec R are ρ ,I,D ρ ,I,D i+1 i I,D glued together along the common open set Spec R using the trivial automorphism σ ,I,D i,i+1 or the automorphism θ in the cases D=∅ or D = D respectively. After localizing p ,p i,+ i+1,− at h, we have a commutative diagram θ ◦ψ p ,q i i,+ S S ρ ,I,∅ ρ ,I,D i ρ ,+ θ ◦ψ p ,p ρ ,+ i,+ i+1,− i p ,q i+1,− S S σ ,I,∅ σ ,I,D i,i+1 i,i+1 ρ ,− i+1 ρ ,− i+1 S S ρ ,I,∅ ρ ,I,D i+1 i+1 θ ◦ψ p ,q i+1 i+1,+ Here the maps ψ are the ones deﬁned in Proposition 2.5 and (2.8). This shows that ρ,± the isomorphisms θ ◦ ψ between Spec S ¯ and Spec S are compatible with the p i ρ ,I,∅ ρ ,I,D i,+ i i gs gs ¯ ¯ gluings, and hence give an isomorphism between U \ ({0}× T ) and U \ ({0}× T ). I,D I,∅ Now V satisﬁes Serre’s condition S . Since X and X are ﬂat deformations of n 2 I I gs V × T , by Lemma 2.10 the above isomorphism extends across the codimension two set gs {0}× T , giving the desired isomorphism between U and U . I I We now need to use the above observations along the Gross-Siebert locus to obtain results about deformations away from the Gross-Siebert locus. For the remainder of the section, we work with data (Y, D), η, P, but now as in Assumptions 3.13. Furthermore, √ √ can we take D = D . Thus if we take I an ideal with either I = Gor I = m,weobtain 132 MARK GROSS, PAUL HACKING, AND SEAN KEEL aﬂat family X → Spec R =: S , and in the former case, X → S restricts to the open I I I I I gs subscheme of Spec R whose underlying open subset is T , giving the family over the thickening of the Gross-Siebert locus. With J = m or G, let f : X → S denote the formal deformation determined by J J J N+1 the deformations X N+1 → Spec R N+1 for N ≥ 0. Thus S = Spf(lim k[P]/J ) is the J J J ←− formal spectrum of the J-adic completion of k[P], X is a formal scheme, and X → S J J J is an adic ﬂat morphism of formal schemes. We refer to [G60] for background on formal schemes. Let Z := Sing(f ) ⊂ X denote the singular locus of f : X → S .Thus Z ⊂ X is I I I I I I I I a closed embedding of schemes. Since the singular locus is compatible with base-change, n n the singular loci Z ⊂ X determine a closed embedding Z ⊂ X which we refer to as J J J J the singular locus of f : X → S . J J J Again, with J = m or G, we have a section s: S → X = S × V given by s(t) = J J J n o o o t ×{0} for t ∈ S . We write X := X \ s(S ) ⊂ X and X ⊂ X , X ⊂ X for the induced J J J J I J J I J open embeddings. Lemma 4.5. — In the above situation, there exists 0 = g ∈ k[P] such that Supp(g · O ) is contained in s(S ). In particular, f (g · O ) is a coherent sheaf on S . J J∗ Z J Proof. — We can write an explicit open covering {U } of X in the two cases J = m i,J [D ] 2 or J = G, as follows. Write a = z and m =−D .Inthe case J = m, i i (4.1) U = V X X − a X ⊂ A × (G ) × S . i,J i−1 i+1 i m X J i X ,X i i−1 i+1 In the case J = G, i −1 2 U = V X X − a X 1 + b X ⊂ A × (G ) × S , i,J i−1 i+1 i ij m X J i i i X ,X i−1 i+1 [E ] ij with b = z as usual. ij We now use the charts U to compute the singular locus explicitly. In the case i,J J = m, the singular locus Z of U /S is given by i,J i,J J Z = V(X , X , a ) ⊂ U . i,J i−1 i+1 i i,J Hence if we deﬁne g = a ··· a then Supp(g · O ) is contained in s(S ). 1 n Z J Similarly, if J = G, the structure sheaf of the singular locus of U is annihilated by i,J g := a (b − b ).(Here (b − b ) is the discriminant of the polynomial f (X ) := i i ij ik ij ik i =k j =k (X + b ). It is a linear combination of f and f with coefﬁcients in k[{b }][X ].See i ij ij i [L02], p. 200–204.) So we can take g = g ··· g . 1 n The support of g · O is a closed subset of s(S ), hence proper over S . It follows Z J J that f (g · O ) is coherent by [G61], 3.4.2. J∗ Z J MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 133 can Theorem 4.6. —Let (Y, D), η, P satisfy Assumptions 3.13,and take D = D . Then the maps k[P]→ f O and O → f O are not injective, so the generic ﬁbre of f is m Z S m Z m ∗ m m ∗ m N+1 smooth in the sense of Deﬁnition 4.2. This also implies that for I = m and N $ 0, the map k[P]/I → f O is not injective. I∗ Sing(f ) Proof. — By Lemma 4.5 there exists 0 = g ∈ k[P] such that Supp(g · O ) ⊂ s(S ). Z G gp Let E be the subgroup of P generated by P \ G, so that U = Spec k[P + E] is an open subset of Spec k[P].Denoteby S the open subset of S isomorphic to the completion of U along the subscheme deﬁned by G + E. This is the formal thickening of the Gross- gs Siebert locus T .ByProposition 4.3 there then exists 0 = h ∈ k[P+ E] such that Supp(h· O −1 o ) is disjoint from s(S ∩ U). By multiplying h by a monomial whose exponent Z ∩f (S ) G G lies in P \ G, we can assume that h ∈ k[P].Thus gh · O has support in the closed subset s(S \ (S ∩ U)). Since this sheaf is coherent, there exists a non-zero element k ∈ m ⊂ G G k[P] such that ghk · O = 0. Noting by construction that ghk ∈ k[P],wehave k[P]→ (Z , O ) is not injective, hence the composition k[P]→ (Z , O ) → (Z , O ) G Z G Z m Z G G m is not injective. Since k[P]⊂ (S , O ), O → f O is not injective. m S S m∗ Z m m m 5. The relative torus The ﬂat deformations X can → Spec k[P]/I produced by the canonical scatter- I,D ing diagram have a useful special property: there is a natural torus action on the total space X can compatible with a torus action on the base. The meaning of this action I,D will be clariﬁed in Part II, where we will prove that our family extends naturally, in the positive case, to a universal family of Looijenga pairs (Z, D) together with a choice of isomorphism D → D ,where D is a ﬁxed n-cycle. The torus action then corresponds to ∗ ∗ changing the choice of isomorphism. D n Fixing the pair (Y, D) as usual, D = D +···+ D ,let A = A be the afﬁne space 1 n D D with one coordinate for each component D .Let T be the diagonal torus acting on A , i.e., the torus T whose character group D D χ T = Z is the free module with basis e ,..., e . D D 1 n Deﬁnition 5.1. — We deﬁne a canonical map w : A (Y) → χ(T ) given by C → (C · D )e . i D Suppose P ⊂ A (Y) is a toric submonoid containing NE(Y).Wethenget an action of T on Spec k[P],aswellason Spec k[P]/I for any monomial ideal I, and hence also on Spf(k[P]) for any completion of k[P] with respect to a monomial ideal. 134 MARK GROSS, PAUL HACKING, AND SEAN KEEL We can also deﬁne a unique piecewise linear map w : B → χ T ⊗ R with w(0) = 0and w(v ) = e ,for v the primitive generator of the ray ρ . i D i i Theorem 5.2. —Let I be an ideal for which X can → Spec k[P]/I is deﬁned. Then T I,D acts equivariantly on X can → Spec k[P]/I. Furthermore, each theta function ϑ ,q ∈ B(Z),isan I,D q eigenfunction of this action, with character w(q). Proof. — It’s enough to check this on the open subset X ⊂ X can.Wehavea can I,D I,D cover of X by open sets the hypersurfaces can I,D U ⊂ A × (G ) × Spec R ρ ,I m X I i X ,X i i−1 i+1 given by the equation −D [D ] i i X X = z X f , i−1 i+1 ρ i i can where f is the function attached to the ray ρ in D .Ifweact on X with weight w(v ) ρ i j j p can and on z with weight w(p) (for p ∈ P),thenwenotethatfor every (d, f ) ∈ D ,every monomial in f has weight zero by the explicit description of f in Deﬁnition 3.3.Inpar- d d ticular, the equation deﬁning U is clearly T -equivariant, and each of the monomials ρ ,I is an eigenfunction. Now X is obtained by gluing U ⊂ U with U ⊂ U ,us- can ρ ,σ ,I ρ ,I ρ ,σ ,I ρ ,I I,D i i,i+1 i i+1 i,i+1 i+1 can ing scattering automorphisms of D , and these open sets are naturally identiﬁed with 2 D (G ) × Spec R . The scattering automorphisms commute with the T action, by m I X ,X i i+1 the fact that the scattering functions have weight zero. Thus T acts equivariantly on can X → Spec k[P]/I. I,D Now we check our canonical global function ϑ is an eigenfunction, with charac- ter w(q). By construction, given a broken line γ , the weights of monomials attached to adjacent domains of linearity are the same, since the functions in the scattering diagram are of weight zero. Thus the weight of Mono(γ ) only depends on q. This weight can be determined by ﬁxing the base point Q in a cone σ which contains q, in which case the broken line for q which doesn’t bend and is wholly contained in σ yields the monomial ϕ (q) z , which has weight w(q).Thus ϑ is an eigenfunction with weight w(q). 6. Extending the family over boundary strata Here we prove Theorem 0.1 and Theorem 0.2. Let us review what we know so far. can For any pair (Y, D), we know that D is consistent by Theorem 3.8. Thus, if the number n of irreducible components of D satisﬁes n ≥ 3, Theorem 2.28 gives the construction of MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 135 f : X → Spec R of Theorem 0.1. The algebra structure on A has structure constants I I I given by counts of broken lines as in Theorem 2.34.The T equivariance is given by Theorem 5.2. If furthermore (Y, D) has a toric model, then the smoothness statement follows from Theorem 4.6. We will give a proof of Theorem 0.2 and the remaining cases of 0.1 by ﬁrst proving Theorem 0.2 in the case that we know that we have the desired algebra structure on A , and then bootstrap to the general case for both theorems. 6.1. Theorem 0.2 in the case that (Y, D) has a toric model. — As usual, let P be the toric monoid associated to a strictly convex rational polyhedral cone σ ⊂ A (Y) which P 1 R contains the Mori cone NE(Y) .Wehave m = P\{0}. For a monomial ideal I ⊂ Pwe deﬁne A := R · ϑ I I q q∈B(Z) can where R = k[P]/I. We take throughout D = D . Assumptions 6.1. — For any monomial ideal I with I = m, the multiplication rule of The- orem 2.34 deﬁnes an R -algebra structure on A ,sothat A ⊗ R = H (V , O ). I I I R m n V I n Note we have already shown that Assumptions 6.1 hold if n ≥ 3by Theorems 2.28, 2.34 and 3.8. Let ⊂ B(Z) be a ﬁnite collection of integral points such that the corresponding functions ϑ generate the k-algebra H (V , O ).