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The Neron model for families of intermediate Jacobians acquiring “algebraic” singularities

The Neron model for families of intermediate Jacobians acquiring “algebraic” singularities THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS ACQUIRING ^ALGEBRAIC" SINGULARITIES by HERBERT CLEMENS i. Introduction Let V be an irreducible complex projective manifold of dimension 2m — i. Let {^Jae s be an algebraic family of algebraic (m — i)-cycles on V whose members Zg are all homologically equivalent. In Appendix A of his paper " Periods of integrals on algebraic manifolds. III 9 ? ([G]$ p. 165), P. Griffiths defines an analytic map, called the Abel-Jacobi homomorphism (1.1 ) S-^J(V). r-FwjHpm-l/y. Q-p Here J(V) ==-—————————— is the Jacobian variety of V, and (1.1) is defined Him-lC^ Z) by picking a basepoint SQ e S and sending f \ r^ f\ (I-2) ^L- Jz,. Next let X be a complex projective manifold of dimension 2m, let A be the unit disc, and let (1.3 ) V,, ^eA, be an analytic family of divisors on X which are irreducible and non-singular as long as t ={= o. Suppose (1.4 ) {ZJ,es, is an algebraic family of algebraic (m — i)-cycles on V< for each t e A. Suppose further that, for fixed t, all cycles Z^, s eS,, are homologous to one another. Finally suppose that y = U ({Qxs< ) te^ " ' ' 217 6 HERBERT CLEMEN S is a smooth analytic variety with (1.5) y-b. everywhere of maximal rank. We make no assumptions about properness of (1.5) or connectivity of fibres. Let / * -.A* = (A —{o} ) be the bundle of complex tori over the punctured disc whose fibre over t is J(V,). Then for every section T: A -> y of (1.5) there is a commutative diagram of proper morphisms V ^ /* (1.6) defined fibrewise by (i . 2) with ^ = r(f). The point of this paper is to complete (1.6) to a commutative diagram ('.7) where / is obtained from </* by filling in over t = o with a commutative complex Lie group. The complex Lie group in question is the fibre over t = o of an analytic analogue of the Neron minimal model. We will be able to carry out this program only after putting some severe restrictions on the family (1.3). We devote the rest of the introduction to explaining these restrictions. Let ^=^({QxV,) and (1.8) (A: y-*^A*. We pull-back the bundle (1.8) via the universal covering map ^-^A * u \-> t = e 2" 1'" to obtain ^ : "^ -> S. Abusing notation we write y-1^) = ¥„ == V, = (x-^) whenever t = ^. The derived bundles R^_^(Z) and R^-^^Z) are trivial and we will denote their modules of global sections, taken modulo torsion, by H, and Hz respectively. Also H° = H2 ® C. The natural isomorphism Hz S H^(V«^) - H^_,(V< ) = H^(VJ s H^ 218 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 7 is not the identity map on H^ but rather the monodromy isomorphism T,: Hz ->Hz. Let T* be the adjoint of T» with respect to the natural unimodular pairing H^xH^ Z (Y.<°) ~>^^ Our first major assumption is that (1.9) (T, - I)2 == (T-- I)2 = o. Let N, == log T,, N* == log T*. We then have a filtration {W J on H? defined by W^H^o W^H^CkerNJ1 W^^ == (image NJ1 Wg^ =HZ. This filtration is called the asymptotic weight filtration. Under the identification H° == H2"*-1^; C) the Hodge filtration on H2"1"1^; C) induces a filtration F; on H0. This filtration /V/ varies with u e A, but there is a well-defined filtration (i. 10) F^ = lim exp(- t^N*)F: on H® called the asymptotic Hodge filtration. In [S], W. Schmid shows that the array (H^ W«, F'^) is a w^rf jyorf^ structure such that N': H 0^ ^ is a morphism of mixed Hodge structures of type (— i, — i). In fact in this situation which " comes from geometry ", we have that N-: H^W^.^W^ is an isomorphism over Q^. Our second major assumption is: (1.11) The Hodge structure of weight 2m on H^Wg^.i induced by (1.10) is of pure type (m, m). Finally we make an assumption which is not essential but will simplify the exposition: (1.12) All of H2"1"1^; C) is primitive cohomology. 219 8 HERBERT CLEMEN S 2. Growth of normal functions Let (2.1) T: A -> y be a section of the fibration y ^ A considered in (1.7). Let Z,SV, , feA, be the corresponding family of (m - i) -cycles. By Kleiman's smoothing theorem ([KJ; p. 297), we can assume that, if t + o, (a. 2) Z, = Z; - Z;' where Z; and Z;' are smooth and do not meet. We can resolve the family (2.3) ,U^({QxV<) sAx X along Vo so that: i) the fibre over t = o is a normal crossing variety in a smooth ambient space of dimension 2m', ii) the proper transform 2 of .SA^x^) is smooth and meets the fibre over zero transversely. In ([G1]; p. 245), we explicitly construct an action of the semigroup [o, i] x R on the resolved ambient space which is equivariant with the action {r,Q).t=re^t of [o, i] x R on A. It is easy to see that this action can be defined so as to respect Z since it is constructed first locally and then pieced together via fixations which can be constructed to be compatible with Z. As before, let (2.4) u =—.logf 2TO and take, for some fixed u^, a (2w - i)-chain ^ such that ^ = Z,. For (r, 6) e [o, i] x R, define I^-M).!