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Extremal metrics and K-stability

Extremal metrics and K-stability Abstract We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson, which relate to the case of Kähler–Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it. 1. Introduction Much progress has recently been made in understanding the relation between stability and the existence of special metrics on smooth algebraic varieties. Such a relation was conjectured by Yau for Kähler–Einstein metrics. One of the problems is to find the appropriate notion of stability. An example is K-stability, introduced by Tian [11] (see also Donaldson [3]), which is conjectured to be equivalent to the existence of a Kähler–Einstein metric or, more generally, a Kähler metric of constant scalar curvature. In this paper, we define a modification of K-stability and formulate a conjecture relating it to the existence of extremal metrics in the sense of Calabi [1]. These metrics are defined to be critical points of the L2 norm of the scalar curvature defined on Kähler metrics in a fixed cohomology class. See also Mabuchi [7] for the relation between stability and extremal metrics. The definition of K-stability in [3] involves considering test-configurations of the variety which are degenerations into possibly singular and non-reduced schemes. For every test- configuration there is an induced ℂ*-action on the central fibre, and the stability condition is that the corresponding generalised Futaki invariant is non-negative (using the convention in [10] for the Futaki invariant). Our modification essentially consists of working ‘orthogonally’ to a maximal torus of automorphisms of the variety. This means that we consider only test-con-fi-gu-ra-tions which commute with this torus, and the ℂ*-action induced on the central fibre is orthogonal to the torus with respect to an inner product, to be defined. Since this orthogonality condition is not very natural, it is more convenient instead to modify the Futaki invariant as follows. The maximal torus contains a distinguished ℂ*-action χ, generated by the extremal vector field (see Futaki and Mabuchi [6]). This induces an action on the central fibre, which we also denote by χ. Denote by α the ℂ*-action on the central fibre induced by the test-configuration, and write F(α) for the Futaki invariant. The modified Futaki invariant is then   1.1 The variety is K-stable relative to the maximal torus if is non-negative for all test-configurations commuting with the torus, and zero only for test-configurations arising from a ℂ*-action on the variety. Conjecture 1.1 A polarised variety admits an extremal metric in the class of the polarisation, if and only if it is K-stable relative to a maximal torus. In the next section we will give the definitions of the Futaki invariant and the inner product. In Section 3 we motivate Conjecture 1.1 by a result in the finite-dimensional framework of moment maps and stability. Finally, in Section 4 we give an example to support Conjecture 1.1. 2. Basic definitions We first recall the definition of the generalized Futaki invariant from Donaldson [3]. Let V be a polarised scheme of dimension n, with a very ample line bundle ℒ. Let α be a ℂ*-action on V with a lifting to ℒ. This induces a ℂ*-action on the vector space of sections for all integers k ≥ 1. Let dk be the dimension of , and denote the infinitesimal generator of the action by Ak. Denote by wk(α) the weight of the action on the top exterior power of . This is the same as the trace . Then dk and wk(α) are polynomials in k of degree n and n + 1 respectively for k sufficiently large, so we can write   The Futaki invariant is defined to be   The choice of lifting of α to the line bundle is not unique; however, Ak is defined up to the addition of a scalar matrix. In fact, if we embed V into using ℒ, then lifting α is equivalent to giving a ℂ*-action on which induces α on V in . Since the embedding by sections of a line bundle is not contained in any hyperplane, two such ℂ*-actions differ by an action that acts trivially on , that is, one with a constant weight, say λ. We find that for another lifting, the sequence of matrices are related to the Ak by   where I is the identity matrix. A simple computation shows that F(α) is independent of the lifting of α to ℒ. We now define an inner product on ℂ*-actions. Since ℂ*-actions do not naturally form a vector space, this is not really an inner product, but for smooth varieties we will see that it is the restriction of an inner product on a space of holomorphic vector fields to the set of ℂ*-actions. Let α and β be two ℂ*-actions on V with liftings to ℒ. If we denote the infinitesimal generators of the actions on by Ak, Bk, then is a polynomial of degree n + 2 in k. We define the inner product to be the leading coefficient in   Again, this does not depend on the particular liftings of α and β to the line bundle, since we are normalizing each Ak and Bk to have trace zero. Before we proceed, it is worth looking at the case when the variety is smooth. In this case we can consider the algebra of holomorphic vector fields on V which lift to ℒ. This is the Lie algebra of a group of holomorphic automorphisms of V. Inside this group, let G be the complexification of a maximal compact subgroup K. Let be the Lie algebras of G, K. Denoting by the elements in 𝔨 which generate circle subgroups, our inner product on ℂ*-actions gives an inner product on . Since this is a dense subalgebra of 𝔨, the inner product extends to 𝔨 by continuity. We further extend this inner product to 𝔤 by complexification, and compute it differential-geometrically. This is analogous to the computation in [3, Section 2.2] showing the relation between the Futaki invariant as defined here, and the original differential-geometric definition of Futaki [5]. Note that 𝔤 is a space of holomorphic vector fields on V which lift to ℒ. Let v, w be two holomorphic vector fields on V, with liftings v̂, ŵ to ℒ. Let ω be a Kähler metric on V in the class , induced by a choice of Hermitian metric on ℒ. We can then write   where is the horizontal lift of v, is the horizontal lift of w, t is the canonical vector field on the total space of ℒ, defined by the action of scalar multiplication, and f and g are smooth functions on V. As in [3] we see that   so in particular f and g are defined up to an additive constant, and we can normalize them to have