(Then the ϑ , q ∈ generate A as an q n V q I R -algebra if I = m and Assumptions 6.1 hold.) Note for n ≥ 3we can take for the points {v },and for n = 1, 2, one can make a simple choice for , see Section 6.2. Lemma 6.2. — For any monomial ideal J ⊂ P, (J + m ) = J. k>0 Proof. — The inclusion ⊃ is obvious. For the other direction, as the intersection is a monomial ideal, it’s enough to consider a monomial in the intersection. But notice that k k k a monomial is in J + m iff it is either in J or in m . The result follows since m = 0. Assuming 6.1,let A be the collection of monomial ideals J ⊂ P with the following properties: (1) There is an R -algebra structure on A such that the canonical isomorphism of J J R -modules A ⊗ R = A is an algebra isomorphism, for all I = m. I+J J R I+J I+J (2) ϑ , q ∈ generate A as an R -algebra. q J J By the lemma, the algebra structure in (1) is unique if it exists. The algebra struc- √ √ ture on all A determines such a structure on A := lim A , A := lim A . Also, I I J I+J I=m I=m ←− ←− 136 MARK GROSS, PAUL HACKING, AND SEAN KEEL there are canonical inclusions A ⊂ R · ϑ q∈B(Z) A ⊂ R · ϑ J J q q∈B(Z) ˆ ˆ where R is the completion of R at m and R = lim R/(I + J) the inverse limit over all ←− ˆ ˆ ideals I with I = m. Here the direct products are viewed purely as R, R modules. We can also view A := R · ϑ ⊂ R · ϑ . J J q J q q∈B(Z) q∈B(Z) It is clear that A ⊂ A (as submodules of the direct product). Thus (1) holds if and only if J J the following holds: (1 )For each p, q ∈ B(Z), at most ﬁnitely many z ϑ with C ∈ J appear in the product expansion of Theorem 2.34 for ϑ · ϑ ∈ A . p q J Lemma 6.3. —If J ∈ A and J ⊂ J , then J ∈ A. In addition, A is closed under ﬁnite intersections. Proof. — The ﬁrst statement is clear. Now assume J , J ∈ A. It’s clear that (1 ) holds 1 2 for J ∩ J ,so A is an algebra. Moreover we have an exact sequence of k-modules 1 2 J ∩J 1 2 0 → A → A × A → A → 0 J ∩J J J J +J 1 2 1 2 1 2 exhibiting A as the ﬁbre product A × A =: A × A =: A. We now show J ∩J J A J 1 B 2 1 2 1 J +J 2 1 2 this ﬁbre product is a ﬁnitely generated k-algebra. Indeed, note that since the maps A , A → B are surjective, so are the maps A → A .Let {u } be a generating set for 1 2 i i the ideal ker(A → B). Since A is Noetherian, one can ﬁnd a ﬁnite such set. Note 2 i that u˜ := (0, u ) ∈ A. In addition, choose ﬁnite sets {x }, {y } generating A and A i i i j 1 2 as k-algebras. For each of these elements, choose a lift to A, giving a ﬁnite set of lifts {˜ u , x˜ , y˜ }, which we claim generate A. Indeed, given (x, y) ∈ A, one can subtract a poly- i i i nomial in the x˜ ’s to obtain (0, y ). Necessarily y ∈ ker(A → B), and hence we can write i 2 y = f u with f a polynomial in the y ’s. Let f be the same polynomial in the y˜ ’s. Then i i i i i i f u˜ = (0, y ), showing generation. i i Thus A is also a ﬁnitely generated R -algebra. Now the generation state- J ∩J J ∩J 1 2 1 2 ment follows from Lemma 6.4, taking R = R ,S = A ,I = J /J ∩ J ,J = J /J ∩ J , J ∩J J ∩J 1 1 2 2 1 2 1 2 1 2 ={q ,..., q },and themap R[T ,..., T ]→ Sgiven by T → ϑ . 1 m 1 m i q i MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 137 Lemma 6.4. —Let I, J ⊂ R be ideals in a Noetherian ring, with I · J = 0,and let S be a ﬁnitely generated R-algebra, and R[T ,..., T ]→ S an R-algebra map which is surjective modulo 1 m I and J. Then the map is surjective. Proof. — The associated map Spec S → A × Spec R is proper, as can be easily checked using the valuative criterion for properness. Indeed, any map S → Kfor a ﬁeld K factors through either S/IS or S/JS. Since this is a map of afﬁne schemes, S is a ﬁnite R[T ,..., T ]-module. Now we can apply Nakayama’s lemma. 1 m Proposition 6.5. — There is a unique minimal radical monomial ideal I ⊂ P such that (1) min and (2) hold for any monomial ideal J with I ⊂ J. min Proof. — Certainly any ideal J with J = m lies in A. Note that a radical monomial ideal is the complement of a union of faces of P, so there are only a ﬁnite number of such ideals. Suppose I , I are two radical ideals such that J ∈ A for any J with I ⊂ J .Note 1 2 i i i i that any ideal J with I ∩ I ⊂ J can be written as J ∩ J ,with I ⊂ J .(Indeed,we 1 2 1 2 i i can use the primary decomposition of J. If J = p is an intersection of primary ideals, necessarily the prime ideal p contains either I or I for each k.Thenlet J be the k 1 2 1 intersection of those p whose radical contains I and J be the intersection of those p k 1 2 k whose radical contains I .) Thus by Lemma 6.3,J ∈ A. This shows the existence of I . 2 min Proposition 6.6. — Suppose Assumptions 6.1 hold. (1) Suppose the intersection matrix (D · D ) is not negative semi-deﬁnite. Then I = (0) ⊂ i j min k[P]. (2) Suppose F ⊂ σ is a face such that F does not contain the class of every component of D. Then I ⊂ P \ F. min Proof. — We prove both cases simultaneously, writing F := Pin case (1). We claim there exists an effective divisor W = a D with support D such that W· D > 0for allD i i j j contained in F and a > 0for all i. For case (1), see Lemma 6.9.Incase(2),say [D ] ∈ / F. i 1 Then we can take a $ a $···$ a > 0. 1 2 n The algebra structure depends only on the deformation type of (Y, D).ByPropo- sition 4.1 of [GHK12], we may replace (Y, D) by a deformation equivalent pair such that any irreducible curve C ⊂ Y intersects D. Let NE(Y) ⊂ A (Y, R) denote the closure of NE(Y) .Let F := NE(Y) ∩ F, R 1 R R afaceof NE(Y) .Deﬁne = D− W, 0 < 1. Then (Y, ) is KLT (Kawamata log terminal). We claim K + ∼−Wis negative onF \{0}. By construction (K + )· D < 0 Y Y j for [D]∈ F and (K + ) · C < 0for C ⊂ D. Let N be a nef divisor such that F = j Y NE(Y) ∩ N .Then aN − (K + ) is nef and big for a $ 0, and thus some multiple of R Y N deﬁnes a birational morphism g by the basepoint-free theorem [KM98], Theorem 3.3. 138 MARK GROSS, PAUL HACKING, AND SEAN KEEL Thus F is generated by exceptional curves of g. We deduce that (K + ) ∩ F ={0} and (K + ) is negative on F \{0} as claimed. Now by the cone theorem [KM98], Theorem 3.7, NE(Y) is rational polyhedral near F and there is a contraction p: Y → Ysuch thatF is generated by the classes of curves contracted by p. It follows that we can ﬁnd NE(Y) ⊂ σ ⊂ σ such that F is a R P P face of σ . Now the algebra structure for P comes from P by base extension, so (replacing PbyP ) we can assume F = F , and thus that W is positive on F\{0}. Now let J be a monomial ideal with J = P \ F. Consider condition (1 ). By the D C T -equivariance of Theorem 5.2,any z ϑ that appears in ϑ · ϑ has the same weight s p q for T . Thus it is enough to show that the map w: B(Z) × (P \ J) → χ T ,(q, C) → w(q) + w(C) has ﬁnite ﬁbres. It is enough to consider ﬁbres of σ(Z) × (P \ J) → χ(T ) for each σ ∈ .Notethat σ(Z)× P is the set of integral points of a rational polyhedral cone, and w max is linear on this set. Thus it is enough to check that ker(w)∩ (σ (Z)× F) = 0. So suppose we have q ∈ σ(Z),C ∈ Fwith w(q) + w(C) = 0. Say σ = σ .Then q = av + bv , i,i+1 i i+1 for a, b ∈ Z .Wehave ≥0 w(q) + w(C) = ae + be + (C · D )e ; D D j D i i+1 j thus if this is zero, we have C· D ≤ 0for all j.Inparticular, W· C ≤ 0. Since W is positive on F\{0},C = 0. Now necessarily a = b = q = 0. This proves (1 ). For (2), let A ⊂ A be the subalgebra generated by the ϑ , q ∈ .Fix aweight J q w ∈ χ(T ).Toshow A = A it is enough to show that the ﬁnite set z ϑ ∈ A | (q, C) ∈ B(Z) × (P \ J) of weight w q J is contained in A (since the z ϑ give a k-basis of A ). We argue by decreasing induction q J on ord (C) (see Deﬁnition 2.21). Since the set of possible (q, C) is ﬁnite, there is an upper bound on the possible ord ’s. So the claim is vacuously true for large ord .Consider m m z · ϑ ,with ord (C) = h. Since the ϑ generate A modulo m,wecan ﬁnd a ∈ A such p m q J that ϑ = a + m with m ∈ m · A . Moreover, we can assume a,and thus m, is homogeneous for the T action. Now C C C z ϑ = z a + z m. p MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 139 C D C Clearly z m is a sum of terms z ϑ of weight w and ord (D)> h,so z m ∈ A by induc- q m tion. Remark 6.7. — Suppose p: Y → Y is a contraction such that some component of D is not contracted by p.Let F bethe face of NE(Y) generated by classes of curves contracted by p.Then NE(Y) is rational polyhedral near F. (This follows from the cone theorem, cf. the proof of Proposition 6.6.) In particular there exists a rational polyhedral cone σ ⊂ A (Y, R) such that NE(Y) ⊂ σ and σ coincides with NE(Y) near F. P 1 R P P R Corollary 6.8. —Theorem 0.2 holds if D has n ≥ 3 irreducible components. Proof. — Immediate from Proposition 6.6. 6.2. Proof of Theorems 0.1 and 0.2 in general. — We now consider an arbitrary Looijenga pair (Y, D), along with a toric monoid P with NE(Y) ⊂ P ⊂ A (Y, Z).Let τ : (Y , D ) → (Y, D) be a toric blowup such that (Y , D ) has a toric model p : (Y , D ) → ¯ ¯ (Y, D).Wehavethe map τ : A (Y , Z) → A (Y, Z). We can ﬁnd a strictly convex ratio- ∗ 1 1 nal polyhedral cone σ with NE Y ⊂ σ ⊂ A Y , R P 1 which has a face F spanned by the τ -exceptional curves, and which surjects under τ onto σ ⊂ A (Y, R). For any monomial ideal I ⊂ Pwith I = m,let I ⊂ P be the in- P 1 verse image of I under τ .Then I is the prime monomial ideal associated to the face F. Since the exceptional curves are a proper subset of D we have I ∈ A(Y ) by Proposi- tion 6.6.Notethat Spec k[P]/I is naturally a closed subscheme of Spec k[P ]/I ,via the map induced by the surjection τ : P → P. Now restrict the family X → Spec k[P ]/I ∗ I to Spec k[P]/I. This gives an algebra structure on A := k[P]/I ϑ . I q q∈B(Z) We now verify Assumptions 6.1. First, we show that the multiplication rule of this algebra is the one described in Theorem 2.34. The argument is just as in the proof of Proposi- can can tion 3.12:Wehave B = B and take ψ := τ : P → P. Note ψ(D ) = D (Y ,D ) (Y,D) ∗ (Y ,D ) (Y,D) (i.e., the rays are the same, and we apply ψ to the decoration function). This does not literally give a bijection on broken lines (because different exponents in the decoration of can aray in D could map to the same exponent under ψ ). However, by Equation (3.4), (Y ,D ) 140 MARK GROSS, PAUL HACKING, AND SEAN KEEL with z a point close to q, ⎛ ⎞ ⎛ ⎞ ⎝ ⎠ ⎝ ⎠ c(γ )c(γ ) = ψ c γ ψ c γ 1 2 1 2 (γ ,γ ) (γ ,γ ) 1 2 1 2 γ ∈ξ γ ∈ξ γ γ 1 1 2 2 Limits(γ )=(q ,z) Limits(γ )=(q ,z) i i i i s(γ )+s(γ )=q s(γ )+s(γ )=q 1 2 1 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ψ c γ c γ , 1 2 ⎜ ⎟ ⎜ ⎟ (γ ,γ ) 1 2 ⎝ ⎠ Limits(γ )=(q ,z) s(γ )+s(γ )=q 1 2 can where ξ denotes the set of all broken lines γ for D such that ψ ◦ γ = γ as paths γ i i i (Y ,D ) i and the monomials attached to ψ(γ ) differ from those attached to γ only in the k-valued coefﬁcients (see the proof of Proposition 3.12). This implies the claim. Next we need to check that the ﬁbre over the zero stratum of Spec k[P] is V .In case n ≥ 3, this is straightforward from the multiplication rule. Indeed, modulo m,every broken line contributing to the multiplication rule is a straight line, and furthermore it cannot cross any ray of . From this one sees that A = R [ ]. m m The cases n = 1 and 2 require special attention. We will do the case of n = 1, as n = 2 is similar (and simpler). We cut B = B along the unique ray ρ = ρ ∈ ,and (Y,D) 1 consider the image under a set of linear coordinates ψ on B \ ρ . This identiﬁes B \ ρ with a strictly convex rational cone in R .Let w, w be the primitive generators of the two boundary rays. Modulo m the decoration on every scattering ray is trivial, so every broken line is straight. Moreover, no line can cross ρ (or the attached monomial becomes trivial modulo m by the strict convexity of ϕ). Now it follows for any x ∈ B(R)\ ρ and any q ∈ (B\ ρ)(Z) there is a unique (straight) broken line with Limits = (q, x), while there are exactly two (straight) broken lines with Limits = (v, x), v = v —under ψ these become two distinct straight lines with directions w, w . Performing a toric blowup of (Y, D) to get n = 3 can be accomplished by subdividing the cone generated by w and w along the rays generated by w + w and 2w + w . Then by Theorem 0.2 in the case n = 3, we see that A is generated over k by ϑ = ϑ = ϑ ,ϑ ,ϑ v w w w+w 2w+w where we abuse notation and use the same symbol for an integer point in the convex cone generated by w and w , and the corresponding point in B(Z). Now applying the multiplication rule of Theorem 2.34 one checks easily the equalities: ϑ · ϑ = ϑ + ϑ v w+w 2w+w w+2w ϑ · ϑ = ϑ = ϑ . 