^ where u == u, + IQ + ^\ Then ^ has as its boundary the algebraic cycle Z, with t = e2'"". ' / 220 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 9 Lemma (2.5). — The cycle (Fy^^ — FJ eH2^_i(V<; Z) has zero intersection number with any cycle which is invariant under the monodromy transformation T,: H^i(V,;Z)->H,,_,(V,;Z) Yl-^(i,i).Y . Proof. — The cycle (F^i — FJ, by construction, bounds in the ambient space of the resolution of the family (2.3). Therefore, by the Local Invariant Cycle Theorem ([G1]; p. 230)5 this cycle integrates to zero against any invariant element of H2"1"1^; C). Since the Poincar^ duals of the invariant cocycles are the invariant cycles, the lemma is proved. Next let <o(^) be a section of the canonical prolongation of FmR2m-l^(^ where (JL : ^ ->• A* is our family (1.8) and the canonical prolongation is as established in ([D]; pp. 91-92). Lemma (2.6). — Th e integrals j>«)=J^) are all of the form a{t)logt+f{t) where a(t) and f(t) are holomorphic on A. Proof. — The idea of the proof is to make the integrals in question into period functions, i.e. integrals over cycles, for some two-parameter family of varieties. Let X as in (1.3) denote the ambient variety for the family {V^ } and let U< cX, ^eA, be an analytic family of very ample hypersurfaces such that: i) if t + o, U( is smooth and meets V^ transversely, ii) Z^(U,nV<); iii) Z, is homologous to zero in U^. That such TJ( exist is an application of the results of [K]. (See the Columbia University Thesis of Spencer Bloch, 1971, pp. 9-10.) If we choose the family TJ( sufficiently amply, there will be a two-parameter family W^, (^)EAXA, of hypersurfaces in X such that: i) W(( (.) is smooth and irreducible for (^, t' ) e A* x A*, ii) W,;o)=U,uV,. 2 io HERBERTCLEMENS Next choose chains Sy c U( such that ^ = Z, and define F^ ^ = 1^ — S,,. Finally because of the simple nature of the degenerations W^ ^ "~^^(<,o) we can form a continuously varying family of cycles r^) c W^, such that Hr^r^=r^. Now the assumption (1.12) that the differential u(t) on V< is primitive implies that there is a two-parameter family of differentials (o^eF-H^W^C) such that: i) 6)(^, 0) = G)(^) + 0 ; v< u, ii) the family G)(^ t' ) extends over A X A to give a section of the canonical prolongation of the Hodge bundle for the two-variable degeneration (1). In fact, let <P(^ = r^,.^ — F^^. Then <p(u.u'+i) == ?(u,u') and» ^ ^^ lim f (of/, f ) == o. <^oJq>(^) 0 / Therefore the integrals f <o(^')-«'J "(^') </ 1 (»,*<') •"*'(»*,«') are well-defined functions of the variables u and f = e^'. Also since periods with respect to the canonical prolongation have at most logarithmic growth, these functions must be holomorphic along A* X {o}, and have value Jr,^) at (^ o). Finally we use logarithmic growth of periods with respect to the canonical prolongation once again, this time in the ^-direction. This allows us to conclude that a(t)=^-.( <o^) 2m Jr r •'^tt+l"1!* is bounded at t == o, since, by Lemma (2.5), it is a well-defined function of t. So f co(^~^)logf J ly is well-defined and so also bounded at t == o and the lemma is proved. (1) For a more complete discussion of this point, see the Appendix. 222 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS n 3. The Neron model Recall that in § i we defined N,: Hz^Hz. Now (image N,) is not in general a direct summand of H^ so we enlarge it: (3.1) Ey^ === {< p e Hz : some integral multiple of 9 lies in (image NJ } is called the module of vanishing cycles. Also (3.a) E^=(kerN.) is called the module of invariant cycles. Eyan is totally isotropic for the intersection pairing on Hg, in fact, the intersection bilinear form is identically zero on EinvXE^. So it easy to see that there is a splitting of H^ (3.3) Hz=L®ECE^ such that E®Eyan == E^y, and the intersection pairing is unimodular on L€)E^ and on E, and these two symplectic modules are orthogonal. Furthermore we can adjust the definition of L so that L is totally isotropic. Let {9,} be a basis for E^n satisfying (1.12). Let {8^, e/}?=i be a symplectic basis for E and let {Xj c L be such that {X p <pj}jLi is a symplectic basis for L©E^. The Riemann relations imply that if {o\(^), ^,)} is a framing of P'R^'^C) for (JL as in (1.8), then the matrix -j^) s^w (3.4) .L,^) J^). is invertible for each t =t= o. So we can normalize the choice of the framing to make (3.4) be the identity matrix for each t 4= o. (Notice that the matrix (3.4) is well-defined as a function of t since the cycles ^ and e^ are invariant.) Now if <( * " denotes Poincar^ dual, we can write any framing {<x)^), ^(t)} of pnH^-^V^; C) for t=^ as (3.5) i) S(^))^+?(^^))8?+^^ ii) 2(^^))^+S(^^))8?+2;(J^^^^^^^ The elements (3.5) therefore frame F^ c H° (see (1.10)). 222 12 HERBERT CLEMEN S Now Schmid's theory and our normalization of (3.4) imply a considerable amount about the entries in (3.5). First of all we wish to compute FS as in (i. 10). To do this, suppose (3.6) N.