zero integral over V. We would like to show that   where we have assumed that f and g have zero integral over V. Making use of the identity , it is enough to show this when v = w. Furthermore, we can assume that v generates a circle action, since is dense in 𝔨. We can find the leading coefficients of dk, and for this circle action using the equivariant Riemann–Roch formula, in the same way as was done in [3]. We find that these leading coefficients are given by   respectively. If we normalize f to have zero integral over V, then we obtain the formula for the inner product that we were after. This inner product on holomorphic vector fields has also been defined by Futaki and Mabuchi in [6], where it is shown that it depends only on the Kähler class, and not on the specific representative chosen. This can also been seen from the fact that it can be defined algebro-geometrically, just like the Futaki invariant. We next recall the notion of a test-configuration from [3], and introduce the modification that we need. Definition 2.1 A test-configuration for (V, L) of exponent r consists of a ℂ*-equivariant flat family of schemes (where ℂ* acts on ℂ by multiplication) and a ℂ*-equivariant ample line bundle ℒ over 𝒱. We require that the fibres be isomorphic to (V, Lr) for t ≠ 0, where . The test-configuration is called a product configuration if . We say that the test-configuration is compatible with a torus T of automorphisms of (V, L), if there is a torus action on which preserves the fibres of , commutes with the ℂ*-action, and restricts to T on for t ≠ 0. With these preliminaries we can state the main definition. Definition 2.2 A polarised variety (V, L) is K-stable relative to a maximal torus of automorphisms if for all test-configurations compatible with the torus, and equality holds only if the test-configuration is a product configuration. Here we denote by and χ̃ the ℂ*-actions induced on the central fibre of the test-configuration (χ̃ being induced by the extremal ℂ*-action χ in the chosen maximal torus) and is defined as in equation (1.1). 3. Moment map and stability The aim of this section is to describe a result in the finite-dimensional picture of moment maps and stability, which motivates Conjecture 1.1. First we introduce the necessary notation. Let X be a finite-dimensional Kähler variety. When the results of this section are formally applied to the problem of extremal metrics, X will be an infinite-dimensional space of complex structures on a variety V. The details will be discussed at the end of this section. Denote the Kähler form by ω, and let ℒ be a line bundle over X with first Chern class represented by ω. Suppose that a compact connected group K acts on X by holomorphic transformations, preserving ω, and that there is a moment map for the action   where is the dual of the Lie algebra of K. This allows us to define an action of 𝔨 on sections of ℒ as follows. Choose a Hermitian metric on ℒ such that the corresponding unitary connection has curvature form given by . If induces a vector field v on X, and is the corresponding Hamiltonian function given by the composition   then ξ acts on ℒ via , where v̄ is the horizontal lift of v, and t is the vertical vector field generating the U(1)-action on the fibres (see Donaldson and Kronheimer [4, Section 6]). Suppose that there is a complexification G of the group K, with Lie algebra . The action of 𝔨 on X and ℒ extends to actions of 𝔤 by complexification. We will assume that this infinitesimal action gives rise to an action of G on the pair . This is the situation studied in geometric invariant theory, and the main definition that we need is the following one. Definition 3.1 A point x ∈ X is stable for the action of G on if, for a choice of lifting of x, the set Gx̃ is closed in ℒ This notion is what some authors call polystability, and usually stability requires in addition that the point in question have a zero-dimensional stabilizer in G. The relation between the moment map and stability is given by the following well-known result (see, for example, the work by Mumford, Fogarty and Kirwan [9]). Proposition 3.2 A point x ∈ X is stable for the action of G on if and only if there is an element g ∈ G such that . We would like to extend this characterisation of the G-orbits of zeros of the moment map to G-orbits of critical points of the norm squared of the moment map. More precisely, suppose that there is a non-degenerate inner product on 𝔨, invariant under the adjoint action. We also assume that on the Lie algebra 𝔱 of a maximal torus in K, the inner product is rational when restricted to the kernel of the exponential map. Since any two maximal tori are conjugate, it follows that the inner product is rational in this sense on the Lie algebra of any other torus in K as well. Using the inner product, identify 𝔨 with and from now on consider the moment map as a map from X to 𝔨. The norm squared of the moment map then defines a function ,   We will show that the G-orbits of critical points of f are characterized by stability with respect to the action of a certain subgroup of G. First of all, by differentiating f, we find that x is a critical point if and only if the vector field induced by μ(x) vanishes at x. In particular, the minima of the functional are given by x with μ(x) = 0, and the other critical points have nontrivial isotropy groups containing the group generated by μ(x). This is a circle subgroup, by the following lemma. Lemma 3.3 If x ∈ X is a non-minimal critical point of f, then μ(x) generates a circle subgroup of K. Proof Let , and denote by T the closure of the subgroup of K generated by β. This is a compact connected Abelian Lie group, and hence it is a torus. Letting 𝔱 be the Lie algebra of T, the moment map μT for the action of T on X is given by the composition of μ with the orthogonal projection from 𝔨 to 𝔱. Since by definition, , we find that . Let be an integral basis for the kernel of the exponential map from 𝔱 to T. Because of the rationality assumption on the inner product, what we need to show is that is rational for all i. Since is the Hamiltonian function for the vector field induced by vi, we know that vi acts on the fibre ℒx via . Since exp(vi) = 1, we find that fi(x) must be an integer. □ Now we define the subgroups of G which will feature in the stability condition. For a torus T in G with Lie algebra 𝔱, define two subalgebras of 𝔤:   Denote the corresponding