2w+w w+2w 3w+3w w+w MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 141 It follows that 2 3 ϑ · ϑ · ϑ = ϑ + ϑ 2w+w v w+w 2w+w w+w 2 3 and thus A = k[x, y, z]/(xyz − x − z ), which is isomorphic to the ring of sections H C, O(m) m≥0 for a line bundle O(1) of degree one on an irreducible rational nodal curve C of arith- metic genus 1. Thus Spec A = V . m 1 Combining this with Propositions 6.5 and 6.6, this proves Theorems 0.1 and 0.2 hold for all (Y, D) except for the smoothness statement of Theorem 0.1. To show smoothness, note that if m denotes the maximal monomial ideal of P , X → S the formal deformation provided by Theorem 0.1 for the pair (Y , D ) with the toric model, we know that k[P]→ H (Z , O ) is not injective by Theorem 4.6. m Z Now choose (see the beginning of the proof of Proposition 6.6) a divisor A = a D with i i a ≥ 0for all i and A relatively τ -ample, so that A · D > 0for anyD contracted by τ . j j A D D This determines a one-parameter subgroup T = G of T via the map χ(T ) → Z given by e → a . D i Let J = P \ F, so that [C]∈ J if and only if C is not contracted by τ .Thusif [C]∈ F, A [C] A T acts on z with weight C.A > 0, and for q ∈ B(Z),T acts with non-negative weight since a ≥ 0for all i. It then follows that the map 0 0 H (Z , O ) → H (Z , O ) J Z m Z is injective because every component of Z has a limit point in Z under the T action. J m So we conclude that k[P]→ H (Z , O ) is not injective. J Z gp F Now F is generated by the classes of the D contracted by τ.Let T := gp D Hom(F, G ). The composition F ⊂ A (Y , Z) → χ(T ) is a primitive embedding, be- m 1 cause the intersection matrix of F ⊂D ,..., D is unimodular, where D ,..., D are 1 r 1 r the irreducible components of the boundary of Y . So the corresponding composition D F F D F T → Hom(A (Y , Z), G ) → T admits a splitting T → T .By T -equivariance, 1 m the restriction of the family X /S to the open subscheme of S deﬁned by T ⊂ S is J J J J isomorphic to a direct product of X /S (coming from (Y, D), P) with T . In particular, m m X /S has smooth generic ﬁbre. m m 6.3. The case that (Y, D) is positive. Lemma 6.9. — The following are equivalent for a Looijenga pair (Y, D): (1.1) There exist integers a ,..., a such that ( a D ) > 0. 1 n i i (1.2) There exist positive integers b ,..., b such that ( b D ) · D > 0 for all j . 1 n i i j 142 MARK GROSS, PAUL HACKING, AND SEAN KEEL (1.3) Y \ D is the minimal resolution of an afﬁne surface with (at worst) Du Val singularities. (1.4) There exist 0 < c < 1 such that −(K + c D ) is nef and big. i Y i i If any of the above equivalent conditions hold, then so do the following: (2.1) The Mori cone NE(Y) is rational polyhedral, generated by ﬁnitely many classes of rational curves. Every nef line bundle on Y is semi-ample. (2.2) The subgroup G of Aut(Pic(Y),·,·) ﬁxing the classes [D ] is ﬁnite. (2.3) The union R ⊂ Y of all curves disjoint from D is contractible. Deﬁnition 6.10. — We say a Looijenga pair (Y, D) is positive if it satisﬁes any of the equivalent conditions (1.1)–(1.4) of the above lemma. Proof.—We have K + c D = (K + D) − (1 − c )D =− (1 − c )D Y i i Y i i i i so (1.2) implies (1.4), and (1.2) obviously implies (1.1). ⊥ ⊥ If (1.1) holds then (D ,·,·),where D ={H ∈ Pic Y| H· D = 0 ∀i},isnegative deﬁnite, by the Hodge Index Theorem, and this implies (2.2) and (2.3). Suppose (1.4) holds. By the basepoint-free theorem [KM98], 3.3, the linear system m b D = −m K + c D i i Y i i deﬁnes a birational morphism for m ∈ N sufﬁciently large, with exceptional locus the union R of curves disjoint from D. Adjunction shows R is a contractible conﬁguration of (−2)-curves, which gives (1.3). (2.1) follows from the cone theorem [KM98], 3.7. We show (1.1) implies (1.2). By the Riemann-Roch theorem, if W is a Weil divisor (on any smooth surface) and W > 0 then either W or −W is big (i.e., the rational map given by |nW| is birational for sufﬁciently large n). So, possibly replacing the divisor by its negative, we may assume W = a D is big. Write i i W = a D = W + (−a )D . i i i i a >0 −a >0 i i Thus W is big, and replacing W by nW , we may assume all a ≥ 0and |W| deﬁnes a birational (rational) map. Subtracting off the divisorial base-locus (which does not affect the rational map) we may further assume the base locus is at most zero dimensional. Now W = b D is effective, nef and big, and supported on D. We show we may assume that i i in addition b > 0and W · D > 0for each i.If W · D > 0, then we may assume b > 0 i i i i (by adding D to W if necessary). Now consider the set S ⊂{1,..., n} of components D of D such that W · D = 0. By connectedness of D we ﬁnd b > 0for each i ∈ S. i i i Thus Supp(W) = D. By the Hodge index theorem the intersection matrix (D · D ) is i j i,j∈S MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 143 negative deﬁnite. Hence there exists a linear combination E = α D ,with α ∈ Z for i i i i∈S each i ∈ S, such that E · D < 0for each i ∈ S. Now replacing W by W − E, we obtain W · D > 0for each i = 1,..., n. Finally we show (1.3) implies (1.1). Since U = Y \ D is the resolution of an afﬁne variety U with Du Val singularities, we have U = Y \ D where Y is a normal projective surface and D is a Weil divisor such that D is the support of an ample divisor A. Let −1 π : Y → Y be a resolution of singularities such that π| −1 : π (U ) → U is the reso- π (U ) lution U → U . Furthermore, we can assume the inclusion U ⊂ Y extends to a birational ˜ ˜ ˜ ˜ morphism f : Y → Y. Let D be the inverse image of D under π,so Y \ D = U. The ∗ ∗ ∗ divisor π A has support D. So we can write π A = f ( a D ) + μ E where the E i i j j j 2 ∗ 2 2 are the f -exceptional curves and a ,μ ∈ Z.Then ( a D ) ≥ (π A) = A > 0. i j i i Corollary 6.11. —Let (Y, D) be a positive Looijenga pair. Let P = NE(Y). The multi- can D plication rule Theorem 2.34 applied with D = D determines a ﬁnitely generated T -equivariant R = k[P]-algebra structure on the free R-module A = R · ϑ . q∈B(Z) Furthermore, Spec A → Spec R is a ﬂat afﬁne family of Gorenstein semi-log canonical (SLC) surfaces with central ﬁbre V , and smooth generic ﬁbre. Any collection of ϑ whose restrictions generate A/m = n q H (V , O ) generate A as an R-algebra. In particular the ϑ generate for n ≥ 3. n V v n i Proof. — Everything but the singularity statement follows from Theorem 0.2.The Gorenstein SLC locus in the base is open, and T -equivariant. Taking a big and nef divi- sor H = a D with a > 0and H· D > 0for all i, we obtain a one-parameter subgroup i i i i D D of T given by the map χ(T ) → Z, e → a . By deﬁnition of the weights, the weights D i [C] of z for C ∈ NE(Y) and ϑ for p ∈ B(Z) are all non-negative. Thus the corresponding torus T = G gives a contracting action. In particular, since V is Gorenstein and SLC, H m n all ﬁbres are Gorenstein and SLC. Moreover, the map 0 0 H (Z, O ) → H (Z , O ) Z m Z is injective, where Z is the singular locus of the family X → S. But since R → H (O , O ) is not injective, as shown in the proof of Theorem 0.1, we deduce that Z Z m m the map R → H (Z, O ) is not injective. Letting f be in the kernel of the map, the ﬁbres over Spec R \ V(f ) are smooth. In Part II we will prove that when D is positive, our mirror family admits a canon- ical ﬁbrewise T -equivariant compactiﬁcation X ⊂ (Z, D). The restriction (Z, D) → T := Pic(Y) ⊗ G comes with a trivialization D → D × T . We will show that (Z, D) Y m ∗ Y 144 MARK GROSS, PAUL HACKING, AND SEAN KEEL is the universal family of Looijenga pairs (Z, D ) deformation equivalent to (Y, D) to- gether with a choice of isomorphism D → D . Now for any positive pair (Z, D ) to- Z ∗ Z gether with a choice of isomorphism φ: D → D , our construction equips the com- Z ∗ plement U = Z \ D with canonical theta functions ϑ , q ∈ B (Z). We will give a Z q (Z,D) characterisation in terms of the intrinsic geometry of (Z, D ). Changing the choice of isomorphism φ changes ϑ by a character of T = Aut (D ), the identity component of q ∗ Aut(D ). Here we illustrate with two examples: Example 6.12. — Consider ﬁrst the case (Y, D) a5-cycle of (−1)-curves on the (unique) degree 5 del Pezzo surface, Example 3.7.Inthis case T = T = Pic(Y)⊗ G , Y Z m andthusbythe T -equivariance, all ﬁbres of the restriction X → T are isomorphic. We consider the ﬁbre over the identity e ∈ T , thus specializing the equations of Example 3.7 by setting all z = 1. It’s well known that these equations deﬁne an embedding of the 5 5 original U = Y\ D into A —ifwetakethe closurein P (for the standard compactiﬁcation 5 5 A ⊂ P ) one checks easily we obtain Y with D the hyperplane section at inﬁnity. Now it is easy to compute the zeroes and poles: (ϑ ) = E + D − D − D v i i i+2 i−2 (indices mod 5). In particular {ϑ = 0}= E ∩ U ⊂ U, which characterizes ϑ up to v i v i i scaling. Example 6.13. —Now let (Y, D = D + D + D ) be (the deformation type of) 1 2 3 a cubic surface together with a triangle of lines. Let X ⊂ Spec(k[NE(Y)]) × A be the canonical embedding given by ϑ := ϑ , i = 1, 2, 3. In this case, as we shall see in Part II, i v the scattering diagram is particularly beautiful, with every ray d of rational slope occur- ring, with precisely six curves on the cubic surface contributing to f . We will also show in Part II that the mirror is given by the equation D 2 E D π H D +D +D i ij i 1 2 3 ϑ ϑ ϑ = z ϑ + z z ϑ + z − 4z . 1 2 3 i i i j π Here the E are the interior (−1)-curves meeting D ,and thesum over π is the sum over ij i ¯ ¯ ¯ all possible toric models π : Y → Yof (Y, D) to a pair (Y, D) isomorphic to P with its toric boundary. (Such π are permuted simply transitively by the Weyl group W(D ) by [L81], Prop. 4.5, p. 283.) The same family, in the same canonical coordinates, was discov- ered by Oblomkov [Ob04]. As we learned from Dolgachev, after a change of variables (in A ), and restricting to T (the locus over which the ﬁbers have at worst Du Val singular- ities) this is identiﬁed with the universal family of afﬁne cubic surfaces (the complement to a triangle of lines on projective cubic surface) constructed by Cayley in [C1869]. The 3 3 universal family of cubic surfaces with triangle is obtained as the closure in A ⊂ P .In particular, as in the ﬁrst example, our mirror family compactiﬁes naturally to the universal MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 145 family of Looijenga pairs deformation equivalent to the original (Y, D). There is again a geometric characterisation of ϑ (up to scaling): The linear system |−K − D|=|D + D | i Y i j k (here {i, j, k}={1, 2, 3}) is a basepoint free pencil. It deﬁnes a ruling π : Y → P which 1 1 restricts to a double cover D → P .Let {a, b}⊂ P be the branch points of π| .Let i D 1 1 p = π(D + D ) ∈ P . There is a unique point q ∈ P \{a, b, p} ﬁxed by the unique in- j k 1 ∗ interchanging a and b and ﬁxing p.Let Q = π (q) ∈|D + D | be the volution of P j k corresponding divisor. The curve Q ⊂ P is a smooth conic. In Part II we will show Proposition 6.14. — (ϑ ) = Q − D − D . i j k 7. Looijenga’s conjecture In this section we apply the main construction of this paper to give a proof of Looijenga’s conjecture on smoothability of cusp singularities, Theorem 0.5. The simple conceptual idea is explained in the introduction. Here we give the rather involved details. 