(^==2^^. Then (3.7) f, - "Sff^f •>>j y s J<S>j. is a well-defined function of t = e^, and F^ is obtained by replacing f in (3.5) by the operator (3.7) and taking limits as t ->-o. } Now our assumption in (1.11) is that (^ + W^_i) 2 L* = SC^* for L as in (3.3). But we have arranged that the (f^(t)) = KroneckerS,. and (J,^^) = ° in (3-5). So replacing J^ by (3.7), all enS-ies in (3.5 i)) stay bounded as t -> o. Thus in particular (3.8) L"*^) =mjiu+ (holo. fn. oft). Also, the fact that F^ n W^_i induces a Hodge structure on E* implies that all entries in (3-5) ") are bounded and therefore holomorphic functions of t at (== o. So we can rewrite (3.5) as follows: (3.9) i) «\(f) == \* + 2 (w<,M + "„(<))¥; + 2 (<o,/(<))e? ") ^W = 8; + S (^,(f))y* + S (^(f))s? where aU functions of t on the right-hand-side are holomorphic at t = o. We were able to replace m^ with my in the above formula because my = (\.N,X,) == (X..T.X,) = (T^-^.X^) =(-N.\.X,) =^.N.X.)=OT,,. From (3.9) and the characterization of the canonical prolongation in ([Z]; p. 189), one concludes that the framing (3.9) of (3.10) FOTR!!B-1(A.(C) is in fact a framing defining the canonical prolongation of Deligne. Therefore Lemma (2.6) applies to the families of differentials {^(f), •»)&(<)}. Also if we use the dual basis to (3.9) to frame the dual bundle to (3 10) then the "Jacobian bundle" ^•-^A* 224 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 13 in (1.6) can be described as the bundle whose fibre over t is obtained by dividing the affine space C1"4'8 by the lattice generated by the columns of the matrix n o {m^u+^(t)) (o)^))1 lo I W) (T^))J where, as always, u = —. log t. Let 2TCZ (3.11 ) ^'^A be the analytic fibration obtained by filling in over t == o with the quotient of C^8 by the partial lattice generated by the columns of I o (^(o))T (3-") 0 I (YW(0))J (The fact that these columns are indeed independent over R follows from the fact that [I("y]^(o))] is the period matrix for the Hodge structure of weight 2m — i on ^2m-il^2m-2 in ^e asymptotic mixed Hodge structure.) The fibration (3.11) is as in ([Z]; p. 191). Our next step is to enlarge the fibre over t == o in (3.11) as is done in the construction of the Neron model associated to the degeneration of a family of abelian varieties. The purpose is the same, namely, so that the sections of / * —^ A* considered in S 2 f^jro extend over t = o. To accomplish this we refer to (3.3) and define (3.13) L={XeL@)Q^:X has integral intersection number with each element of N^L}. Then the group L/L is naturally the dual of the group Ey^/N,L and so has order equal to det(w,.) where (Wy) is the non-degenerate symmetric matrix in (3.6). Furthermore the natural map (3.14) N,: L/L-^E^/N.L is an isomorphism. Since each element X e L is a section of R^_^(QJ which is invariant modulo elements of Ram-i^^)? h g^^ a well-defined section (3.15) ^^r' '^- Such a section is zero if and only if X e L. Thus we have an isomorphism of L/L with a group of sections (3.15). From now on we will denote this group simply as (3.16) ^. 225 14 HERBERT CLEMEN S We are now ready to define the Neron model associated to the degeneration (3.17) r-^^ of complex tori. We take \<S\ copies of^'n^s.n ) and index them by the elements of <S. We identify a point x in the fibre of ^ over t 4= o with a point y in the fibre of ^ over the same t if and only if (3.18) ^-j==^-^ in J(V(). The result is a smooth complex manifold (3-19) /-^^ whose restriction to A* is (3.17), and whose fibre Jo over t == o fits into the exact sequence (3.20) o ->Jo^Jo-> ^ -^o where Jo is the fibre of / f over t = o. By Lemma (2.6), if (o(^) is any section of the canonical prolongation of FmR^-^C!), ^en J^^LX)^) is a well-defined function of t holomorphic at t = o for each X e L. Thus (3-21) "JN.W is a well-defined section of (3.19) which extends over t = o. In fact (3.21) gives a section of (3.19) which passes through the same component of Jo that f does, th^t J x is, the component given by g = . • , j /^ So now suppose we have an analytic family of algebraic (m — i)-cycles 7 _ y y ^.( —- Z.( — Z.( as in (2.2). Let ^u=Z;-~Z;' as before. By Lemma (2.5)3 (3.22) r,,.i-r,=<p6E^ so that there is X e L such that N,(X) == <p. By Lemma (2.6), the integrals j^w-uf^t) are bounded holomorphic functions of t for G)(^) e{co,(/), T]^)}. Thus: -^ THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 15 Theorem (3.23). — The Abel'Jacobi map A-->^ '^-J. extends over t = o to a section of ^-^ A whose value at t = o lies in the component of Jo ^w by s-L^ with N,x== r^i-r^ . Since all our constructions can be carried out holomorphically with respect to auxiliary parameters we conclude: Corollary (3.24). — The diagram (1.6) extends to a commutative diagram y -^ / \ / This last corollary is just the analytic analogue of the " universal property " of the Neron model in the algebraic case. Appendix The purpose of this appendix is to justify the assertion, made during the proof of Lemma (2.6), that the integrals ( A- 1 ) J^W appear as coefficients, along t' = o, of sections of the canonical prolongation of (A.a) {F2H3(W(,,,))^,)g^^» with respect to a flat basis of (A.3) {H^W^,^}^,^^,^. It is this fact that allows us to conclude from ([D]$ pp. 91-92) that the normal functions (A.i) have at worst logarithmic growth. Let TT : S x H -> A' x A' (u, u^ h> (t, t') 227 16 HERBERT CLEMEN S be the universal covering map as in § 2. Using TT*, we pull the bundle (A. 3) back to ^/ /^/ ^ a trivial bundle on A X A whose global sections will be denoted by K . There are two commuting, nilpotent endomorphisms of K^, namely N = logarithm of monodromy around t •== o, N' = logarithm of monodromy around t' = o. Notice that W/< Q\ = V^ u U<, the union of two smooth manifolds meeting transversely. So (N')2 = o, in fact, the topological part of our analysis in § 3 applies to the one- variable degeneration ^-^.O). We want to apply the Gattini-Kaplan-Schmid theory of asymptotic mixed Hodge structures to the two-parameter family (A. 3). This theory says, first of all, that K32 has a mixed Hodge structure whose weight filtration is defined by the nilpotent endo- morphism N + N', and whose Hodge filtration is given by (A.4) P== lim exp(~MN-M/N/)^(F*H3(W.^)). v .»/ (<,r)->(o,o) A v / \ \ ((,<}// If H56 is as in (i .8)-(i . n) , then there is a subquotient of K^ which can be identified with H21. Namely let kernel^ : K2 -. K2) ^A.5 j ^3 -(^g^)®^) nK ^ The filtrations on Hz ® C induced by the weight and Hodge filtrations on K21 ® C define a mixed Hodge structure on (A. 5) which is isomorphic to (the mixed Hodge structure on) (A.6) H^OG^ Here Hz has the mixed Hodge structure in (i .8)-(i .11), and Gz has the asymptotic mixed Hodge structure for the family {U^}^^ constructed in the proof of Lemma (2.6). Now let 9,* and s? be as in (3.5). These are flat, N-invariant sections of {H3(V()z}(eA* ^d can be extended to flat sections of (A. 3) which are both N and N' invariant. We compute the limit (A. 4) in two steps, first letting t ' go to zero and then letting t go to zero. As in § 3, we see that after the first step, the vector space Ff = lim exp(-^^^^(W^) can be written as the direct sum of two subspaces, (A.7) M,=(Ffn{SZ<p;+SZ^}1) and a second subspace, which we will call (A.8) L,, 228 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 17 which lies in (ker N') ® C and gives F2!!3^) in the weight three graded quotient of the asymptotic mixed Hodge structure associated to the one-parameter family (A.9) W^->W(,o)=V,uU,. The important point is that L = lim exp(— ^N)L( and M = lim exp(— !/N)M( is a direct-sum decomposition of F2 (see (A. 4)) . This is because i) dim L< == ]- dim IP(V() == dim{SCy; + SCs?} = dim F^Wa^0)) $ ii) M c{SC(pj + SCe?}1 and so, by § 3, M n (ker N') ® C projects to zero in FWCH0)). Therefore there is a framing co(^) of L( which extends to a partial framing co(^ t' ) of the canonical prolongation of (A. 2) . These are the differentials which occur in the proof of Lemma (2.6). The differentials u{t) + o V< U< framing L( are well-defined modulo F2( (image N')®C ) by the fact that they are dual to the basis {9,}, {s^} of F^We^0))*. More precisely, these differentials map isomorphically to a framing of L( under the natural morphism of mixed Hodge structures H3^ u U() -> (asymptotic mixed Hodge structure for the family (A. 9)) . For r^ as in (A. i) , cTy is algebraic so that the integral (A. i ) will occur as the coefficient of an algebraic basis element of (i.n.geN-)^- "'ty^'. . §3 {hyperplane section} So if we change the framing co(^) by an element of F^imageN')®^, the value of the coefficient is unchanged. 3 18 HERBERTCLEMENS REFERENCES [C 1] CLEMENS, H., Degeneration of Kahler manifolds. Duke Math. Journal, 44 (1977), 215-290. [C2] CLEMENS, H., Double Solids, Advances in Math. (to appear). [D] DELIGNE, P., equations Differentielles a Points Singuliers Reguliers, Springer Lecture Notes in Math., 163 (1970). [G] GRIFFITHS, P. A., Periods of integrals on algebraic manifolds, III, Publ. Math. I.H.E.S., 38 (1970), 125-180. [K] KLEIMAN, S., Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Publ. Math. I.H.E.S., 36 (1969), 281-298. [S] SCHMID, W., Variation of Hodge structure : The singularities of the period mapping, Inventiones math., 22 (i973)» an-S^- [Z] ZUCKER, S., Generalized intermediate Jacobians and the theorem on normal functions, Inventiones math., 33 (1976), 185-222. Department of Mathematics, The University of Utah, Salt Lake City, Utah 84112. Manuscrit refu Ie 12 janvier 1981. Rivisi Ie j mai 1982. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

The Neron model for families of intermediate Jacobians acquiring “algebraic” singularities

Clemens, Herbert
Publications mathématiques de l'IHÉS , Volume 58 (1) – Sep 11, 2008
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Springer Journals
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Copyright © 1983 by Publications Mathématiques de l’I.H.É.S.