connected subgroups by GT and . Then GT is the identity component of the centraliser of T, and is a subgroup isomorphic to the quotient of GT by T. Working on the level of the compact subgroup K, if , then the same formulae define Lie algebras and subgroups of K, such that   We can now write down the stability condition that we need. Definition 3.4 Let T be a torus in G fixing x. We say that x is stable relative to T if it is stable for the action of on . The main result of this section is the following theorem. Theorem 3.5 A point x in X is in the G-orbit of a non-minimal critical point of f if and only if it is stable relative to a maximal torus which fixes it. Before giving the proof, consider the effect of varying the maximal compact subgroup of G. If we replace K by a conjugate gKg−1 for some g ∈ G and we replace ω by , then we obtain a new compact group acting by sympectomorphisms. The associated moment map μg is related to μ by   3.1 where we identify the Lie algebra of gKg−1 with . Using the inner product on induced from the bilinear form on 𝔤, define the function . This satisfies fg(gx) = f(x) by (3.1) and the -invariance of the bilinear form, so in particular the critical points of fg are obtained by applying g to the critical points of f. Proof of Theorem 3.5 Suppose first that x is in the G-orbit of a critical point of f. By replacing K with a conjugate if necessary, we can assume that x itself is a critical point, so μ(x) fixes x. By Lemma 3.3 we obtain a circle action fixing x, generated by . Choose a maximal torus T fixing x, containing this circle. Since the moment map for the action of on X is the composition of μ with the orthogonal projection from 𝔨 to , we see that . By Proposition 3.2 this implies that x is stable for the action of . Conversely, suppose that x is stable for the action of for a maximal torus T which fixes x. Choose a maximal compact subgroup K of G containing T. Then is a maximal compact subgroup of , and using the assumption on x, Proposition 3.2 implies that y = gx is in the kernel of the corresponding moment map μT, for some . Then, for the moment map corresponding to K, μ(y) is contained in 𝔱 and therefore fixes y. This means that y is a critical point of f. □ We will now explain how the formula (1.1) arises. Since Gx fixes x, the action on the fibre defines a map . The derivative at the identity gives a linear map , which we denote by −Fx in order to match with the sign of the Futaki invariant. We say that is the weight of the action of α on ℒx. According to the Hilbert–Mumford numerical criterion for stability (see [9]), we have the following necessary and sufficient condition for a point x to be stable: for all one-parameter subgroups in , the weight on the central fibre is negative, or equal to zero if fixes x. Here x0 is defined to be . In other words, the condition is that   with equality if and only if x is fixed by the one-parameter subgroup. It is in-con-venient to restrict atten-tion to one-parameter subgroups in because the orthogonality condition is not a natural one for test-configurations. We would therefore like to be able to consider one-parameter subgroups in GT and adapt the numerical criterion. For a one-parameter subgroup in GT generated by we consider the one-parameter subgroup in generated by the orthogonal projection of α onto , which we denote by . We have   where is an orthonormal basis for 𝔱. Since and x is fixed by T, the central fibre for the two one-parameter groups generated by α and is the same; the only difference is the weight of the action on this fibre. Since is linear, we obtain   The extremal vector field χ is defined to be the element in 𝔱 dual to the functional Fx, restricted to 𝔱, under the inner product. In other words,   If we now choose the orthonormal basis βi such that , then the previous formula reduces to   If we define this expression to be , then the stability condition is equivalent to for all one-parameter subgroups generated by , with equality only if the one-parameter subgroup fixes x. We therefore obtain the following theorem. Theorem 3.6 A point x ∈ X is in the G-orbit of a critical point of f if and only if for each one-parameter subgroup of G generated by an element we have   with equality only if α fixes x. Here, T is a maximal torus fixing x and χ is the corresponding extremal vector field. To conclude this section, we explain why this result motivates Conjecture 1.1. The main idea is the infinite-dimensional picture described in Donaldson [2], in which the scalar curvature arises as a moment map. We start with a symplectic manifold M with symplectic form ω, and assume for simplicity that H1(M) = 0. The space X is the space of integrable complex structures on M compatible with ω. Then X is endowed with a natural Kähler metric. Together with ω, the points of X define metrics on M, so X can also be thought of as a space of Kähler metrics on M. The group K is the identity component of the group of symplectomorphisms of M. This acts on X, preserving the symplectic form. The Lie algebra 𝔨 of K can be identified with , the space of smooth real-valued functions on M with zero integral, using the Hamiltonian construction (we use the condition H1(M) = 0 here). The dual can also be identified with using the L2 pairing. Then a moment map for the action of K of X is given by mapping a point in X to the scalar curvature function of the corresponding metric on M. The complexification 𝔤 of 𝔨 is with the L2 product. The corresponding group G does not exist, but we can consider a foliation of X generated by the action of 𝔤 on X, whose leaves would be the orbits of G. As explained in [2], these leaves correspond to metrics on M in a fixed Kähler class. The problem of finding critical points of the norm squared of the moment map in a ‘G-orbit’ is therefore the problem of finding extremal metrics in a Kähler class. By the finite-dimensional result in this section, we expect that an analogous stability condition will characterize the Kähler classes which contain an extremal metric. Fixing an element J ∈ X, elements of the Lie algebra 𝔤 give rise to vector fields on . If we identify 𝔤 with , then, given an element with both f and g real-valued, the corresponding vector field is Vf + JVg. Here Vf and Vg are the Hamiltonian vector fields corresponding to f and g. The L2 inner product on 𝔤 is therefore an inner product on a space of vector fields (not necessarily holomorphic) on M. What we did in Section 2 was to shift attention to the central fibre of a test-configuration, on which the vector fields (or rather, the ℂ*-actions which they generate) are holomorphic, so we can compute the inner product algebro-geometrically and thereby give a purely algebro-geometric definition of the stability condition. 