7.1. Duality of cusp singularities. — We review the notion of dual cusp singularities. By deﬁnition, a cusp is a normal surface singularity for which the exceptional locus of the minimal resolution is a cycle of rational curves. The self-intersections of these ex- ceptional curves determine the analytic type of the singularity, see [L73]. Cusps have a quotient construction due to Hirzebruch [Hi73] which we explain here. See also [L81], III, Section 2 for this point of view. Let M = Z ,and let T ∈ SL(M) be a hyperbolic matrix, i.e., T has a real eigen- value λ> 1. Then T determines a pair of dual cusps as follows: Let w ,w ∈ M be 1 2 R eigenvectors with eigenvalues λ = 1/λ, λ = λ, chosen so that w ∧ w > 0 (in the stan- 1 2 1 2 dard counter-clockwise orientation of R ). Let C, C be the strictly convex cones spanned by w ,w and w ,−w ,and let C, C be their interiors, either of which is preserved by 1 2 2 1 T. Let U , U be the corresponding tube domains, i.e., C C U := z ∈ M | Im(z) ∈ C /M ⊂ M /M = M ⊗ G . C C C m T acts freely and properly discontinuously on U , U .Write Y , Y for the holomor- C C C C phic hulls of U /, U /,where is the group generated by T. These each have one C C additional point, p ∈ Y , p ∈ Y ,and (Y , p), (Y , p ) are normal surface germs of C C C C cusps. Deﬁnition 7.1. — (Y , p) and (Y , p ) are dual cusp singularities. C C All cusps (and their duals) arise this way. Remark 7.2. — If M is identiﬁed with its dual by choosing an isomorphism M = Z, the cone C is identiﬁed with the dual cone of C. In this way C / and C/ are dual 146 MARK GROSS, PAUL HACKING, AND SEAN KEEL integral afﬁne manifolds, which suggests that the duality between the corresponding cusps is a form of mirror symmetry. To resolve the cusp singularities of Y , say, one considers the convex hull of integral points in C, and let v , i ∈ Z, be the integral points in ∂, listed so that v i i−1 and v are the integral points adjacent to v on ∂.Let be the inﬁnite fan with two- i+1 i dimensional cones generated by v ,v for i ∈ Z. Note that T acts on , and thus acts by i−1 i translation T(v ) = v for some integer n, which we can take to be positive by reversing i i+n the ordering of the v if necessary. Then X is a toric variety with an inﬁnite chain of P ’s, and T acts on X . There is a tubular neighbourhood N of this inﬁnite chain of P ’s on which the group generated by T acts properly discontinuously, see [AMRT75], p. 48. Then N/ is a minimal resolution of singularities of a neighbourhood of the singularity of Y . Note the exceptional divisors D , i taken modulo n,with D corresponding to the C i i ray of generated by v ,satisfy (1.2). Thus if there is a Looijenga pair (Y, D ) with 2 2 D = D + ··· + D and (D ) = D for each i, the corresponding afﬁne manifold with 1 n i i singularities is precisely B=| |/ by Example 1.10,and B = C/. In fact, the dual cusp singularity can be described directly from the cone C and the polyhedron : Lemma 7.3. —Let T, C, and the v be as above, giving a cusp singularity p ∈ Y .Let Z i C be the toric variety (only locally of ﬁnite type) associated to .Let E ⊂ Z be the toric boundary of Z,an inﬁnite chain of smooth rational curves corresponding to the boundary of . Then there exists a tubular neighbourhood E ⊂ N ⊂ Z such that the action on induces a properly discontinuous action on N.Let F ⊂ X denote the quotient of E ⊂ N by .So F is a cycle of smooth rational curves. Then F ⊂ X can be contracted to a singularity p ∈ X, which is a copy of the dual cusp p ∈ Y .Moreover, X is obtained from the minimal resolution of p ∈ X by contracting all the (−2)-curves. Proof.—Let be the normal fan for the polytope and C the closure of its support. We observe that C coincides with the dual of C, together with the induced -action. By Remark 7.2, it follows that X is a partial resolution of a copy of the dual cusp. The surface X has Du Val singularities of type A. Indeed, v is a vertex of iff m := −D > 2. The corresponding point of Z is smooth if m = 3 and a singularity of i i type A if m > 3 (by direct calculation using v + v = m v ). Also K is relatively m −3 i i−1 i+1 i i ˜ ample over X by Lemma 7.4 below. Indeed, the vectors u := v − v are the primitive i i i−1 integral vectors in the direction of the edges of ,and u − u = v + v − 2v = (m − 2)v , i+1 i i+1 i−1 i i i so the lines v + R· (u − u ) = R· v all meet at the origin. We deduce that X is obtained i i+1 i i from the minimal resolution of X by contracting all (−2)-curves as claimed. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 147 Lemma 7.4. —Let P ⊂ R be a rational convex polygon and X the associated toric surface. Fix an orientation of the boundary of P and let e , e , e denote oriented consecutive edges of the boundary 0 1 2 of P.Let C ⊂ X denote the component of the toric boundary associated to the bounded edge e ⊂ P.Let v = e ∩ e and v = e ∩ e be the vertices of e .Let u , u , u ∈ Z denote the primitive integral 0 0 1 1 1 2 1 0 1 2 vectors in the direction of e , e , e . Then K · C > 0 if and only if the lines v + R(u − u ) and 0 1 2 X 0 1 0 v + R(u − u ) meet on the opposite side of e to P. 1 2 1 1 2 2 Proof.—Write M = Z ⊂ R for the lattice of characters of the torus of X. Choose an orientation M ' Z and use it to identify M with its dual lattice N. Let U ⊂ X denote the union of the two toric afﬁne open subsets corresponding to the vertices v and v of P. Then U is a toric open neighbourhood of C ⊂ X. Then, under this identiﬁcation anduptoasign,the fan of U in N consists of the two cones u , u , u , u R 0 1 R 1 2 R ≥0 ≥0 and their faces. The condition on the lines v + R(u − u ) and v + R(u − u ) in the 0 1 0 1 2 1 statement is equivalent to the condition that the primitive generator u of the central ray of the fan lies on the same side of the afﬁne line spanned by the primitive generators u and u of the two outer rays of as the origin 0 ∈ N. Now by [R83], 4.3, this condition is equivalent to K · C > 0. Because this quotient construction is analytic we will have to deal with convergence issues to show that our construction extends to this analytic situation. 7.2. Cusp family. — In this subsection, we ﬁx the following. Let (Y, D) be a rational surface with anti-canonical cycle, now over the ﬁeld k = C.Weobtain (B, ),with having one-dimensional cones ρ and two-dimensional cones σ as usual. i i,i+1 We assume that the intersection matrix (D · D ) is negative deﬁnite. Let i j 1≤i,j≤n f : Y → Y be the contraction (in the analytic category) of D ⊂ Y to a cusp singular- ity q ∈ Y . We assume that f is the minimal resolution of Y ,thatis, D ≤−2for all i.We further assume that n ≥ 3 to avoid additional technical issues of the ﬂavour dealt with in Section 6.2. The case of Looijenga’s conjecture with n ≤ 2 is in fact trivial, see the proof of Theorem 7.13. Let L be a nef divisor on Y such that NE(Y) ∩ L =D ,..., D . R 1 n R ≥0 ≥0 Here the subscript R denotes the real cone in A (Y, R) generated by the given elements ≥0 1 or set. (Indeed, if Y is projective we can take L = h A for A an ample divisor on Y . In general, let A be an ample divisor on Y. There exist unique a ∈ Q such that L := A + a D is orthogonal to D for each j . By negative deﬁniteness of D ,..., D ,we i i j 1 n have a > 0for each i. It follows that L is nef.) Let σ ⊂ A (Y, R) be a rational polyhedral cone containing NE(Y),P = σ ∩ P 1 P A (Y, Z) the associated toric monoid. We assume that σ is strictly convex and σ ∩ L 1 P P is a face of σ .Let m = P \{0} and J = P \ P ∩ L ⊂ P, the radical monomial ideal P 148 MARK GROSS, PAUL HACKING, AND SEAN KEEL associated to the face σ ∩ L of σ . We will write S = Spec k[P], and for any monomial P P ideal I, we write S = Spec k[P]/I. We take the multivalued piecewise linear function ϕ as usual to have bending pa- rameter κ =[D ]∈ P. We wish to build a deformation of the n-vertex over S with ρ,ϕ ρ I I = J. However, this is already a problem over S because none of the κ lie in J. Thus J ρ,ϕ we can’t apply the results of Section 2 directly as the standard open sets U will not glue ρ,J compatibly because of issues involving triple intersections. To deal with this, we need to shrink these open sets. This procedure is carried out as follows. Theorem 7.5. —Fix R > 1. There exists an analytic open neighbourhood S of 0 ∈ S and an analytic ﬂat family f : X → S together with a section s: S → X satisfying the following properties: J J J J J (1) The general ﬁbre X of f is a Stein analytic surface with a unique singularity s(t) ∈ X J,t J J,t isomorphic to the dual cusp to q ∈ Y . (2) For each ray ρ ∈ there is an open analytic subset V ⊂ X and open analytic embed- i ρ ,J J dings V ⊂ (X , X , X ) ∈ U ||X | < R|X |, |X | < R|X | ⊂ U ρ ,J i−1 i i+1 ρ ,J i−1 i i+1 i ρ ,J i i i where −D [D ] 2 ρ i U := V X X − z X ⊂ A × (G ) × S ρ ,J i−1 i+1 m X J i i X ,X i i−1 i+1 such that (a) X := X \ s(S ) = V . J ρ,J J J ρ∈ (b) V ∩ V =∅ unless ρ = ρ or ρ and ρ are the edges of a maximal cone σ ∈ . ρ,J ρ ,J (3) The restriction of X /S to S N+1 is identiﬁed with an analytic neighbourhood of the vertex J J+m can in the restriction of the family X N+1 /S N+1 given by Theorem 2.28, (1) with D = D , m m for each N ≥ 0. Proof. — (1) We use the notation of Example 1.10, so that the pair (Y, D) deter- mines an inﬁnite fan in M with the primitive generators of the rays being the v for R i ˜ ˜ i ∈ Z.Wealsohave T ∈ SL(M) acting on the fan .Wehave B=| |/,where is the group generated by T. Now as in Section 7.1,let ⊂ M be the convex hull of the points v ∈ M. Thus R i ˜ ˜ is an inﬁnite convex polytope. Let be the subdivision of induced by .Inwhat follows, we will build a Mumford degeneration Z/S with special ﬁbre Z the stable toric J 0 variety associated to .Inother words, Z will be the union of the toric surfaces associ- ated to the maximal polytopes in . S is the afﬁne toric variety associated to the face σ := σ ∩ L of σ ,and P := J bdy P P bdy σ ∩ A (Y, Z) contains the classes of the components of the boundary of Y. We deﬁne bdy 1 gp a piecewise linear convex function ϕ˜:| |→ P ⊗ R by restriction of a piecewise bdy MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 149 linear convex function ϕ˜ on | |. This function is the single-valued representative for ϕ on the universal cover of B , and as such, is deﬁned up to an integral linear function by specifying its bending parameter κ =[D ]∈ P ρ ,ϕ˜ i mod n bdy if ρ = R v . i ≥0 i Then ϕ˜ determines a Mumford degeneration: this is a slight generalization of Sec- tion 1.3.One deﬁnes gp := (m, r)| m ∈ , r∈˜ ϕ(m) + σ ⊂ M ⊕ P ⊗ R . bdy R Z bdy Let C() be the closure of gp (sm, sr, s)| (m, r) ∈ , s ≥ 0 ⊂ M ⊕ P ⊗ R ⊕ R. R Z bdy gp Then C[C() ∩ (M ⊕ P ⊕ Z)] has a natural grading given by the last coordinate, and bdy the degree zero part of this ring is easily seen to contain C[P ].Thusweobtainthe bdy Mumford family determined by ϕ˜ as gp Z := Proj C C() ∩ M ⊕ P ⊕ Z → Spec C[P ]= S . bdy J bdy One sees easily that the ﬁbre over 0 ∈ S of Z → S has inﬁnitely many compo- J J nents indexed by the 2-cells of the subdivision of ,eachofwhichisacopy of the blowup of A at the origin. The general ﬁbre is a toric surface (only locally of ﬁnite type) containing an inﬁnite chain of smooth rational curves, which specializes to the union of the exceptional curves of the blowups in Z . By construction acts on Z over S (because 0 J ϕ˜ is -invariant modulo integral afﬁne functions). Let E ⊂ Z/S be the family of curves described above (the relative toric boundary). The group acts properly discontinuously on a tubular neighbourhood N of E ⊂ Z(cf. [AMRT75], p. 48). Let p: (F ⊂ X) → S denote the quotient of (E ⊂ N) → S by . The divisor F ⊂ X is Cartier and the dual of its normal bundle is relatively ample over a neighbourhood of 0 ∈ S . Indeed, the special ﬁbre X is a union of n irreducible J 0 components each isomorphic to a tubular neighbourhood of the exceptional locus in the blowup of A ,and F ⊂ X is the cycle of n smooth rational curves formed by the 0 0 exceptional curves of the blowups. Hence the normal bundle of F in X has degree 0 0 1 ⊗k −1 on each component of F . Moreover, we have R p (N ) = 0for each k > 0. 