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Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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10.1007/BF02953770
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Abstract

THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS ACQUIRING ^ALGEBRAIC" SINGULARITIES by HERBERT CLEMENS i. Introduction Let V be an irreducible complex projective manifold of dimension 2m — i. Let {^Jae s be an algebraic family of algebraic (m — i)-cycles on V whose members Zg are all homologically equivalent. In Appendix A of his paper " Periods of integrals on algebraic manifolds. III 9 ? ([G]$ p. 165), P. Griffiths defines an analytic map, called the Abel-Jacobi homomorphism (1.1 ) S-^J(V). r-FwjHpm-l/y. Q-p Here J(V) ==-—————————— is the Jacobian variety of V, and (1.1) is defined Him-lC^ Z) by picking a basepoint SQ e S and sending f \ r^ f\ (I-2) ^L- Jz,. Next let X be a complex projective manifold of dimension 2m, let A be the unit disc, and let (1.3 ) V,, ^eA, be an analytic family of divisors on X which are irreducible and non-singular as long as t ={= o. Suppose (1.4 ) {ZJ,es, is an algebraic family of algebraic (m — i)-cycles on V< for each t e A. Suppose further that, for fixed t, all cycles Z^, s eS,, are homologous to one another. Finally suppose that y = U ({Qxs< ) te^ " ' ' 217 6 HERBERT CLEMEN S is a smooth analytic variety with (1.5) y-b. everywhere of maximal rank. We make no assumptions about properness of (1.5) or connectivity of fibres. Let / * -.A* = (A —{o} ) be the bundle of complex tori over the punctured disc whose fibre over t is J(V,). Then for every section T: A -> y of (1.5) there is a commutative diagram of proper morphisms V ^ /* (1.6) defined fibrewise by (i . 2) with ^ = r(f). The point of this paper is to complete (1.6) to a commutative diagram ('.7) where / is obtained from </* by filling in over t = o with a commutative complex Lie group. The complex Lie group in question is the fibre over t = o of an analytic analogue of the Neron minimal model. We will be able to carry out this program only after putting some severe restrictions on the family (1.3). We devote the rest of the introduction to explaining these restrictions. Let ^=^({QxV,) and (1.8) (A: y-*^A*. We pull-back the bundle (1.8) via the universal covering map ^-^A * u \-> t = e 2" 1'" to obtain ^ : "^ -> S. Abusing notation we write y-1^) = ¥„ == V, = (x-^) whenever t = ^. The derived bundles R^_^(Z) and R^-^^Z) are trivial and we will denote their modules of global sections, taken modulo torsion, by H, and Hz respectively. Also H° = H2 ® C. The natural isomorphism Hz S H^(V«^) - H^_,(V< ) = H^(VJ s H^ 218 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 7 is not the identity map on H^ but rather the monodromy isomorphism T,: Hz ->Hz. Let T* be the adjoint of T» with respect to the natural unimodular pairing H^xH^ Z (Y.<°) ~>^^ Our first major assumption is that (1.9) (T, - I)2 == (T-- I)2 = o. Let N, == log T,, N* == log T*. We then have a filtration {W J on H? defined by W^H^o W^H^CkerNJ1 W^^ == (image NJ1 Wg^ =HZ. This filtration is called the asymptotic weight filtration. Under the identification H° == H2"*-1^; C) the Hodge filtration on H2"1"1^; C) induces a filtration F; on H0. This filtration /V/ varies with u e A, but there is a well-defined filtration (i. 10) F^ = lim exp(- t^N*)F: on H® called the asymptotic Hodge filtration. In [S], W. Schmid shows that the array (H^ W«, F'^) is a w^rf jyorf^ structure such that N': H 0^ ^ is a morphism of mixed Hodge structures of type (— i, — i). In fact in this situation which " comes from geometry ", we have that N-: H^W^.^W^ is an isomorphism over Q^. Our second major assumption is: (1.11) The Hodge structure of weight 2m on H^Wg^.i induced by (1.10) is of pure type (m, m). Finally we make an assumption which is not essential but will simplify the exposition: (1.12) All of H2"1"1^; C) is primitive cohomology. 219 8 HERBERT CLEMEN S 2. Growth of normal functions Let (2.1) T: A -> y be a section of the fibration y ^ A considered in (1.7). Let Z,SV, , feA, be the corresponding family of (m - i) -cycles. By Kleiman's smoothing theorem ([KJ; p. 297), we can assume that, if t + o, (a. 2) Z, = Z; - Z;' where Z; and Z;' are smooth and do not meet. We can resolve the family (2.3) ,U^({QxV<) sAx X along Vo so that: i) the fibre over t = o is a normal crossing variety in a smooth ambient space of dimension 2m', ii) the proper transform 2 of .SA^x^) is smooth and meets the fibre over zero transversely. In ([G1]; p. 245), we explicitly construct an action of the semigroup [o, i] x R on the resolved ambient space which is equivariant with the action {r,Q).t=re^t of [o, i] x R on A. It is easy to see that this action can be defined so as to respect Z since it is constructed first locally and then pieced together via fixations which can be constructed to be compatible with Z. As before, let (2.4) u =—.logf 2TO and take, for some fixed u^, a (2w - i)-chain ^ such that ^ = Z,. For (r, 6) e [o, i] x R, define I^-M).!