An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I

By T. Mabuchi. 2004

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Recently, Donaldson proved asymptotic stability for a polarized algebraic manifold M with polarization class admitting a Kahler metric of constant scalar curvature, essentially when the linear algebraic part H of Aut(M) is semisimple. The purpose of this paper is to give a generalization of Donaldson’s result to the case where the polarization class admits an extremal Kahler metric, even when H is not semisimple. 0. Introduction For a connected polarized algebraic manifold (M , L) with an extremal Kahler metric in the polarization class c1(L)R, we consider the Kodaira embedding 8m = 8jL m j : M ,! P (Vm), m 1, where P (Vm) denotes the set of all hyperplanes in Vm := H 0(M , O(Lm)) through the origin. For the identity component Aut(M) of the group of holomorphic automorphisms of M , let H denote its maximal connected linear algebraic subgroup. Replacing the ample holomorphic line bundle L by some positive integral multiple of L if necessary, we may fix an H -linearization of L , i.e., a lift to L of the H -action on M such that H acts on L as bundle isomorphisms covering the H -action on M , and may further assume that the natural H -equivariant maps prm : m O V1 ! Vm , m = 1, 2, : : : , are surjective (cf. [18], Theorem 3). In this paper, applying a method in [15], we shall generalize a result in Donaldson [3] about stability to extremal Kahler cases: 2000 Mathematics Subject Classification. Primary 14L24, 32Q15; Secondary 32Q20, 53C25. 116 T. MABUCHI Main Theorem. For a polarized algebraic manifold (M , L) as above with an extremal Kahler metric in the polarization class, there exists an algebraic torus T in H such that the image 8m(M) in P (Vm) is stable relative to T (cf. Section 2 and [14]) for m 1. In particular in [16], by an argument as in [3], an extremal Kahler metric in a fixed integral Kahler class on a projective algebraic manifold M will be shown to be unique1 up to the action of the group H . Fix once for all an extremal Kahler metric !0 in the polarization class in Main Theorem. By a result of Calabi [1], the identity component K of the group of isometries of (M , !0) is a maximal compact connected subgroup of H . For the identity component Z of the center of K , we consider the complexification Z of Z in H . Then we shall see that Main Theorem is true for T = Z (cf. Section 1). One may ask why relative stability in place of ordinary stability has to be considered in our study. The reason why we choose relative stability is because, in general, the obstruction in [13] to asymptotic semistability does not vanish (cf. [17]). Thus, as to the group action on Vm related to stability, we must replace the full special linear group SL(Vm) of Vm by its subgroup Gm(T ) (see (1.3)), where the algebraic torus T in Z is chosen in such a way that the obstruction vanishes when restricted to Gm(T ), i.e., G 0m(T ) fixes Mm (cf. Section 1). Note also that Gm(T ) is a direct product of special linear groups. To see why we choose such a group Gm(T ) in place of SL(Vm), we may compare our stability with that of holomorphic vector bundles. Recall that a holomorphic vector bundle splitting into a direct sum of stable vector bundles often appears in the boundary of a compactified moduli space of stable vector bundles. Similarly for our stability of manifolds, a splitting phenomenon occurs for Vm in (1.2). Roughly speaking, we consider the moduli space of all M’s with fixed decomposition data (1.2), where same type of construction of moduli spaces occurs typically for the Hodge decomposition in the variation of Hodge structures. We now explain the difficulty which we encounter in applying the method of [15]. Such a difficulty comes up when we use the estimate of Phong and Sturm [21]. By applying a stability criterion in [15] of Hilbert-Mumford’s type, we write the vector space pm as an orthogonal direct sum pm = p 0 m p00 m (cf. Section 3), and then check the stability of Mm along the orbits of the one-parameter subgroups in Gm(T ) generated by elements of p00 m . Though p and p00 m are transversal by the equality p0m = pm \ p, we further need the orthogonality of p and p00 m in order to apply directly the estimate in [21]. Since such an orthogonality does not generally hold, we are in 1For uniqueness of extremal Kahler metrics, Chen and Tian recently obtain a more general result without any projectivity condition. HITCHIN-KOBAYASHI CORRESPONDENCE FOR MANIFOLDS, II 117 trouble, but still the situation is not so bad (see (3.17), (3.18)), and this overcomes the difficulty. 1. Reduction of Main Theorem In this section, by introducing necessary notation, we reduce the proof of Main Theorem to showing Theorems A and B below. Throughout this paper, we fix once for all a pair (M , L) of a connected projective algebraic manifold M and an ample holomorphic line bundle L over M as in the introduction. For Vm in the introduction, we put Nm := dimC Vm 1, where the positive integer m is such that Lm is very ample. Let n and d be respectively the dimension of M and the degree of the image Mm := 8m(M) in the projective space P (Vm). Fixing an H -linearization of L as in the introduction, we consider the associated representation: H ! PGL(Vm). Pulling it back by the finite unramified cover: SL(Vm) ! PGL(Vm), we obtain an isogeny (1.1) : H ! H , where H is an algebraic subgroup of SL(Vm). On the other hand, for an algebraic torus T in H , the H -linearization of L naturally induces a faithful representation H ! GL(Vm), and this gives a T -action on Vm for each m. Then we have a finite subset 0m = f 1, 2, : : : , m g of the free Abelian group Hom(T , C ) of all characters of T such that the vector space Vm = H 0(M , O(Lm)) is uniquely written as a direct sum