0 ∗ F/X 1 ∨ ⊗k Indeed, by cohomology and base change it sufﬁces to show that H ((N ) ) = 0, F /X 0 0 and this follows from Serre duality. Now by a relative version of Grauert’s contractibility criterion [F75], Thm. 2, taking global sections of the structure sheaf deﬁnes a contraction p: X → X /S to a family of Stein analytic spaces with exceptional locus F. The general J J ﬁbre X of X /S is the dual cusp by Lemma 7.3. The section s : S → X takes x ∈ S to J,t J J J J J the cusp of X . J,x 150 MARK GROSS, PAUL HACKING, AND SEAN KEEL We now show that X /S is ﬂat and the special ﬁbre is the neighbourhood of the J J n-vertex obtained by contracting F ⊂ X . The key point is that R p O is a locally free 0 0 ∗ ˜ O -module, cf. [W76], Theorem 1.4(b). Indeed, we have R p O (−F) = 0 by cohomology and base change, the theorem on formal functions, and the vanishing 1 ∨ ⊗k H ((N ) ) = 0for k > 0 used above. So, pushing forward the exact sequence F /X 0 0 0 → O (−F) → O → O → 0 ˜ ˜ X X we obtain 1 1 R p O = R p O ' O . ∗ ˜ ∗ F S X J Recall that S is a toric variety, so in particular Cohen-Macaulay. Let t ,..., t be a regular J 1 r sequence at 0 ∈ S of length dim S and write J J S = V(t ,..., t ) ⊂ S , 1 i J i i i i ˜ ˜ X = X| i ,and let X /S be the family over S deﬁned by O i = p O i . Arguing as above S X ∗ X J J J J J we ﬁnd that R p O i ' O . Pushing forward the exact sequence ∗ ˜ X S ·t i+1 0 → O i−→O i−→O i+1−→0 ˜ ˜ ˜ X X X we deduce that the natural map (7.1) O i /t O i → O i+1 i+1 X X X J J is an isomorphism. Hence by the local criterion of ﬂatness [Ma89], Ex. 22.3, p. 178, it r r r sufﬁces to show that X /S is ﬂat with special ﬁbre the n-vertex. But S is the spectrum of J J J an Artinian C-algebra, so this follows from [W76], Theorem 1.4(b). o [D ] i mod n (2) Write Z = Z \ E, and m =−D , a = z for i ∈ Z.Wehaveanopen i i i mod n covering Z = U , i,J i∈Z where i 2 U = V x x − a x ⊂ A × (G ) × S . i,J i−1 i+1 i m x J i i x ,x i−1 i+1 Similarly, we have an open covering Z = U , i,J i∈Z MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 151 where m −2 U = V x x − a x ⊂ A × S , i,J i J i−1 i+1 i x ,x ,x i−1 i+1 ¯ ¯ E ∩ U = V(x ) ⊂ U , i,J i i,J and U \ E = U via x = x x , x = x x . (Note that m =−D ≥ 2 by assump- i,J i,J i−1 i i+1 i i i−1 i+1 i tion.) Recall that the inﬁnite cyclic group acts on Z/S , there is a -invariant tubular neighbourhood N ⊂ ZofE ⊂ Z on which the action is properly discontinuous, and F ⊂ X is obtained as the quotient of E ⊂ Nby . In terms of the open covering above the action is given by U → U , x → x . Note that the map U ∩ N → Xis not an open i,J i+n,J j j+n i,J embedding (because, for example, U contains the general ﬁbre of Z /S ). Fix R > 1. i,J J We deﬁne W ⊂ U ∩ Nby i i,J W = (x , x , x ) ∈ U ∩ N||x | < R|x |, |x | < R|x | i i−1 i i+1 i,J i−1 i i+1 i and similarly deﬁne W ⊂ U ∩ Nby i i,J W = x , x , x ∈ U ∩ N | x < R, x < R . i i i,J i−1 i+1 i−1 i+1 Then W \ E = W . i i The W cover the special ﬁbre E of E/S (using R > 1). The open set W ⊂ i 0 J i ˜ ˜ Nis -invariant, the quotient (N, E) → (X, F) by is a covering map, and p: X → X is proper with exceptional locus F. Hence we may assume (passing to an analytic neighbourhood S of 0 ∈ S and s(0) ∈ X )that N = W . J J i By Lemma 7.6 below there exists δ> 0such that W ∩ (x , x , x )||x | <δ ⊂ |x | < 1 ∀j i i−1 i i+1 i j for each i. We replace W by W ∩{|x | <δ},and modify W similarly. Then as above we i i i i may assume that N = W ,and N ⊂{|x | < 1 ∀i}.Weclaim that W ∩ W =∅ for all i i i j j > i + 1. It sufﬁces to work on the general ﬁbre of Z /S , which is an algebraic torus. The coordinate functions x are characters of this torus (up to a multiplicative constant). By construction we have |x | < 1for each i on N. Hence, shrinking the base S ,wemay m −2 i 2 assume that |a x | < 1/R for each i.The relation x x = a x i−1 i+1 i gives the inequality |x /x | < 1/R |x /x |. i+1 i i i−1 152 MARK GROSS, PAUL HACKING, AND SEAN KEEL Combining such inequalities we obtain j−i (7.2) |x /x | < 1/R |x /x | for j > i. j+1 j i+1 i Now |x /x | < RonW and |x /x | < RonW ,so |x /x | >(1/R )|x /x | on W ∩ i+1 i i j−1 j j j j−1 i+1 i i W .For j > i + 1 this contradicts the inequality (7.2), hence W ∩ W =∅ as claimed. j i j It follows that W embeds in X ,using thefactthatwehaveassumed n ≥ 3. Let i J V denote its image, with indices now understood modulo n.Thus X = V and the i i inverse image of V is the (disjoint) union of W such that j ≡ i mod n.Wehave V ∩ V = i j i j ∅ for j = i − 1, i, i + 1 by our claim above. So, writing V := V for ρ the ray of ρ,J i corresponding to D , the condition (2)(b) is satisﬁed. (3) We have the open covering X = V and open embeddings V ⊂ U ,and i i i,J an open covering X = U N+1 . The restrictions of U /S and U N+1 /S N+1 to N+1 i,m i,J J i,m m N+1 S are identiﬁed, and the gluing maps coincide. It follows that the restriction of J+m N+1 N+1 X /S is identiﬁed with a neighbourhood of the vertex in the restriction of X /S J J m m using Lemma 2.10. Lemma 7.6. — We use the notation of the proof of Theorem 7.5(2). Let x be the coordinate functions on Z = U . There exists δ> 0 such that on each open set U if |x | <δ, |x | < i,J i,J i i−1 R|x |, |x | < R|x |,and |z | < 1 for all p ∈ P then |x | < 1 for all j . i i+1 i bdy j Proof. — The points v , v ,and v are consecutive integral points on the bound- i−1 i i+1 ary of the inﬁnite convex polytope with asymptotic directions w ,w . It follows that 1 2 w = α v + β (v − v ) and w = α v + β (v − v ) for some α ,β ,α ,β ∈ R . 1 i1 i i1 i−1 i 2 i2 i i2 i+1 i i1 i1 i2 i2 >0 Note that the ratios β /α ,β /α only depend on i modulo n (because w ,w are eigen- i1 i1 i2 i2 1 2 vectors of T and T(v ) = v ). Let μ be the maximum of the ratios β /α ,β /α for i i+n i1 i1 i2 i2 −μ i = 1,..., n.Let δ = R .If j > i then v = αv + β(v − v ) with β/α < β /α . j i i+1 i i2 i2 p α β The coordinate function x can be written as z x (x /x ) on V ,where p ∈ P .Thus j i+1 i i bdy α β p |x | <δ R < 1for |x | <δ and |z | < 1. Thesameistrue for j < i by symmetry. j i 7.3. Thickening of the cusp family. — We continue to work with the setup at the be- ginning of Section 7.2,with (Y, D),L, σ ,P, m and J as given there. Theorem 7.7. —Let p : X → S be the analytic family of Theorem 7.5. Possibly after J J replacing S by a smaller neighbourhood of 0 ∈ S and X by a smaller neighbourhood of s(S ) ⊂ X , J J J J J independent of the choice of I below, the following holds. Let I ⊂ P be a monomial ideal such that I = J and let S ⊂ S denote the induced thickening of S ⊂ S . There is an inﬁnitesimal deformation I J I J N+1 f : X → S of f : X → S such that for each N > 0 the restriction to Spec k[P]/(I + m ) is I I J J I J identiﬁed with an analytic neighbourhood of the vertex in the restriction of the family X N+1 /S N+1 given m m can by Theorem 2.28, (1) applied with D = D . can Proof. — As usual, D is the canonical scattering diagram on B associated to the pair (Y, D). Note that the hypotheses (I) and (II) of Theorem 3.8 are satisﬁed for the MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 153 ideal J (but not (III)). This is because we can take L = A+ a D with A ample on Y and i i a > 0, and any A -class β intersects the D non-negatively. Then for any k there are only i i 1 1 a ﬁnite number of A -classes β such that L · β< k.Inaddition,there areno A -classes β with β · L = 0. can Let D be the scattering diagram obtained by reducing D modulo I as follows. can For each ray d of D , we truncate the attached function f by removing monomial terms lying in I · k[P ], and we discard the ray if the truncated function equals 1. Because of (II), D has only ﬁnitely many rays d and the attached functions f areﬁnitesumsof monomials. Because of (I), D is empty if I = J. We use the scattering diagram D to deﬁne a complex analytic space X /S as I I follows. Recall from (2.7) the description of the schemes U /S for ρ = ρ ∈ : ρ ,I I i −D [D ] ρ 2 U = V X X − z X f ⊂ A × (G ) × S . ρ ,I i−1 i+1 ρ m X I i i X ,X i i−1 i+1 We have open subsets U , U ⊂ U deﬁned by X = 0and X = 0re- ρ ,σ ,I ρ ,σ ,I ρ ,I i−1 i+1 i i−1,i i i,i+1 i spectively. We have canonical identiﬁcations (7.3)U = U = U ρ ,σ ,I σ ,σ ,I ρ ,σ ,I i i,i+1 i,i+1 i,i+1 i+1 i,i+1 where U = (G ) × S . σ ,σ m I i,i+1 i,i+1 X ,X i i+1 Recall that we have an open covering X = V and open analytic embeddings ρ,J J ρ∈ V ⊂ U .Write ρ,J ρ,J V := (V ∩ U ) ∩ (V ∩ U ) ⊂ U σ ,σ ,J ρ ,J ρ ,σ ,J ρ ,J ρ ,σ ,J σ ,σ ,J i,i+1 i,i+1 i i i,i+1 i+1 i+1 i,i+1 i,i+1 i,i+1 where we use the identiﬁcation (7.3). Let V ⊂ V ,V ⊂ V denote ρ ,σ ,J ρ ,J ρ ,σ ,J ρ ,J i i,i+1 i i+1 i,i+1 i+1 the open subsets corresponding to V under (7.3). Let V ,V , etc., be the σ ,σ ,J ρ ,I ρ ,σ ,I i,i+1 i,i+1 i i i,i+1 inﬁnitesimal thickenings of these open sets determined by the thickenings U of U . ρ ,I ρ ,J i i Let θ : U → U be the gluing isomorphism deﬁned as in Section 2.2. γ,D ρ ,σ ,I ρ ,σ ,I i+1 i,i+1 i i,i+1 can Note that as the canonical scattering diagram D is trivial modulo J, θ restricts to γ,D the identiﬁcation (7.3) modulo J, and thus restricts to an isomorphism V → V . ρ ,σ ,I ρ ,σ ,I i+1 i,i+1 i i,i+1 Gluing the V via these isomorphisms we obtain an inﬁnitesimal deformation X /S of ρ,I I I X /S . Note that there are no triple overlaps of the V by Theorem 7.5(2)(b), hence no ρ,J J J compatibility condition for the gluing automorphisms. It is clear from the construction that the families X /S and X N+1 /S N+1 are compatible. m m I I We deﬁne sections ϑ ∈ (X , O ) for q ∈ B(Z), compatible with the sections q X of Theorem 2.28, (2). We proceed as in the algebraic case: we ﬁrst deﬁne a local sec- tion Lift (q) for each choice of basepoint Q ∈ B \ Supp(D) on a corresponding open Q 0 154 MARK GROSS, PAUL HACKING, AND SEAN KEEL patch of X using the broken lines construction. The new difﬁculty here is that the func- tions Lift (q) are not algebraic, even over the unthickened locus S . Indeed, by deﬁnition Lift (q) = Mono(γ ) is a formal sum of monomials corresponding to broken lines γ for q with endpoint Q. Note that with our current choice of ideal J, this sum is always inﬁnite, as is already evident in Example 2.18. So we must prove convergence. This is done in Section 7.3.1, see Propositions 7.10 and 7.11. Once this convergence is proved, we observe that these patch to give well-deﬁned can o global sections. This follows from the consistency of D and compatibility of X /S with I I X /S N+1 for N ≥ 0. N+1 m We deﬁne an inﬁnitesimal thickening X /S of X /S by O = i O where I J X ∗ X I J I i: X ⊂ X is the inclusion. Then X /S is ﬂat by Lemma 2.29 and the existence of the J I J I lifts ϑ . 