^ where u == u, + IQ + ^\ Then ^ has as its boundary the algebraic cycle Z, with t = e2'"". ' / 220 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 9 Lemma (2.5). — The cycle (Fy^^ — FJ eH2^_i(V<; Z) has zero intersection number with any cycle which is invariant under the monodromy transformation T,: H^i(V,;Z)->H,,_,(V,;Z) Yl-^(i,i).Y . Proof. — The cycle (F^i — FJ, by construction, bounds in the ambient space of the resolution of the family (2.3). Therefore, by the Local Invariant Cycle Theorem ([G1]; p. 230)5 this cycle integrates to zero against any invariant element of H2"1"1^; C). Since the Poincar^ duals of the invariant cocycles are the invariant cycles, the lemma is proved. Next let <o(^) be a section of the canonical prolongation of FmR2m-l^(^ where (JL : ^ ->• A* is our family (1.8) and the canonical prolongation is as established in ([D]; pp. 91-92). Lemma (2.6). — Th e integrals j>«)=J^) are all of the form a{t)logt+f{t) where a(t) and f(t) are holomorphic on A. Proof. — The idea of the proof is to make the integrals in question into period functions, i.e. integrals over cycles, for some two-parameter family of varieties. Let X as in (1.3) denote the ambient variety for the family {V^ } and let U< cX, ^eA, be an analytic family of very ample hypersurfaces such that: i) if t + o, U( is smooth and meets V^ transversely, ii) Z^(U,nV<); iii) Z, is homologous to zero in U^. That such TJ( exist is an application of the results of [K]. (See the Columbia University Thesis of Spencer Bloch, 1971, pp. 9-10.) If we choose the family TJ( sufficiently amply, there will be a two-parameter family W^, (^)EAXA, of hypersurfaces in X such that: i) W(( (.) is smooth and irreducible for (^, t' ) e A* x A*, ii) W,;o)=U,uV,. 2 io HERBERTCLEMENS Next choose chains Sy c U( such that ^ = Z, and define F^ ^ = 1^ — S,,. Finally because of the simple nature of the degenerations W^ ^ "~^^(<,o) we can form a continuously varying family of cycles r^) c W^, such that Hr^r^=r^. Now the assumption (1.12) that the differential u(t) on V< is primitive implies that there is a two-parameter family of differentials (o^eF-H^W^C) such that: i) 6)(^, 0) = G)(^) + 0 ; v< u, ii) the family G)(^ t' ) extends over A X A to give a section of the canonical prolongation of the Hodge bundle for the two-variable degeneration (1). In fact, let <P(^ = r^,.^ — F^^. Then <p(u.u'+i) == ?(u,u') and» ^ ^^ lim f (of/, f ) == o. <^oJq>(^) 0 / Therefore the integrals f <o(^')-«'J "(^') </ 1 (»,*<') •"*'(»*,«') are well-defined functions of the variables u and f = e^'. Also since periods with respect to the canonical prolongation have at most logarithmic growth, these functions must be holomorphic along A* X {o}, and have value Jr,^) at (^ o). Finally we use logarithmic growth of periods with respect to the canonical prolongation once again, this time in the ^-direction. This allows us to conclude that a(t)=^-.( <o^) 2m Jr r •'^tt+l"1!* is bounded at t == o, since, by Lemma (2.5), it is a well-defined function of t. So f co(^~^)logf J ly is well-defined and so also bounded at t == o and the lemma is proved. (1) For a more complete discussion of this point, see the Appendix. 222 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS n 3. The Neron model Recall that in § i we defined N,: Hz^Hz. Now (image N,) is not in general a direct summand of H^ so we enlarge it: (3.1) Ey^ === {< p e Hz : some integral multiple of 9 lies in (image NJ } is called the module of vanishing cycles. Also (3.a) E^=(kerN.) is called the module of invariant cycles. Eyan is totally isotropic for the intersection pairing on Hg, in fact, the intersection bilinear form is identically zero on EinvXE^. So it easy to see that there is a splitting of H^ (3.3) Hz=L®ECE^ such that E®Eyan == E^y, and the intersection pairing is unimodular on L€)E^ and on E, and these two symplectic modules are orthogonal. Furthermore we can adjust the definition of L so that L is totally isotropic. Let {9,} be a basis for E^n satisfying (1.12). Let {8^, e/}?=i be a symplectic basis for E and let {Xj c L be such that {X p <pj}jLi is a symplectic basis for L©E^. The Riemann relations imply that if {o\(^), ^,)} is a framing of P'R^'^C) for (JL as in (1.8), then the matrix -j^) s^w (3.4) .L,^) J^). is invertible for each t =t= o. So we can normalize the choice of the framing to make (3.4) be the identity matrix for each t 4= o. (Notice that the matrix (3.4) is well-defined as a function of t since the cycles ^ and e^ are invariant.) Now if <( * " denotes Poincar^ dual, we can write any framing {<x)^), ^(t)} of pnH^-^V^; C) for t=^ as (3.5) i) S(^))^+?(^^))8?+^^ ii) 2(^^))^+S(^^))8?+2;(J^^^^^^^ The elements (3.5) therefore frame F^ c H° (see (1.10)). 