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Lower bounds on the Calabi functional (2005, background): “argument does not need any special constructions (as in [ 14 ]), in the case when the manifold has holomorphic vector fields.”
Blowing up and desingularizing constant scalar curvature Kähler manifolds (2004, background): “We know for example, by Mabuchi’s extension of Donaldson’s work [40], [17], that if an integral class is represented by an extremal Kähler metric, then the underlying algebraic manifold is asymptotically stable in a sense which depends on the structure of the automorphism group which preserves the c”
K-stability of constant scalar curvature polarization (2008, methodology): “of δ0 in [10], Step 5, the proof of Lemma 3.4 in [ 9 ] allows us to make a perturbation of hm via the action of exp(p ′′ ) (see [10] for the definition of p ′′ ) to obtain a critical point for the Chow norm.”
An obstruction to the existence of constant scalar curvature K (2004, background): “of (X,Lr) can be “balanced” for r ≫ 0 [Do2]. This implies it is asymptotically Chow stable [Zh, P, Wa], which in turn implies that (X,L) is K-semistable (see for example Theorem 3.9 of [RT]). Mabuchi [Mab] has extended this to manifolds with non-discrete automorphism group satisfying a mild conditio”
Lectures on Stability and Constant Scalar Curvature (2008, background): “In the particular case when L = K−1X , it is known that the boundedness of the Mabuchi K-energy from below on c1(L) implies that infω∈c1(L)‖R(ω)− µn‖L2 = 0 (see e.g. [7, 110, 104]).”
Blowing up Kähler manifolds with constant scalar curvature, II (2005, result): “Recent years have seen some spectacular works, inspired by the analogy with the Hitchin-Kobayashi correspondence for vector bundles, relating the existence of canonical metrics to different stability notions of manifolds [29], [8], [21], [18].”
Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, background): “Then the following is equivalent for a metric φ in E1(−KX) : • φ minimizes the modified Ding functional DV • φ minimizes the modified Mabuchi functional MV • φ is a weak Kähler-Ricci soliton for (X,V ).”
Calabi flow and projective embeddings (2008, background): “The restriction on Aut(X,L) in Donaldson’s Theorem 3 has been relaxed by Mabuchi [22, 23 ], whilst the picture in [13] has been related to the Deligne pairing by Phong–Sturm in [25, 26], an approach which also leads to a sharpening of some estimates.”
Stability of extremal Kahler manifolds (2004, methodology): “D G ] 1 0 A pr 2 00 4 STABILITY OF EXTREMAL KÄHLER MANIFOLDS TOSHIKI MABUCHI Dedicated to Professor Shoshichi Kobayashi on his seventieth birthday”
Scalar curvature and projective embeddings, II (2004, unknown): “D G ] 1 4 Ja n 20 05 Scalar curvature and projective embeddings, II S. K. Donaldson March 29, 2022”