7.3.1. Convergence of lifts. — Let C ⊂ M be the closure of the support | | of the fan , a closed convex cone. Let w ,w be generators of C. Then w ,w are eigen- 1 2 1 2 −1 vectors of T with eigenvalues λ ,λ for some λ ∈ R. We may assume that λ> 1. Let ˜ ˜ π : B → B denote the universal cover of B .So B is identiﬁed with C = Int(C),with 0 0 0 0 ˜ ˜ deck transformations given by the action of =T on C. Let P , ϕ˜,and B (Z) denote ˜ ˜ the pullbacks of P , ϕ,and B (Z).Let D denote the scattering diagram on B induced 0 0 gp by D. We ﬁx a trivialization of P as the constant sheaf with ﬁbre P ⊕ M. The behaviour of the broken lines γ is best studied by passing to the universal ˜ ˜ ˜ ˜ cover B of B .Let Q ∈ B , q ∈ B (Z), and choose lifts Q ∈ B , q˜ ∈ B (Z). Then a broken 0 0 0 0 0 0 line γ on B for q with endpoint Q lifts uniquely to a broken line γ˜ on B for T (q˜) with 0 0 endpoint Q, for some N ∈ Z, and the attached monomials are identiﬁed via P = π P . Note that T (q˜) approaches R · w as N→∞ and R · w as N→−∞. ≥0 2 ≥0 1 ˜ ˜ ˜ If γ is a broken line for a point q˜ ∈ B (Z) with endpoint Q ∈ B then γ : (−∞, 0] 0 0 ˜ ˜ ˜ → B is a piecewise linear path in B = C with initial direction −˜ q, ending at Q, and 0 0 ˜ ˜ crossing all the rays of D between R q˜ and R Q in order. Let t ,..., t ∈ (−∞, 0) ≥0 ≥0 1 l denote the points where γ is not afﬁne linear. Each point γ(t ) lies on a ray d of D and i i the change γ (t + )− γ (t − ) in the direction of γ as it crosses d is an integral multiple i i i of the primitive generator of d . Moreover this multiple is positive because each ray of the canonical scattering diagram is an outgoing ray in the terminology of Deﬁnition 2.13.So the path γ is “convex when viewed from the origin”. It is convenient for the convergence calculation to decompose the monomials for broken lines as follows. Let γ : (−∞, 0]→ B be a broken line, t ∈ (−∞, 0] a point such that γ is afﬁne linear near t and γ(t) lies in the interior of a maximal cone σ of ,and cz the monomial attached to the domain of linearity of γ containing t.Here c ∈ k and gp q ϕ˜ (m) q−˜ ϕ (m) σ σ q ∈ P ⊂ P ⊕ M. We write cz = az ,where m = r(q) ∈ Mand a = cz is a ϕ˜ monomial in k[P]. We also use the same decomposition for the monomials occurring in ˜ ˜ the scattering functions f for d ∈ D.Let d be a ray in D with primitive integral generator m ∈ Mand τ = τ the smallest cone of containing d.Then f − 1 is a sum of monomials d d MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 155 q q ϕ˜ (m) cz where c ∈ k and q ∈ P ,0 = −r(q) ∈ d. We write cz = az where m = r(q) and ϕ˜ q−˜ ϕ (m) a = cz is a monomial in k[P]. The scattering diagram D on B is ﬁnite (because we have reduced modulo I), and ˜ ˜ ˜ D is its inverse image under π : B → B .Thus D has only ﬁnitely many -orbits of 0 0 rays. Moreover, the -action on the scattering functions f is induced by the given action ϕ˜ (m) on M and the trivial action on k[P] as follows: writing f = 1 + a z as above, R d m ϕ˜ (T(m)) T(τ ) f = 1 + a z . T(d) m ˜ ˜ Lemma 7.8. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal ˜ ˜ ˜ cone σ ∈ . We consider broken lines γ on B for T (q˜),some N ∈ Z, with endpoint Q, such that Mono(γ ) ∈ / I · k[P ].Let k ≥ 0 be such that (k + 1)J ⊂ I. ϕ˜ (1) The number of bends of γ is at most k. N k (2) The number of broken lines for T (q˜) with endpoint Q is O(|N| ). N ϕ˜ (m ) σ γ (3) For γ a broken line for T (q˜) with endpoint Q write Mono(γ ) = a z where a is γ γ a monomial in k[P]. Assume that |z | < < 1 for all p ∈ P. Then |N| [D ] |a |= O z . ρ∈ dim ρ=1 Proof. — (1) At a bend t ∈ (−∞, 0] of γ the attached monomial c z is replaced by i i q q q q i+1 i the monomial c z = cz · c z where cz is a term in a positive power of the scattering i+1 i function f associated to the ray d containing γ(t ). In particular cz ∈ J · k[P ]. Since d i ϕ˜ Mono(γ ) ∈ / I · k[P ] and (k + 1)J ⊂ I it follows that there are at most k bends. ϕ˜ (2) Such a broken line crosses O(|N|) scattering rays. If γ is a broken line for T (q˜) ϕ( ˜ T (q˜)) then the initial attached monomial is speciﬁed, equal to z . At a scattering ray d, let u denote the primitive generator of d, f = f the attached function, and let cz be the monomial attached to the incoming segment of the broken line. Then the possible continuations of the broken line past d correspond to the monomial terms in f ,where d =|r(q) ∧ u| is the index of the sublattice of M generated by r(q) and u.Notethat since f ≡ 1mod J · k[P ] the number of monomial terms in f not lying in I · k[P ] ϕ˜ ϕ˜ τ τ d d is bounded independent of d . Further, since there are a ﬁnite number of -orbits of scattering rays, and the -action preserves monomials, there is a bound on the number of monomial terms independent of the ray d. Thus for a broken line γ for T (q˜) there O(|N|) are = O(|N| ) choices of how it may bend by (1). So the total number of broken lines is O(|N| ). (3) By symmetry we may assume that N ≥ 0. Let d ∈ D be a scattering ray, f = f the attached function, and γ a broken line that crosses d. Suppose ﬁrst that d is contained ϕ˜ (m) in the interior of a maximal cone σ of .Let az be the monomial attached to the incoming segment of γ near d.Let u be the primitive generator of d. Then the outgo- ϕ˜ (m ) ing monomial a z is obtained from the incoming monomial by multiplication by a 156 MARK GROSS, PAUL HACKING, AND SEAN KEEL monomial term in f ,where d =|m∧ u|.Write f = 1+ f +···+ f , a sum of monomials. 1 r Since f ≡ 1mod J · k[P ] we have d i i (7.4) f ≡ f ··· f mod I · k[P ]. 1 r σ i ,..., i 1 r i +···+i ≤k 1 r The multinomial coefﬁcient d d! := i ,..., i i !··· i !(d − i −···− i )! 1 r 1 r 1 r k s is bounded by d . The direction of the scattering ray d is u = T (β) where 0 ≤ s ≤ Nand β ∈ M is chosen from a ﬁnite set. The vector m ∈ Mis of the form N s m = T (q˜) − T α i=1 where l ≤ k,0 ≤ s ≤ Nfor each i,and the α ∈ M are chosen from a ﬁnite set. Indeed, as i i q d in the proof of (2), for the monomial terms cz occurring in the powers f of the function f = f attached to a given scattering ray d,onlyﬁnitelymanyexponents q ∈ P occur d ϕ˜ (working modulo I· k[P ]). So there are only ﬁnitely many possible changes of exponent ϕ˜ q for the attached monomial cz of a broken line at a scattering ray modulo the action of . 2 2 Now identify M = Z and let (·( denote the standard norm on M = R .Then 2N d =|m ∧ u|≤(m(·(u(= O λ . So, the coefﬁcient a ∈ k[P] of the outgoing monomial is given by a = c · z · a where 2kN 2kN p c ∈ C, p ∈ P, and |c|= O(λ ).Thus |a |= O(λ )·|a| for |z | < 1. Next, let ρ ∈ be a ray and σ , σ the maximal cones containing ρ . Suppose γ + − ϕ˜ (m) is a broken line that crosses ρ , travelling from σ to σ .Let a z be the monomial − + − ϕ˜ (m ) attached to the incoming segment of γ near ρ and a z the monomial attached to the outgoing segment. By the deﬁnition of ϕ˜ , −n ,m ϕ˜ (m) [D ] ϕ˜ (m) σ ρ σ − + z = z z , where n ∈ N is primitive, annihilates ρ , and is positive on σ .Write d := −n , m;note ρ + ρ d =|u ∧ m| > 0where u ∈ M is the primitive generator of ρ.If γ does not bend at [D ] d [D ] d ϕ˜ (m ) ρ ρ σ ρ then a = (z ) · a . In general a = (z ) · a where a z is obtained from + − + − − ϕ˜ (m) a z as above (by applying the scattering automorphism associated to ρ and selecting a monomial term). [D ] We need to show the exponent d =|u ∧ m| > 0of z in the previous paragraph 2N is large for some lift ρ˜ of any given ray ρ ∈ . This will allow us to absorb the O(λ ) MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 157 factors coming from bends of γ and obtain the estimate (3). Let γ be a broken line for T (q˜).Then N s m = T (q˜) − T α γ i i=1 as above, where 0 ≤ l ≤ k,N ≥ s ≥··· ≥ s ≥ 0, and the α lie in a ﬁnite set. Write 1 l i s = Nand s = 0. Choose j such that s − s ≥ N/(k + 1). Now consider the exponent 0 l+1 j j+1 N s d =|u ∧ m| for m = T (q˜) − T α given by the monomial attached to the segment i=1 of the broken line between bends j and j + 1. Let ρ˜ ∈ be the lift of ρ ∈ between bends j and j + 1 which is closest to bend j + 1 (such a lift exists if N is sufﬁciently s s −s j+1 j j+1 large). Let u = T β be the primitive generator of ρ˜.Then |u ∧ m|=|β ∧ T m | −s where m = T (m). Writing m = μ w + μ w , we see that |μ | is bounded since w 1 2 1 1 2 1 −1 has eigenvalue λ < 1. Also, μ is bounded away from zero by Lemma 7.9.Now s −s −(s −s ) s −s j j+1 j j+1 j j+1 u ∧ m = β ∧ T m = μ (β ∧ w )λ + μ (β ∧ w )λ , 1 2 1 2 where s − s > N/(k + 1),so j j+1 N/(k+1) |u ∧ m| > c · λ for some constant c > 0. [D ] Combining our results now gives, when |z | < 1for all p ∈ P , the estimate bdy N/(k+1) k c·λ 2kN [D ] a = O λ · z , ρ∈ dim ρ=1 where the ﬁrst factor bounds the contribution associated to bends of γ and the second factor bounds the contribution associated to rays ρ of the fan crossed by γ , as described in the preceding two paragraphs. This implies the estimate (3) in the statement, using [D ] |z | < 1. Indeed, the above expression is of the form bN aN c·λ λ · x [D ] N where a, b, c > 0and λ> 1 are constants, and x = |z |. This is bounded by Cx for 0 ≤ x < < 1, for some constant C (depending on ). (To see this, we may assume x = 0, take logarithms, and establish an inequality bN (7.5) aNlog λ + cλ log x ≤ log C + Nlog x. We have − log x > − log> 0. Rearranging (7.5), we require that, for some choice of C, bN aNlog λ ≤ log C + (− log x) cλ − N 158 MARK GROSS, PAUL HACKING, AND SEAN KEEL for all N. This holds because bN aNlog λ ≤ (− log x) cλ − N for N sufﬁciently large.) Lemma 7.9. —Let A ⊂ R be a ﬁnite set and λ ∈ R, λ> 1.For k ∈ N let S ⊂ R be the setofrealnumbers s ofthe form s = c λ where l ≤ kand c ∈ A,n ∈ Z for each i. Then S i i i ≥0 k i=1 is discrete for each k. Proof. — Proof by induction on k.Wehave S ={0}. Suppose S is discrete. We 0 k have S = λ (S + A). Since λ> 1 we deduce that S discrete. k+1 k k+1 n≥0 For Propositions 7.10 and 7.11 below, the assertions hold after possibly replacing X by a smaller neighbourhood of s(0) ∈ X (independent of I, Q and q). J J Proposition 7.10. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal cone σ of . Each term of the formal sum Lift (q) = Mono(γ ) is an analytic function on V Q σ,σ,I and the sum deﬁnes an analytic function on V . σ,σ,I Proof.—Recall that V is an inﬁnitesimal thickening of the reduced complex σ,σ,I analytic space V .Write σ,σ,J V := (X , X )||X | < R|X |,|X | < R|X | ⊂ (G ) × S. σ,σ 1 2 1 2 2 1 m X ,X 1 2 Then V is a reduced complex analytic space containing V as a locally closed sub- σ,σ σ,σ,I space. We show that the sum Lift (q) converges (uniformly on compact sets) to an ana- lytic function on a neighbourhood of V in V . σ,σ,I σ,σ ˜ ˜ ˜ Let Q ∈ B be a lift of Q and σ˜ the lift of σ containing Q. Let u , u be the primitive 0 1 2 generators of σ˜ (a basis of M) such that the orientation of u , u agrees with that of w ,w . 1 2 1 2 ϕ˜ (u ) σ˜ i Let X = z , i = 1, 2, be the associated coordinate functions on V ,sothat i σ,σ,I V ⊂ (X , X )||X | < R|X |,|X | < R|X | ⊂ (G ) × S . σ,σ,I 1 2 1 2 2 1 m X ,X I 1 2 α α ϕ˜ (m) 1 2 σ˜ For m ∈ M, writing m = α u + α u ,wehave z = X X . 1 1 2 2 1 2 As already noted, broken lines γ on B for q with endpoint Q lift uniquely to ˜ ˜ broken lines on B for T (q˜) with endpoint Q, for some N ∈ Z, and the attached ϕ˜ (m ) σ˜ γ monomials are identiﬁed. Write Mono(γ ) = a z and m = α u + α u . Clearly γ γ 1 1 2 2 α α 2 ±1 ±1 Mono(γ ) = a X X ∈ k[P][X , X ] is an analytic function on V . Also write m = γ σ,σ γ 1 2 1 2 μ w + μ w . By Lemma 7.12,(1), μ and μ are bounded below (using the symmetric 1 1 2 2 1 2 statement interchanging w and w if T (q˜)∈Q,w ). The points u and u are ad- 1 2 1 R 1 2 ≥0 jacent integral points on the boundary of the inﬁnite convex polytope with asymptotic MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 159 directions w ,w . It follows that w = β u + γ (u − u ) and w = β u + γ (u − u ), 1 2 1 1 2 1 1 2 2 2 1 2 2 1 for some β ,β ,γ ,γ > 0. Hence 1 2 1 2 μ μ α α 1 2 ϕ˜ (m ) 1 2 β γ β γ σ˜ γ 1 1 2 2 z = X X = |X | |X /X | |X | |X /X | . 2 1 2 1 2 1 1 2 Now |X /X | < R, |X /X | < RonV . Thus, as we have chosen δ in Lemma 7.6 so 1 2 2 1 σ,σ −γ /β −γ /β ϕ˜ (m ) 1 1 2 2 γ σ˜ that 0 <δ < min(R , R ),if μ ,μ are both positive, |z | is bounded for 1 2 |X |,|X | <δ. On the other hand, suppose μ , say, is negative. Then if |X | >δ > 0, we 1 2 1 2 have μ β μ β μ 1 1 1 1 1 β γ −μ γ −μ γ 1 1 1 1 1 1 |X | |X /X | < δ |X /X | < δ R . 2 1 2 2 1 Since β and γ are ﬁxed and μ is bounded below, the above quantity is bounded. 1 1 1 β γ μ 2 2 2 Similarly, if μ is negative, (|X | |X /X | ) is bounded provided |X | >δ > 0. Thus 2 1 2 1 1 ϕ˜ (m ) σ˜ γ in any event, |z | is bounded for 0 <δ < |X |,|X | <δ. (We will only obtain uniform 1 2 convergence of the series Lift (q) on compact subsets of V .) By Lemma 7.8,(3),if Q σ,σ p |N| |z | < < 1for all p ∈ Pwe have |a |= O( ). By Lemma 7.8, (2), the number of N k broken lines for T (q˜) is O(|N| ). Combining, we deduce that Lift (Q) = Mono(γ ) is convergent on the open analytic subset V of V deﬁned by |X |,|X | <δ and σ,σ 1 2 σ,σ |z | < 1for all p ∈ P, for some δ> 0 (independent of I and q). After replacing X by an analytic neighbourhood of the vertex s(0) ∈ X , we may assume that V ⊂ V . J σ,σ,I σ,σ Proposition 7.11. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal cone σ of and let ρ be an edge of σ . Consider the formal sum Lift (q) = Mono(γ ).For Q sufﬁciently close to ρ each term of the sum is an analytic function on V and the sum deﬁnes an analytic ρ,I function on V . ρ,I Proof.—Write ρ = ρ , without loss of generality assume σ = σ ,sothat σ ∈ i i,i+1 i−1,i is the other maximal cone containing ρ .Let Q, ρ˜ , σ˜ , σ˜ be compatible lifts to i i i,i+1 i−1,i B .Let u , u , u be the primitive generators of ρ˜ and the remaining edges of σ˜ 0 i i−1 i+1 i i−1,i and σ˜ , and write X , X , X for the corresponding coordinates on V .So i,i+1 i i−1 i+1 ρ,I −D [D ] V ⊂ (X , X , X )||X | < R|X |, |X | < R|X | ⊂ V X X − z X f ρ,I i−1 i i+1 i−1 i i+1 i i−1 i+1 ρ ⊂ A × (G ) × S . m X X ,X i I i−1 i+1 Deﬁne −D [D ] V = (X , X , X )||X | < R|X |, |X | < R|X | ⊂ V X X − z X f ρ i−1 i i+1 i−1 i i+1 i i−1 i+1 ρ ⊂ A × (G ) × S. m X X ,X i i−1 i+1 We assume that the orientation of u , u is thesameasthatof w ,w . i−1 i+1 1 2 We ﬁrst consider broken lines γ lying in the cone generated by u and w .Write i 2 ϕ˜ (m ) σ˜ γ i,i+1 Mono(γ ) = a z ,and m = αu + α u = μ w + μ w . By Lemma 7.12,(1), γ γ i + i+1 1 1 2 2 160 MARK GROSS, PAUL HACKING, AND SEAN KEEL |μ | is bounded, and μ > 0 for all but ﬁnitely many γ . By Lemma 7.12,(2), α ≥ 0, so 1 2 + + ±1 Mono(γ ) = a X X ∈ k[P][X , X , X ] is analytic on V for each γ . In particular γ i−1 i+1 ρ i i+1 i we may assume in what follows (discarding ﬁnitely many terms Mono(γ ))that μ > 0. Writing w =−β u + γ u and w = β u + γ (u − u ),wehave β ,β ,γ ,γ > 0 1 1 i+1 1 i 2 2 i 2 i+1 i 1 2 1 2 and μ μ 1 2 ϕ˜ (m ) −β γ β γ γ 1 1 2 2 σ˜ z = |X | |X | |X | |X /X | . i+1 i i i+1 i −γ /β 2 2 Recall that |X /X |,|X /X | < RonV . Note then that for 0 <δ < R ,the i+1 i i−1 i ρ second factor on the right is bounded for |X | <δ as we are taking μ > 0. If μ < 0, i 2 1 then we have −β γ −β μ (−β +γ )μ 1 1 1 1 1 1 1 |X | |X | < R |X | i+1 i i which is bounded for δ < |X | <δ for any small δ > 0. Finally, if μ > 0, we use the i 1 equation for V , which gives −1 D −1 −[D ] |X | =|X |·|X | ·|f | · z . i+1 i−1 i ρ The function f on V restricts to the constant function 1 over S . Hence we may impose ρ ρ J the condition |f | >δ for any small δ > 0. Note that 2 β μ 2 −β μ 1 1 1 1 −β μ D −1 −[D ] β μ β μ (1+D ) −β μ [D ] 1 1 ρ ρ 1 1 1 1 ρ 1 1 ρ |X | = |X ||X | |f | z < R |X | |f | z . i+1 i−1 i ρ i ρ Thus we see that in any event, if δ < |X | <δ, |f | >δ and |z | < < 1for all p ∈ P, i ρ then ϕ˜ (m ) [D ] σ˜ γ ρ z · z is bounded, where c = β · sup({μ }, 0) is a constant. Now by Lemma 7.8, (3), again using 1 1 |z | < < 1for all p ∈ P, then ϕ˜ (m ) |N| σ˜ γ Mono(γ ) = a z = O . N k Recall that the number of broken lines for T (q˜) is O(|N| ) (Lemma 7.8, (2)). We deduce that the sum Mono(γ ) over broken lines γ lying in u ,w is uniformly convergent i 2 R ≥0 on compact sets for |X | <δ, f = 0, and |z | < 1for all p ∈ P, where δ> 0 is independent i ρ of I and q. Symmetrically, if we choose a basepoint Q ∈ σ sufﬁciently close to ρ,wecan i−1,i use the same argument for broken lines lying in u ,w ending at Q . Such a broken i 1 R ≥0 line will have Mono(γ ) = a X X with α ≥ 0. Thesameargumentasabove shows γ − i−1 i that the sum Mono(γ ) over all such broken lines γ is uniformly convergent on com- pact sets for |X | <δ, f = 0, and |z | < 1for all p ∈ P, where δ> 0 is again independent i ρ of I and q. However, the statement we are trying to prove involves the lift at Q, not Q .For MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 161 this purpose, we will omit terms coming from broken lines with α = 0from the above sum, as such terms will also arise in the above analysis at Q. To deal with this issue, note that given such a broken line γ ending at Q with α > 0, we get a ﬁnite number of broken lines γ ,1 ≤ j ≤ s, ending at a point Q ∈˜ σ − j i,i+1 sufﬁciently close to ρ˜ as follows. Extend γ until it reaches ρ˜ (which can be done since i i α > 0). Then applying the automorphism associated to crossing ρ˜ to Mono(γ ) gives − i α s α j − − α α a X X f , which we write as a X X .The j th monomial in the sum gives a γ γ i−1 i ρ j=1 j i−1 i new broken line γ bending at ρ˜ (unless α = α, in which case we just extend the ﬁnal line j i j α j ϕ˜ (m ) − γ σ˜ i,i+1 j segment of γ ), with new attached monomial a X X = a z .Aslongas Q lies γ γ j i−1 i j in the cone generated by ρ and −m , by extending or shortening the last line segment i γ of γ and applying a homothety, one can obtain a broken line γ ending at Q .Thusfor j j ˜ ˜ Q sufﬁciently close to ρ˜ , we obtain broken lines γ ,...,γ ending at Q .However,the i 1 s ˜ ˜ choice of Q may depend on γ . To see there is a choice of Q which works for all γ , note that for all but a ﬁnite number of γ , m lies in the half-space R · w + R · w γ ≥0 1 2 by Lemma 7.12, (1). Further, since γ bends at ρ˜ , m ∈ Z · u + Z · u .Thus m = j i γ i <0 i+1 γ β u + β u for β> 0, β < 0, and m = (β − l)u + β u for some l > 0 with a bound i + i+1 + γ i + i+1 only depending on f mod I. From this it follows that there cannot be a sequence of γ and γ constructed from γ as above so that the cones generated by u and −m get smaller j i γ and smaller. ˜ ˜ Thus we see that taking Q sufﬁciently close to ρ˜ , the broken lines ending at Q contained in w , Q with α > 0 give in the above fashion broken lines ending at 1 R − ≥0 ˜ ˜ ˜ Q contained in w , Q , and every such broken line ending at Q clearly arises in this 1 R ≥0 way. Furthermore, there are no broken lines contained in w , Q with α = 0, and 1 R − ≥0 thus all broken lines ending at Q have been accounted for. Now consider again a broken line γ contained in w , Q with α > 0. By 1 R − ≥0 construction, Mono(γ ) = Mono(γ )f . We wish to understand the contribution of j=1 ρ Mono(γ ) to Lift (q), and to do so, we write Mono(γ ) in terms of X , X using j Q j i i+1 j=1 −D [D ] the relation X X = z X in k[P ] (see Proposition 2.5). So we can write i−1 i+1 ϕ˜ i ρ˜ −α D +α −α − α [D ] − ρ α − ρ − Mono(γ ) = a z X X f . j γ i+1 i ρ This deﬁnes a (possibly rational) function on V . On the other hand, Mono(γ ) = − α a X X deﬁnes a holomorphic function on V , and using the equation γ ρ i−1 i −D [D ] (7.6)X X = z X f i−1 i+1 ρ which is satisﬁed on V ,wehave −α D +α −α − α [D ] − ρ α − ρ − Mono(γ ) = a z X X f i+1 i ρ 162 MARK GROSS, PAUL HACKING, AND SEAN KEEL as a function on V . Thus we see that Mono(γ ) and Mono(γ ) coincide as func- ρ j tions on V , in the above interpretation. (Essentially we are just using the fact that the relation (7.6) encodes the automorphism associated to crossing ρ ). Thus the fact that Mono(γ ) deﬁnes an analytic function on V for broken lines γ ending at Q con- ρ,I tained in u ,w implies that the sum Mono(γ ) over all broken lines γ ending at i 1 R ≥0 γ ˜ ˜ Q and contained in Q,w is analytic. This implies Lift (q) is analytic on V . 1 R Q ρ,I ≥0 ˜ ˜ ˜ Lemma 7.12. —Let Q ∈ B \ Supp(D) be a point contained in the interior of a maximal ˜ ˜ ˜ cone σ˜ ∈ . Consider broken lines γ on B for T (q˜) with endpoint Q, for all N ∈ Z such that N ϕ˜ (m ) ˜ σ˜ T (q˜)∈Q,w . Write Mono(γ ) = a z ,and m = μ w + μ w . 2 R γ γ 1 1 2 2 ≥0 (1) |μ | is bounded, and μ is positive for all but ﬁnitely many γ . In particular, μ ,μ are 1 2 1 2 bounded below. (2) Let u , u be generators of σ˜ with the same orientation as w ,w .Thenfor Q sufﬁciently 1 2 1 2 close to ρ˜ := R · u ,m lies in the half space R · u + R · u for each γ . ≥0 1 γ 1 ≥0 2 Proof.—(1)Wemay assume thelift q˜ of q is chosen so that T (q˜) ∈Q,w 2 R ≥0 if and only if N ≥ 0. Note that the rays spanned by w ,w are irrational so μ ,μ = 0. 