222 12 HERBERT CLEMEN S Now Schmid's theory and our normalization of (3.4) imply a considerable amount about the entries in (3.5). First of all we wish to compute FS as in (i. 10). To do this, suppose (3.6) N.(^==2^^. Then (3.7) f, - "Sff^f •>>j y s J<S>j. is a well-defined function of t = e^, and F^ is obtained by replacing f in (3.5) by the operator (3.7) and taking limits as t ->-o. } Now our assumption in (1.11) is that (^ + W^_i) 2 L* = SC^* for L as in (3.3). But we have arranged that the (f^(t)) = KroneckerS,. and (J,^^) = ° in (3-5). So replacing J^ by (3.7), all enS-ies in (3.5 i)) stay bounded as t -> o. Thus in particular (3.8) L"*^) =mjiu+ (holo. fn. oft). Also, the fact that F^ n W^_i induces a Hodge structure on E* implies that all entries in (3-5) ") are bounded and therefore holomorphic functions of t at (== o. So we can rewrite (3.5) as follows: (3.9) i) «\(f) == \* + 2 (w<,M + "„(<))¥; + 2 (<o,/(<))e? ") ^W = 8; + S (^,(f))y* + S (^(f))s? where aU functions of t on the right-hand-side are holomorphic at t = o. We were able to replace m^ with my in the above formula because my = (\.N,X,) == (X..T.X,) = (T^-^.X^) =(-N.\.X,) =^.N.X.)=OT,,. From (3.9) and the characterization of the canonical prolongation in ([Z]; p. 189), one concludes that the framing (3.9) of (3.10) FOTR!!B-1(A.(C) is in fact a framing defining the canonical prolongation of Deligne. Therefore Lemma (2.6) applies to the families of differentials {^(f), •»)&(<)}. Also if we use the dual basis to (3.9) to frame the dual bundle to (3 10) then the "Jacobian bundle" ^•-^A* 224 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 13 in (1.6) can be described as the bundle whose fibre over t is obtained by dividing the affine space C1"4'8 by the lattice generated by the columns of the matrix n o {m^u+^(t)) (o)^))1 lo I W) (T^))J where, as always, u = —. log t. Let 2TCZ (3.11 ) ^'^A be the analytic fibration obtained by filling in over t == o with the quotient of C^8 by the partial lattice generated by the columns of I o (^(o))T (3-") 0 I (YW(0))J (The fact that these columns are indeed independent over R follows from the fact that [I("y]^(o))] is the period matrix for the Hodge structure of weight 2m — i on ^2m-il^2m-2 in ^e asymptotic mixed Hodge structure.) The fibration (3.11) is as in ([Z]; p. 191). Our next step is to enlarge the fibre over t == o in (3.11) as is done in the construction of the Neron model associated to the degeneration of a family of abelian varieties. The purpose is the same, namely, so that the sections of / * —^ A* considered in S 2 f^jro extend over t = o. To accomplish this we refer to (3.3) and define (3.13) L={XeL@)Q^:X has integral intersection number with each element of N^L}. Then the group L/L is naturally the dual of the group Ey^/N,L and so has order equal to det(w,.) where (Wy) is the non-degenerate symmetric matrix in (3.6). Furthermore the natural map (3.14) N,: L/L-^E^/N.L is an isomorphism. Since each element X e L is a section of R^_^(QJ which is invariant modulo elements of Ram-i^^)? h g^^ a well-defined section (3.15) ^^r' '^- Such a section is zero if and only if X e L. Thus we have an isomorphism of L/L with a group of sections (3.15). From now on we will denote this group simply as (3.16) ^. 225 14 HERBERT CLEMEN S We are now ready to define the Neron model associated to the degeneration (3.17) r-^^ of complex tori. We take \<S\ copies of^'n^s.n ) and index them by the elements of <S. We identify a point x in the fibre of ^ over t 4= o with a point y in the fibre of ^ over the same t if and only if (3.18) ^-j==^-^ in J(V(). The result is a smooth complex manifold (3-19) /-^^ whose restriction to A* is (3.17), and whose fibre Jo over t == o fits into the exact sequence (3.20) o ->Jo^Jo-> ^ -^o where Jo is the fibre of / f over t = o. By Lemma (2.6), if (o(^) is any section of the canonical prolongation of FmR^-^C!), ^en J^^LX)^) is a well-defined function of t holomorphic at t = o for each X e L. Thus (3-21) "JN.W is a well-defined section of (3.19) which extends over t = o. In fact (3.21) gives a section of (3.19) which passes through the same component of Jo that f does, th^t J x is, the component given by g = . • , j /^ So now suppose we have an analytic family of algebraic (m — i)-cycles 7 _ y y ^.( —- Z.( — Z.