4. Example The aim of this section is to work out the stability criterion in a special case, and show how it relates to the existence of extremal metrics. Let Σ be a genus two curve, and ℳ a line bundle on it with degree one. The same computation can be carried out when the genus is greater than two and the line bundle has degree greater than one. Define X to be the ruled surface over Σ. Tønnesen-Friedman [12] has constructed a family of extremal metrics on X that does not exhaust the entire Kähler cone. We will show that X is K-unstable (relative to a maximal torus of automorphisms) for the remaining polarisations. Since there are no non-zero holomorphic vector fields on Σ, a holomorphic vector field on X must preserve the fibres. Thus, the holomorphic vector fields on X are given by sections of . Here, means endomorphisms with trace zero. The vector field given by the matrix generates a ℂ*-action β, and it is up to scalar multiple the only one that does (see Maruyama [8] for proofs). Therefore this must be a multiple of the extremal vector field, which is then given by . The destabilising test-configuration is an example of deformation to the normal cone of a subvariety, studied by Ross and Thomas [10], except that we need to take account of the extremal ℂ*-action as well. We consider the polarisation L = C + mS0, where C is the divisor given by a fibre, S0 is the zero section (that is, the image of in X) and m is a positive constant. We denote by S∞ the infinity section, which as a divisor is just S0 − C. Note that β fixes S∞ and acts on the normal bundle of S∞ with weight 1. We make no distinction between divisors and their associated line bundles, and we use the multiplicative and additive notations interchangeably, so for example for an integer k. The deformation to the normal cone of S∞ is given by the blowup   in the subvariety . If we denote the exceptional divisor by E, the line bundle is ample for , where ε is the Seshadri constant of as in [10]. In our case, . Thus we obtain a test-configuration with the ℂ* action induced by π from the product of the trivial action on X and the usual multiplication on ℂ. Denote the restriction of this ℂ*-action to the central fibre (X0, L0) by α. Since the extremal ℂ*-action fixes S∞, we obtain another action on the test-configuration, induced by π from the product of the extremal ℂ*-action on X and the trivial action on ℂ. Let us call the induced action on the central fibre β. We wish to calculate as defined in (1.1). For this we use the following decomposition, with t being the standard coordinate on ℂ:   According to [10] α acts with weight −1 on t; that is, it acts with weight −j on the summand of index j above. Also, β acts on   with weight l, plus perhaps a constant independent of l which we can neglect, since the matrices are normalized to have trace zero in the formula for the modified Futaki invariant. The dimension of this space is k + l − 1, by the Riemann–Roch theorem. Writing Ak and Bk for the infinitesimal generators of the actions α and β on , and dk for the dimension of this space, we can now compute   Using these, we can compute   We obtain   If for a rational c between 0 and m, then the variety is K-unstable (relative to a maximal torus of automorphisms). In [12, p. 23] the condition given for the existence of an extremal metric of the type studied, is that a certain polynomial   is positive for a < γ < ka in the notation of [12], where k is the parameter of the polarization (k = m + 1 in our notation) and a is a constant defined in terms of k. After a change of variables k = m + 1 and , we find that the two conditions are in fact the same, since a quadratic polynomial with rational coefficients cannot have an irrational double root. Acknowledgements I would like to thank my PhD supervisor, Simon Donaldson, for introducing me to this problem, and Richard Thomas for helpful conversations and for his many suggestions improving earlier versions of this paper. I would also like to thank the referee for his comments, and the EPSRC for financial support. References 1 Calabi E..  Yau S. T..  Extremal Kähler metrics,  Seminar on differential geometry ,  1982 Princeton Univ. Press 2 Donaldson S. K..  Atiyah D.,  Iagolnitzer M..  Remarks on gauge theory, complex geometry and four-manifold topology,  Fields medallists lectures ,  1997 Singapore World Scientific(pg.  384- 403) 3 Donaldson S. K..  Scalar curvature and stability of toric varieties,  J. Differential Geom. ,  2002, vol.  62 (pg.  289- 349) 4 Donaldson P. B.,  Kronheimer S. K.. ,  The geometry of four-manifolds Oxford Math. Monogr ,  1990 Oxford Clarendon Press 5 Futaki A..  An obstruction to the existence of Einstein–Kähler metrics,  Invent. Math. ,  1983, vol.  73 (pg.  437- 443) Google Scholar CrossRef Search ADS   6 Futaki T.,  Mabuchi A..  Bilinear forms and extremal Kähler vector fields associated with Kähler classes,  Math. Ann. ,  1995, vol.  301 (pg.  199- 210) Google Scholar CrossRef Search ADS   7 Mabuchi T..  Stability of extremal Kähler manifolds,  Osaka J. Math. ,  2004, vol.  41 (pg.  563- 582) 8 Maruyama M..  On automorphism groups of ruled surfaces,  J. Math. Kyoto University ,  1971, vol.  11 (pg.  89- 112) 9 Mumford D.,  Fogarty J.,  Kirwan F.. ,  Geometric invariant theory ,  1994 Springer 10 Ross J.,  Thomas R. P..  A study of the Hilbert–Mumford criterion for the stability of projective varieties,  2004  preprint, arXiv math.AG/0412519 11 Tian G..  Kähler–Einstein metrics with positive scalar curvature,  Invent. Math. ,  1997, vol.  137 (pg.  1- 37) Google Scholar CrossRef Search ADS   12 Tønnesen-Friedman C..  Extremal Kähler metrics on ruled surfaces,  1997 Odense University  Ph.D. thesis 2000 Mathematics Subject Classification 53C55 (primary), 53C25 (secondary). © 2006 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Extremal metrics and K-stability

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Oxford University Press
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© 2006 London Mathematical Society
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0024-6093