1 2 1 2 Suppose for a contradiction that there is an inﬁnite sequence of broken lines γ such that m = μ w + μ w with μ < 0. Each broken line has at most k bends and there γ 1 1 2 2 2 are a ﬁnite number of -orbits of possible changes α ∈ M of the derivative of γ at a bend; see Lemma 7.8 and its proof. So, passing to a subsequence, we may assume that s s 1 l the bends (in order of increasing t ∈ (−∞, 0])ofeach γ are of types T α ,..., T α 1 l for some ﬁxed α ,...,α ∈ M, l ≤ k,and N ≥ s ≥ s ≥ ··· ≥ s ≥ 0 (depending on γ ). 1 l 1 2 l Say N − s is bounded for i ≤ l and unbounded otherwise. Passing to a subsequence, we may assume that N − s is constant for i ≤ l .Let γ be the broken line obtained by truncating γ after the ﬁrst l bends, and moving by a homothety so that (extending its ﬁnal line segment) γ ends at Q. Then m = T (m) for some ﬁxed m ∈ M. Furthermore, m is obtained from m by adding a positive linear combination of w and w . But since γ γ 1 2 −1 μ < 0and w ,w are eigenvectors of T with eigenvalues λ ,λ, it follows that we must 2 1 2 have m = ν w + ν w with ν < 0. But then for sufﬁciently large N, T (m) does not lie 1 1 2 2 2 in the half-space R · Q + R · w . This is a contradiction because m always lies in this ≥0 2 γ half-space. N s To see that |μ | is bounded, recall that m = T (q˜) − T α where 0 ≤ l ≤ k, 1 γ i i=1 −1 0 ≤ s ≤ N, and the α are selected from a ﬁnite set. Now since Tw = λ w it follows i i 1 1 that |μ | is bounded. (2) Let u be the connected component of B \ Supp (D) contained in σ˜ and ˜ ˜ ˜ containing ρ˜ in its closure. Let Q ∈ u be a point such that Q ∈ρ,˜ Q .Thenif ≥0 N N ˜ ˜ T (q˜)∈Q,w and γ is a broken line for T (q˜) with endpoint Q , we obtain a bro- 2 R ≥0 ken line γ for T (q˜) with endpoint Qand m = m as follows. First apply a homothety γ γ ˜ ˜ to obtain a broken line passing through Q, then truncate at Q. This gives an injective MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 163 ˜ ˜ map between the set of broken lines for T (q˜)∈Q,w ending at Q with the set of 2 R ≥0 such broken lines ending at Q. Now suppose γ is a broken line for T (q˜) with endpoint Qand m not lying in the half-space R · u + R · u . Since m lies in the half-space R · Q + R · w we 1 ≥0 2 γ ≥0 2 ﬁnd m lies in the cone generated by −Q,−u . In particular, m does not lie in the half- γ 1 γ space R · w + R w , so by (1) there are only ﬁnitely many such γ .Now by theabove 1 ≥0 2 construction it follows that for Q sufﬁciently close to ρ˜ there are none. 7.4. Smoothness. — We will now complete the proof of: Theorem 7.13 (Looijenga’s conjecture). — Suppose that k = C and the intersection matrix (D · D ) is negative deﬁnite, so that D ⊂ Y can be contracted to a cusp singularity q ∈ Y . Then i j 1≤i,j≤n the dual cusp to q ∈ Y is smoothable. We continue to work with the setup at the beginning of Section 7.2,with L, σ ,P, m and J as given there. By Theorem 7.7, if I is a monomial ideal with I = J, we obtain a deformation X → S of X → S . For the remainder of the section, we shall write S I J I I J for S .Let f : X → S denote the formal deformation determined by the deformations J J J X N+1 → S N+1 for N ≥ 0. Thus, S is the formal complex analytic space obtained as the J J J completion of S along S , X is a formal complex analytic space, and X → S is an adic J J J J ﬂat morphism. We similarly have the family X → S of formal schemes already studied m m in Section 4. We refer to [G60]and [B78] for background on formal schemes and formal complex analytic spaces. We have a section s: S → X such that, for t ∈ S general, the point s(t) ∈ X J J J J,t o o o on the ﬁbre is the cusp. We write X := X \ s(S ) ⊂ X and X ⊂ X , X ⊂ X for the J J J I J J I J induced open embeddings. Let Z := Sing(f ) ⊂ X denote the singular locus of f : X → S , see Deﬁnition- I I I I I I Lemma 4.1.Thus Z ⊂ X is a closed embedding of schemes or complex analytic spaces. I I n n Since the singular locus is compatible with base-change, the singular loci Z ⊂ X deter- J J mine a closed embedding Z ⊂ X which we refer to as the singular locus of f : X → S . J J J J J Lemma 7.14. — In the above situation, there exists 0 = g ∈ k[P] such that Supp(g · O ) is contained in s(S ). In particular, f (g · O ) is a coherent sheaf on S . J J∗ Z J Proof. — The proof is essentially the same as that of Lemma 4.5.Let U be deﬁned i,J as in (4.1). Then X is a union of open subspaces V , i = 1,..., n,suchthat V is an i,J i,J analytic open subspace of U for each i.Wethentake g = a ··· a as in the proof of i,J 1 n Lemma 4.5, so that Supp(g · O ) is contained in s(S ). So the support of g · O is a Z J Z J J closed subset of s(S ), hence proper over S . It follows that f (g · O ) is coherent by J J J∗ Z [B78], 3.1. 164 MARK GROSS, PAUL HACKING, AND SEAN KEEL Let u(J) denote the natural map u(J): O → f (O ), S J∗ Z J J and u(m) similarly the natural map u(m): O → f (O ). S m∗ Z m m Lemma 7.15. —u(J) is injective if and only if u(m) is injective. Proof.—Let 0 = g ∈ k[P] be the element given by Lemma 7.14.Let K be the kernel of u(J) and K the kernel of g · u(J).Thus K , K are ideal sheaves in O and J S J J J g · K ⊂ K ⊂ K . The local rings of S are domains by Lemma 7.16,so K = 0if and J J J J J only if K = 0. The sheaf K is coherent because the image of g · u(J) is contained in the J J coherent subsheaf f (g · O ) ⊂ f O . J∗ Z J∗ Z J J We claim that the natural map K ⊗ O → K O S J S m m is an isomorphism. Let z ∈ S be the unique point, coinciding with the zero-dimensional torus orbit of S, and let O denote the completion of O at its maximal ideal. Note that S,z S,z O coincides with O and the completion of O at its maximal ideal. It sufﬁces to S,z S ,z S ,z m J show that the map K ⊗ O → K O S,z J,z S ,z m,z is an isomorphism. We have an exact sequence of coherent sheaves 0 → K → O → f (g · O ) S J∗ Z J J J and so an exact sequence of O -modules S,z ˆ ˆ ˆ 0 → K ⊗ O → O → f (g · O ) ⊗ O . S,z S,z J∗ Z z S,z J,z J Now ˆ ˆ ˆ f (g · O ) ⊗ O = (g · O ) ⊗ O = (g · O ) = g · O J∗ Z z S,z Z s(z) S,z Z Z ,s(z) J J J J s(z) where the hats denote completion with respect to the maximal ideal of O .Thus K ⊗ S ,z J J,z O is the kernel of the map S,z ˆ ˆ O → g · O . S,z Z ,s(z) By the base-change property for the singular locus, this map coincides with the corre- sponding map for m. This proves the claim. MIRROR SYMMETRY FOR LOG CALABI-YAU SURFACES I 165 The support of the ideal sheaf K is either empty or S (because the local rings of S are domains and S is connected). So K = 0 if and only if K = 0by the claim. J J J m Lemma 7.16. — The local rings of S are integral domains. Proof. — The completion of the local ring of S at a point z ∈ S is identiﬁed J J with the completion of the local ring of the toric variety S at z. By Serre’s criterion for normality, the completion of a normal Noetherian ring at a maximal ideal is a local normal Noetherian ring; in particular, it is a domain. Since O is a local Noetherian S ,z ring, it is contained in its completion. We deduce that O is a domain. S ,z Proof of Theorem 7.13.—Let f : Y → Y be the contraction of D ⊂ Y. We may assume f is the minimal resolution of Y . We may further assume n ≥ 3. Indeed, the embedding dimension of the dual cusp equals max(n, 3) by [N80], Corollary 7.8, p. 232 and [KM98], Theorem 4.57, p. 143, see also [L81], p. 307. So in particular for n ≤ 3the dual cusp is a hypersurface and thus smoothable. Let L be a nef divisor on Y such that NE(Y) ∩ L =D ,..., D .Let σ ⊂ A (Y, R) be a strictly rational polyhedral R 1 n R P 1 ≥0 ≥0 cone containing NE(Y) such that σ ∩ L is a face of σ .Let P = σ ∩ A (Y, Z) and P P P 1 J = P \ P ∩ L . By Lemma 7.15 and Theorem 4.6, u(J) is not injective. Let x ∈ S be a point lying in the interior of the toric variety S and h ∈ O a J J S ,x nonzero element of the kernel of u(J) near x. By Lemma 7.17 thereisamorphism N+1 v: Spec C[t]/ t → S ∗ N+1 taking the unique point of the domain to x and 0 = v (h) ∈ C[t]/(t ).Let Y/ Spec(C[t] N+1 /(t )) be the pullback of X /S by v and Z ⊂ Y its singular locus. Then Y/ Spec(C[t]/ J J N+1 (t )) is a deformation of the dual cusp singularity. Furthermore, O is annihi- ∗ ∗ N lated by v (h), and the ideal generated by v (h) must contain t ,so O is annihi- lated by t .By[A76], Theorem 5.1, there is an algebraic ﬁnite type deformation N+1 Y / Spec C[[t]] whose restriction to Spec(C[t]/(t )) is locally analytically isomorphic to N+1 Y/ Spec(C[t]/(t )).Let Z ⊂ Y denote the singular locus of Y / Spec C[[t]].Then O is a ﬁnite C[[t]]-module because the ﬁbre Y has an isolated singularity (using [Ma89], N+1 N N N+1 Theorem8.4,p.58).Now O = O /t O and t O = 0, so t O = t O and Z Z Z Z Z Z thus t O = 0 by Nakayama’s lemma. Hence the general ﬁbre of Y / Spec C[[t]] is smooth, and Y / Spec C[[t]] is a smoothing of the dual cusp. Lemma 7.17. —Let A be the completion of a ﬁnitely generated normal Cohen-Macaulay C- algebra at a maximal ideal. Let 0 = a ∈ A. Then there exists N ≥ 0 and a C-algebra map f : A → N+1 C[t]/(t ) such that f (a) = 0. 166 MARK GROSS, PAUL HACKING, AND SEAN KEEL Proof.—Extend a to a regular sequence a, t ,..., t of length dim A. Then the 1 r normalization of A/(t ,..., t ) is a ﬁnite direct sum of copies of C[[t]]. Now the result is 1 r clear. Acknowledgements An initial (and ongoing) motivation for the project was to ﬁnd a geometric com- pactiﬁcation of moduli of polarized K3 surfaces. We received a good deal of initial inspi- ration in this direction from conversations with V. Alexeev. The project also owes a great deal to the ﬁrst author’s collaboration with B. Siebert. We learned a great many things from A. Neitzke, especially about the connections of our work with cluster algebras and moduli of local systems. Our thinking about Looijenga pairs was heavily inﬂuenced by conversations with R. Friedman and E. Looijenga. Many other people have helped us with the project, and discussions with D. Allcock, D. Ben-Zvi, V. Fock, D. Freed, A. Gon- charov, R. Heitmann, D. Huybrechts, M. Kontsevich, A. Oblomkov, T. Perutz, M. Reid, A. Ritter, and Y. Soibelman were particularly helpful. We would also like to thank IHÉS for hospitality during the summer of 2009 when part of this research was done. The ﬁrst author was partially supported by NSF grants DMS-0805328 and DMS-0854987. The second author was partially supported by NSF grant DMS-0968824 and DMS-1201439. The third author was partially supported by NSF grant DMS-0854747. REFERENCES [AC11] D. ABRAMOVICH and Q. CHEN, Stable logarithmic maps to Deligne-Faltings pairs II, arXiv:1102.4531. [A02] V. ALEXEEV, Complete moduli in the presence of semiabelian group action, Ann. Math. (2), 155 (2002), 611– [A76] M. ARTIN, Lectures on Deformations of Singularities, Lectures on Mathematics and Physics, vol. 54, Tata Inst. Fund. Res, Bombey, 1976. [A07] D. AUROUX, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. 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Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA keel@math.utexas.edu Manuscrit reçu le 12 avril 2013 Manuscrit accepté le 10 janvier 2015 publié en ligne le 25 mars 2015.

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