( as in (2.2). Let ^u=Z;-~Z;' as before. By Lemma (2.5)3 (3.22) r,,.i-r,=<p6E^ so that there is X e L such that N,(X) == <p. By Lemma (2.6), the integrals j^w-uf^t) are bounded holomorphic functions of t for G)(^) e{co,(/), T]^)}. Thus: -^ THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 15 Theorem (3.23). — The Abel'Jacobi map A-->^ '^-J. extends over t = o to a section of ^-^ A whose value at t = o lies in the component of Jo ^w by s-L^ with N,x== r^i-r^ . Since all our constructions can be carried out holomorphically with respect to auxiliary parameters we conclude: Corollary (3.24). — The diagram (1.6) extends to a commutative diagram y -^ / \ / This last corollary is just the analytic analogue of the " universal property " of the Neron model in the algebraic case. Appendix The purpose of this appendix is to justify the assertion, made during the proof of Lemma (2.6), that the integrals ( A- 1 ) J^W appear as coefficients, along t' = o, of sections of the canonical prolongation of (A.a) {F2H3(W(,,,))^,)g^^» with respect to a flat basis of (A.3) {H^W^,^}^,^^,^. It is this fact that allows us to conclude from ([D]$ pp. 91-92) that the normal functions (A.i) have at worst logarithmic growth. Let TT : S x H -> A' x A' (u, u^ h> (t, t') 227 16 HERBERT CLEMEN S be the universal covering map as in § 2. Using TT*, we pull the bundle (A. 3) back to ^/ /^/ ^ a trivial bundle on A X A whose global sections will be denoted by K . There are two commuting, nilpotent endomorphisms of K^, namely N = logarithm of monodromy around t •== o, N' = logarithm of monodromy around t' = o. Notice that W/< Q\ = V^ u U<, the union of two smooth manifolds meeting transversely. So (N')2 = o, in fact, the topological part of our analysis in § 3 applies to the one- variable degeneration ^-^.O). We want to apply the Gattini-Kaplan-Schmid theory of asymptotic mixed Hodge structures to the two-parameter family (A. 3). This theory says, first of all, that K32 has a mixed Hodge structure whose weight filtration is defined by the nilpotent endo- morphism N + N', and whose Hodge filtration is given by (A.4) P== lim exp(~MN-M/N/)^(F*H3(W.^)). v .»/ (<,r)->(o,o) A v / \ \ ((,<}// If H56 is as in (i .8)-(i . n) , then there is a subquotient of K^ which can be identified with H21. Namely let kernel^ : K2 -. K2) ^A.5 j ^3 -(^g^)®^) nK ^ The filtrations on Hz ® C induced by the weight and Hodge filtrations on K21 ® C define a mixed Hodge structure on (A. 5) which is isomorphic to (the mixed Hodge structure on) (A.6) H^OG^ Here Hz has the mixed Hodge structure in (i .8)-(i .11), and Gz has the asymptotic mixed Hodge structure for the family {U^}^^ constructed in the proof of Lemma (2.6). Now let 9,* and s? be as in (3.5). These are flat, N-invariant sections of {H3(V()z}(eA* ^d can be extended to flat sections of (A. 3) which are both N and N' invariant. We compute the limit (A. 4) in two steps, first letting t ' go to zero and then letting t go to zero. As in § 3, we see that after the first step, the vector space Ff = lim exp(-^^^^(W^) can be written as the direct sum of two subspaces, (A.7) M,=(Ffn{SZ<p;+SZ^}1) and a second subspace, which we will call (A.8) L,, 228 THE NERON MODEL FOR FAMILIES OF INTERMEDIATE JACOBIANS 17 which lies in (ker N') ® C and gives F2!!3^) in the weight three graded quotient of the asymptotic mixed Hodge structure associated to the one-parameter family (A.9) W^->W(,o)=V,uU,. The important point is that L = lim exp(— ^N)L( and M = lim exp(— !/N)M( is a direct-sum decomposition of F2 (see (A. 4)) . This is because i) dim L< == ]- dim IP(V() == dim{SCy; + SCs?} = dim F^Wa^0)) $ ii) M c{SC(pj + SCe?}1 and so, by § 3, M n (ker N') ® C projects to zero in FWCH0)). Therefore there is a framing co(^) of L( which extends to a partial framing co(^ t' ) of the canonical prolongation of (A. 2) . These are the differentials which occur in the proof of Lemma (2.6). The differentials u{t) + o V< U< framing L( are well-defined modulo F2( (image N')®C ) by the fact that they are dual to the basis {9,}, {s^} of F^We^0))*. More precisely, these differentials map isomorphically to a framing of L( under the natural morphism of mixed Hodge structures H3^ u U() -> (asymptotic mixed Hodge structure for the family (A. 9)) . For r^ as in (A. i) , cTy is algebraic so that the integral (A. i ) will occur as the coefficient of an algebraic basis element of (i.n.geN-)^- "'ty^'. . §3 {hyperplane section} So if we change the framing co(^) by an element of F^imageN')®^, the value of the coefficient is unchanged. 3 18 HERBERTCLEMENS REFERENCES [C 1] CLEMENS, H., Degeneration of Kahler manifolds. Duke Math. Journal, 44 (1977), 215-290. [C2] CLEMENS, H., Double Solids, Advances in Math. (to appear). [D] DELIGNE, P., equations Differentielles a Points Singuliers Reguliers, Springer Lecture Notes in Math., 163 (1970). [G] GRIFFITHS, P. A., Periods of integrals on algebraic manifolds, III, Publ. Math. I.H.E.S., 38 (1970), 125-180. [K] KLEIMAN, S., Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Publ. Math. I.H.E.S., 36 (1969), 281-298. [S] SCHMID, W., Variation of Hodge structure : The singularities of the period mapping, Inventiones math., 22 (i973)» an-S^- [Z] ZUCKER, S., Generalized intermediate Jacobians and the theorem on normal functions, Inventiones math., 33 (1976), 185-222. Department of Mathematics, The University of Utah, Salt Lake City, Utah 84112. Manuscrit refu Ie 12 janvier 1981. Rivisi Ie j mai 1982.

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