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1469-2120
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10.1112/blms/bdl015
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Abstract

Abstract We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson, which relate to the case of Kähler–Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it. 1. Introduction Much progress has recently been made in understanding the relation between stability and the existence of special metrics on smooth algebraic varieties. Such a relation was conjectured by Yau for Kähler–Einstein metrics. One of the problems is to find the appropriate notion of stability. An example is K-stability, introduced by Tian [11] (see also Donaldson [3]), which is conjectured to be equivalent to the existence of a Kähler–Einstein metric or, more generally, a Kähler metric of constant scalar curvature. In this paper, we define a modification of K-stability and formulate a conjecture relating it to the existence of extremal metrics in the sense of Calabi [1]. These metrics are defined to be critical points of the L2 norm of the scalar curvature defined on Kähler metrics in a fixed cohomology class. See also Mabuchi [7] for the relation between stability and extremal metrics. The definition of K-stability in [3] involves considering test-configurations of the variety which are degenerations into possibly singular and non-reduced schemes. For every test- configuration there is an induced ℂ*-action on the central fibre, and the stability condition is that the corresponding generalised Futaki invariant is non-negative (using the convention in [10] for the Futaki invariant). Our modification essentially consists of working ‘orthogonally’ to a maximal torus of automorphisms of the variety. This means that we consider only test-con-fi-gu-ra-tions which commute with this torus, and the ℂ*-action induced on the central fibre is orthogonal to the torus with respect to an inner product, to be defined. Since this orthogonality condition is not very natural, it is more convenient instead to modify the Futaki invariant as follows. The maximal torus contains a distinguished ℂ*-action χ, generated by the extremal vector field (see Futaki and Mabuchi [6]). This induces an action on the central fibre, which we also denote by χ. Denote by α the ℂ*-action on the central fibre induced by the test-configuration, and write F(α) for the Futaki invariant. The modified Futaki invariant is then   1.1 The variety is K-stable relative to the maximal torus if is non-negative for all test-configurations commuting with the torus, and zero only for test-configurations arising from a ℂ*-action on the variety. Conjecture 1.1 A polarised variety admits an extremal metric in the class of the polarisation, if and only if it is K-stable relative to a maximal torus. In the next section we will give the definitions of the Futaki invariant and the inner product. In Section 3 we motivate Conjecture 1.1 by a result in the finite-dimensional framework of moment maps and stability. Finally, in Section 4 we give an example to support Conjecture 1.1. 2. Basic definitions We first recall the definition of the generalized Futaki invariant from Donaldson [3]. Let V be a polarised scheme of dimension n, with a very ample line bundle ℒ. Let α be a ℂ*-action on V with a lifting to ℒ. This induces a ℂ*-action on the vector space of sections for all integers k ≥ 1. Let dk be the dimension of , and denote the infinitesimal generator of the action by Ak. Denote by wk(α) the weight of the action on the top exterior power of . This is the same as the trace . Then dk and wk(α) are polynomials in k of degree n and n + 1 respectively for k sufficiently large, so we can write   The Futaki invariant is defined to be   The choice of lifting of α to the line bundle is not unique; however, Ak is defined up to the addition of a scalar matrix. In fact, if we embed V into using ℒ, then lifting α is equivalent to giving a ℂ*-action on which induces α on V in . Since the embedding by sections of a line bundle is not contained in any hyperplane, two such ℂ*-actions differ by an action that acts trivially on , that is, one with a constant weight, say λ. We find that for another lifting, the sequence of matrices are related to the Ak by   where I is the identity matrix. A simple computation shows that F(α) is independent of the lifting of α to ℒ. We now define an inner product on ℂ*-actions. Since ℂ*-actions do not naturally form a vector space, this is not really an inner product, but for smooth varieties we will see that it is the restriction of an inner product on a space of holomorphic vector fields to the set of ℂ*-actions. Let α and β be two ℂ*-actions on V with liftings to ℒ. If we denote the infinitesimal generators of the actions on by Ak, Bk, then is a polynomial of degree n + 2 in k. We define the inner product to be the leading coefficient in   Again, this does not depend on the particular liftings of α and β to the line bundle, since we are normalizing each Ak and Bk to have trace zero. Before we proceed, it is worth looking at the case when the variety is smooth. In this case we can consider the algebra of holomorphic vector fields on V which lift to ℒ. This is the Lie algebra of a group of holomorphic automorphisms of V. Inside this group, let G be the complexification of a maximal compact subgroup K. Let be the Lie algebras of G, K. Denoting by the elements in 𝔨 which generate circle subgroups, our inner product on ℂ*-actions gives an inner product on . Since this is a dense subalgebra of 𝔨, the inner product extends to 𝔨 by continuity. We further extend this inner product to 𝔤 by complexification, and compute it differential-geometrically. This is analogous to the computation in [3, Section 2.2] showing the relation between the Futaki invariant as defined here, and the original differential-geometric definition of Futaki [5]. Note that 𝔤 is a space of holomorphic vector fields on V which lift to ℒ. Let v, w be two holomorphic vector fields on V, with liftings v̂, ŵ to ℒ. Let ω be a Kähler metric on V in the class , induced by a choice of Hermitian metric on ℒ. We can then write   where is the horizontal lift of v, is the horizontal lift of w, t is the canonical vector field on the total space of ℒ, defined by the action of scalar multiplication, and f and g are smooth functions on V. As in [3] we see that   so in particular f and g are defined up to an additive constant, and we can normalize them to have zero integral over V. We would like to show that   where we have assumed that f and g have zero integral over V. Making use of the identity , it is enough to show this when v = w. Furthermore, we can assume that v generates a circle action, since is dense in 𝔨. We can find the leading coefficients of dk, and for this circle action using the equivariant Riemann–Roch formula, in the same way as was done in [3]. We find that these leading coefficients are given by   respectively. If we normalize f to have zero integral over V, then we obtain the formula for the inner product that we were after. This inner product on holomorphic vector fields has also been defined by Futaki and Mabuchi in [6], where it is shown that it depends only on the Kähler class, and not on the specific representative chosen. This can also been seen from the fact that it can be defined algebro-geometrically, just like the Futaki invariant. We next recall the notion of a test-configuration from [3], and introduce the modification that we need. Definition 2.1 A test-configuration for (V, L) of exponent r consists of a ℂ*-equivariant flat family of schemes (where ℂ* acts on ℂ by multiplication) and a ℂ*-equivariant ample line bundle ℒ over 𝒱. We require that the fibres be isomorphic to (V, Lr) for t ≠ 0, where . The test-configuration is called a product configuration if . We say that the test-configuration is compatible with a torus T of automorphisms of (V, L), if there is a torus action on which preserves the fibres of , commutes with the ℂ*-action, and restricts to T on for t ≠ 0. With these preliminaries we can state the main definition. Definition 2.2 A polarised variety (V, L) is K-stable relative to a maximal torus of automorphisms if for all test-configurations compatible with the torus, and equality holds only if the test-configuration is a product configuration. Here we denote by and χ̃ the ℂ*-actions induced on the central fibre of the test-configuration (χ̃ being induced by the extremal ℂ*-action χ in the chosen maximal torus) and is defined as in equation (1.1). 3. Moment map and stability The aim of this section is to describe a result in the finite-dimensional picture of moment maps and stability, which motivates Conjecture 1.1. First we introduce the necessary notation. Let X be a finite-dimensional Kähler variety. When the results of this section are formally applied to the problem of extremal metrics, X will be an infinite-dimensional space of complex structures on a variety V. The details will be discussed at the end of this section. Denote the Kähler form by ω, and let ℒ be a line bundle over X with first Chern class represented by ω. Suppose that a compact connected group K acts on X by holomorphic transformations, preserving ω, and that there is a moment map for the action   where is the dual of the Lie algebra of K. This allows us to define an action of 𝔨 on sections of ℒ as follows. Choose a Hermitian metric on ℒ such that the corresponding unitary connection has curvature form given by . If induces a vector field v on X, and is the corresponding Hamiltonian function given by the composition   then ξ acts on ℒ via , where v̄ is the horizontal lift of v, and t is the vertical vector field generating the U(1)-action on the fibres (see Donaldson and Kronheimer [4, Section 6]). Suppose that there is a complexification G of the group K, with Lie algebra . The action of 𝔨 on X and ℒ extends to actions of 𝔤 by complexification. We will assume that this infinitesimal action gives rise to an action of G on the pair . This is the situation studied in geometric invariant theory, and the main definition that we need is the following one. Definition 3.1 A point x ∈ X is stable for the action of G on if, for a choice of lifting of x, the set Gx̃ is closed in ℒ This notion is what some authors call polystability, and usually stability requires in addition that the point in question have a zero-dimensional stabilizer in G. The relation between the moment map and stability is given by the following well-known result (see, for example, the work by Mumford, Fogarty and Kirwan [9]). Proposition 3.2 A point x ∈ X is stable for the action of G on if and only if there is an element g ∈ G such that . We would like to extend this characterisation of the G-orbits of zeros of the moment map to G-orbits of critical points of the norm squared of the moment map. More precisely, suppose that there is a non-degenerate inner product on 𝔨, invariant under the adjoint action. We also assume that on the Lie algebra 𝔱 of a maximal torus in K, the inner product is rational when restricted to the kernel of the exponential map. Since any two maximal tori are conjugate, it follows that the inner product is rational in this sense on the Lie algebra of any other torus in K as well. Using the inner product, identify 𝔨 with and from now on consider the moment map as a map from X to 𝔨. The norm squared of the moment map then defines a function ,   We will show that the G-orbits of critical points of f are characterized by stability with respect to the action of a certain subgroup of G. First of all, by differentiating f, we find that x is a critical point if and only if the vector field induced by μ(x) vanishes at x. In particular, the minima of the functional are given by x with μ(x) = 0, and the other critical points have nontrivial isotropy groups containing the group generated by μ(x). This is a circle subgroup, by the following lemma. Lemma 3.3 If x ∈ X is a non-minimal critical point of f, then μ(x) generates a circle subgroup of K. Proof Let , and denote by T the closure of the subgroup of K generated by β. This is a compact connected Abelian Lie group, and hence it is a torus. Letting 𝔱 be the Lie algebra of T, the moment map μT for the action of T on X is given by the composition of μ with the orthogonal projection from 𝔨 to 𝔱. Since by definition, , we find that . Let be an integral basis for the kernel of the exponential map from 𝔱 to T. Because of the rationality assumption on the inner product, what we need to show is that is rational for all i. Since is the Hamiltonian function for the vector field induced by vi, we know that vi acts on the fibre ℒx via . Since exp(vi) = 1, we find that fi(x) must be an integer. □ Now we define the subgroups of G which will feature in the stability condition. For a torus T in G with Lie algebra 𝔱, define two subalgebras of 𝔤:   Denote the corresponding connected subgroups by GT and . Then GT is the identity component of the centraliser of T, and is a subgroup isomorphic to the quotient of GT by T. Working on the level of the compact subgroup K, if , then the same formulae define Lie algebras and subgroups of K, such that   We can now write down the stability condition that we need. Definition 3.4 Let T be a torus in G fixing x. We say that x is stable relative to T if it is stable for the action of on . The main result of this section is the following theorem. Theorem 3.5 A point x in X is in the G-orbit of a non-minimal critical point of f if and only if it is stable relative to a maximal torus which fixes it. Before giving the proof, consider the effect of varying the maximal compact subgroup of G. If we replace K by a conjugate gKg−1 for some g ∈ G and we replace ω by , then we obtain a new compact group acting by sympectomorphisms. The associated moment map μg is related to μ by   3.1 where we identify the Lie algebra of gKg−1 with . Using the inner product on induced from the bilinear form on 𝔤, define the function . This satisfies fg(gx) = f(x) by (3.1) and the -invariance of the bilinear form, so in particular the critical points of fg are obtained by applying g to the critical points of f. Proof of Theorem 3.5 Suppose first that x is in the G-orbit of a critical point of f. By replacing K with a conjugate if necessary, we can assume that x itself is a critical point, so μ(x) fixes x. By Lemma 3.3 we obtain a circle action fixing x, generated by . Choose a maximal torus T fixing x, containing this circle. Since the moment map for the action of on X is the composition of μ with the orthogonal projection from 𝔨 to , we see that . By Proposition 3.2 this implies that x is stable for the action of . Conversely, suppose that x is stable for the action of for a maximal torus T which fixes x. Choose a maximal compact subgroup K of G containing T. Then is a maximal compact subgroup of , and using the assumption on x, Proposition 3.2 implies that y = gx is in the kernel of the corresponding moment map μT, for some . Then, for the moment map corresponding to K, μ(y) is contained in 𝔱 and therefore fixes y. This means that y is a critical point of f. □ We will now explain how the formula (1.1) arises. Since Gx fixes x, the action on the fibre defines a map . The derivative at the identity gives a linear map , which we denote by −Fx in order to match with the sign of the Futaki invariant. We say that is the weight of the action of α on ℒx. According to the Hilbert–Mumford numerical criterion for stability (see [9]), we have the following necessary and sufficient condition for a point x to be stable: for all one-parameter subgroups in , the weight on the central fibre is negative, or equal to zero if fixes x. Here x0 is defined to be . In other words, the condition is that   with equality if and only if x is fixed by the one-parameter subgroup. It is in-con-venient to restrict atten-tion to one-parameter subgroups in because the orthogonality condition is not a natural one for test-configurations. We would therefore like to be able to consider one-parameter subgroups in GT and adapt the numerical criterion. For a one-parameter subgroup in GT generated by we consider the one-parameter subgroup in generated by the orthogonal projection of α onto , which we denote by . We have   where is an orthonormal basis for 𝔱. Since and x is fixed by T, the central fibre for the two one-parameter groups generated by α and is the same; the only difference is the weight of the action on this fibre. Since is linear, we obtain   The extremal vector field χ is defined to be the element in 𝔱 dual to the functional Fx, restricted to 𝔱, under the inner product. In other words,   If we now choose the orthonormal basis βi such that , then the previous formula reduces to   If we define this expression to be , then the stability condition is equivalent to for all one-parameter subgroups generated by , with equality only if the one-parameter subgroup fixes x. We therefore obtain the following theorem. Theorem 3.6 A point x ∈ X is in the G-orbit of a critical point of f if and only if for each one-parameter subgroup of G generated by an element we have   with equality only if α fixes x. Here, T is a maximal torus fixing x and χ is the corresponding extremal vector field. To conclude this section, we explain why this result motivates Conjecture 1.1. The main idea is the infinite-dimensional picture described in Donaldson [2], in which the scalar curvature arises as a moment map. We start with a symplectic manifold M with symplectic form ω, and assume for simplicity that H1(M) = 0. The space X is the space of integrable complex structures on M compatible with ω. Then X is endowed with a natural Kähler metric. Together with ω, the points of X define metrics on M, so X can also be thought of as a space of Kähler metrics on M. The group K is the identity component of the group of symplectomorphisms of M. This acts on X, preserving the symplectic form. The Lie algebra 𝔨 of K can be identified with , the space of smooth real-valued functions on M with zero integral, using the Hamiltonian construction (we use the condition H1(M) = 0 here). The dual can also be identified with using the L2 pairing. Then a moment map for the action of K of X is given by mapping a point in X to the scalar curvature function of the corresponding metric on M. The complexification 𝔤 of 𝔨 is with the L2 product. The corresponding group G does not exist, but we can consider a foliation of X generated by the action of 𝔤 on X, whose leaves would be the orbits of G. As explained in [2], these leaves correspond to metrics on M in a fixed Kähler class. The problem of finding critical points of the norm squared of the moment map in a ‘G-orbit’ is therefore the problem of finding extremal metrics in a Kähler class. By the finite-dimensional result in this section, we expect that an analogous stability condition will characterize the Kähler classes which contain an extremal metric. Fixing an element J ∈ X, elements of the Lie algebra 𝔤 give rise to vector fields on . If we identify 𝔤 with , then, given an element with both f and g real-valued, the corresponding vector field is Vf + JVg. Here Vf and Vg are the Hamiltonian vector fields corresponding to f and g. The L2 inner product on 𝔤 is therefore an inner product on a space of vector fields (not necessarily holomorphic) on M. What we did in Section 2 was to shift attention to the central fibre of a test-configuration, on which the vector fields (or rather, the ℂ*-actions which they generate) are holomorphic, so we can compute the inner product algebro-geometrically and thereby give a purely algebro-geometric definition of the stability condition. 4. Example The aim of this section is to work out the stability criterion in a special case, and show how it relates to the existence of extremal metrics. Let Σ be a genus two curve, and ℳ a line bundle on it with degree one. The same computation can be carried out when the genus is greater than two and the line bundle has degree greater than one. Define X to be the ruled surface over Σ. Tønnesen-Friedman [12] has constructed a family of extremal metrics on X that does not exhaust the entire Kähler cone. We will show that X is K-unstable (relative to a maximal torus of automorphisms) for the remaining polarisations. Since there are no non-zero holomorphic vector fields on Σ, a holomorphic vector field on X must preserve the fibres. Thus, the holomorphic vector fields on X are given by sections of . Here, means endomorphisms with trace zero. The vector field given by the matrix generates a ℂ*-action β, and it is up to scalar multiple the only one that does (see Maruyama [8] for proofs). Therefore this must be a multiple of the extremal vector field, which is then given by . The destabilising test-configuration is an example of deformation to the normal cone of a subvariety, studied by Ross and Thomas [10], except that we need to take account of the extremal ℂ*-action as well. We consider the polarisation L = C + mS0, where C is the divisor given by a fibre, S0 is the zero section (that is, the image of in X) and m is a positive constant. We denote by S∞ the infinity section, which as a divisor is just S0 − C. Note that β fixes S∞ and acts on the normal bundle of S∞ with weight 1. We make no distinction between divisors and their associated line bundles, and we use the multiplicative and additive notations interchangeably, so for example for an integer k. The deformation to the normal cone of S∞ is given by the blowup   in the subvariety . If we denote the exceptional divisor by E, the line bundle is ample for , where ε is the Seshadri constant of as in [10]. In our case, . Thus we obtain a test-configuration with the ℂ* action induced by π from the product of the trivial action on X and the usual multiplication on ℂ. Denote the restriction of this ℂ*-action to the central fibre (X0, L0) by α. Since the extremal ℂ*-action fixes S∞, we obtain another action on the test-configuration, induced by π from the product of the extremal ℂ*-action on X and the trivial action on ℂ. Let us call the induced action on the central fibre β. We wish to calculate as defined in (1.1). For this we use the following decomposition, with t being the standard coordinate on ℂ:   According to [10] α acts with weight −1 on t; that is, it acts with weight −j on the summand of index j above. Also, β acts on   with weight l, plus perhaps a constant independent of l which we can neglect, since the matrices are normalized to have trace zero in the formula for the modified Futaki invariant. The dimension of this space is k + l − 1, by the Riemann–Roch theorem. Writing Ak and Bk for the infinitesimal generators of the actions α and β on , and dk for the dimension of this space, we can now compute   Using these, we can compute   We obtain   If for a rational c between 0 and m, then the variety is K-unstable (relative to a maximal torus of automorphisms). In [12, p. 23] the condition given for the existence of an extremal metric of the type studied, is that a certain polynomial   is positive for a < γ < ka in the notation of [12], where k is the parameter of the polarization (k = m + 1 in our notation) and a is a constant defined in terms of k. After a change of variables k = m + 1 and , we find that the two conditions are in fact the same, since a quadratic polynomial with rational coefficients cannot have an irrational double root. Acknowledgements I would like to thank my PhD supervisor, Simon Donaldson, for introducing me to this problem, and Richard Thomas for helpful conversations and for his many suggestions improving earlier versions of this paper. I would also like to thank the referee for his comments, and the EPSRC for financial support. References 1 Calabi E..  Yau S. T..  Extremal Kähler metrics,  Seminar on differential geometry ,  1982 Princeton Univ. Press 2 Donaldson S. K..  Atiyah D.,  Iagolnitzer M..  Remarks on gauge theory, complex geometry and four-manifold topology,  Fields medallists lectures ,  1997 Singapore World Scientific(pg.  384- 403) 3 Donaldson S. K..  Scalar curvature and stability of toric varieties,  J. Differential Geom. ,  2002, vol.  62 (pg.  289- 349) 4 Donaldson P. B.,  Kronheimer S. K.. ,  The geometry of four-manifolds Oxford Math. Monogr ,  1990 Oxford Clarendon Press 5 Futaki A..  An obstruction to the existence of Einstein–Kähler metrics,  Invent. Math. ,  1983, vol.  73 (pg.  437- 443) Google Scholar CrossRef Search ADS   6 Futaki T.,  Mabuchi A..  Bilinear forms and extremal Kähler vector fields associated with Kähler classes,  Math. Ann. ,  1995, vol.  301 (pg.  199- 210) Google Scholar CrossRef Search ADS   7 Mabuchi T..  Stability of extremal Kähler manifolds,  Osaka J. Math. ,  2004, vol.  41 (pg.  563- 582) 8 Maruyama M..  On automorphism groups of ruled surfaces,  J. Math. Kyoto University ,  1971, vol.  11 (pg.  89- 112) 9 Mumford D.,  Fogarty J.,  Kirwan F.. ,  Geometric invariant theory ,  1994 Springer 10 Ross J.,  Thomas R. P..  A study of the Hilbert–Mumford criterion for the stability of projective varieties,  2004  preprint, arXiv math.AG/0412519 11 Tian G..  Kähler–Einstein metrics with positive scalar curvature,  Invent. Math. ,  1997, vol.  137 (pg.  1- 37) Google Scholar CrossRef Search ADS   12 Tønnesen-Friedman C..  Extremal Kähler metrics on ruled surfaces,  1997 Odense University  Ph.D. thesis 2000 Mathematics Subject Classification 53C55 (primary), 53C25 (secondary). © 2006 London Mathematical Society

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Bulletin of the London Mathematical Society – Oxford University Press

Published: Dec 15, 2006

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