Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy

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1 Li, C. (20) Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy, International Mathematics Research Notices, Vol. 20, No. 9, pp Advance Access publication September, 200 doi:0.093/imrn/rnq52 Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy Chi Li Department of Mathematics, Princeton University, Princeton, NJ , USA Correspondence to be sent to: Based on Donaldson s method, we prove that, for an integral Kähler class, when there is akähler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete. Introduction Let be a compact Kähler manifold of dimension nand fix a Kähler class [ω] on. Define the Kähler potential space K(ω) := {φ C (, R) ; ω + 2π φ > 0} () We will write K for K(ω) if the reference metric ω is clear. K := K/R is the space of Kähler metrics in [ω]. The K-energy functional was defined by Mabuchi [0]. Definition.. For any φ K, letω φ = ω + ( /2π) φ K, define ν ω (ω φ ) = V 0 dt (S(ω φt ) S) dφ t dt ωn φ t Received March 0, 200; Revised June 24, 200; Accepted July 7, 200 Communicated by Prof. Simon Donaldson c The Author 200. Published by Oxford University Press. All rights reserved. For permissions, please journals.permissions@oup.com.

2 262 C. Li φ t is any path connecting 0 and φ in K. S(ω φ ) denotes the scalar curvature of Kähler metric ω φ,and S = V S(ω)ω n = nc () [ω] n [ω] n, V = ω n is the average of scalar curvature, which is independent of chosen Kähler metric in [ω]. The K-energy is well defined, that is, it does not depend on the path connecting 0 and φ. In particular, we can take φ(t) = tφ. AKähler metric of constant scalar curvature is a critical point of K-energy and it is a local minimizer. Now assume the Kähler class is c (L) H 2 (, Z) H, (, R) for some ample line bundle L over. In this note, we prove the following theorem: Theorem.2. Suppose that there is a metric ω of constant scalar curvature in the Kähler class c (L). Thenω minimizes the K-energy in this Kähler class. In the case of Kähler Einstein metrics, this result was proved by Bando and Mabuchi in [] and [2]. In [7], Donaldson proved the above theorem under the assumption that the automorphism group Aut(, L) is discrete. Here Aut(, L) denotes the group of automorphisms of the pair of (, L) modulo the trivial automorphism C (acting by constant scalar multiplication on the fibres). The theorem was proved for more general extremal Kähler metrics on any compact Kähler manifolds by Chen Tian in [5], where the authors used geodesics in infinite dimensional space of Kähler metrics in [ω]. It can also be proved by recent result of Chen Sun [4], where they use geodesic approximation to prove weak convexity of K-energy. In this note, we use Donaldson s method to prove the theorem for every integral class case. The strategy to prove the theorem is finite dimensional approximation. We sketch the idea here. Tian s approximation theorem (Proposition 2.2 and Corollary 2.3) says that K can be approximated by a sequence of finite dimensional symmetric spaces H k. Here H k = GL(Nk, C)/U( ) is the space of Hermitian metrics on the complex vector space H 0 (, L k ).

3 Kähler Metric Obtains the Minimum of K-energy 263 In [7], Donaldson defined a sequence of functional L k on K, which approximate K-energy as k. When restricted to H k, L k is bounded below by the logarithmic of Chow norm. It was known that balanced metric obtains the minimum of Chow norm [6, 23]. In [6], Donaldson already proved, in the case of discrete automorphism group, the existence of balanced metrics, which approximate Kähler metric of constant scalar curvature. Putting these together, he can prove the theorem. Mabuchi [2 4] extended many results of [6] to the case where the varieties have infinitesimal automorphisms. As Mabuchi [] showed, if the automorphism group is not discrete, in general there will be no balanced metrics. Instead, Mabuchi defined T- balanced metrics and T-stability with respect to some torus group contained in Aut(). Donaldson claimed [7] one can use these new techniques to prove the above theorem without assuming that the automorphism group is discrete. In this note, we use the same quantization strategy. But we don t need existence of balanced metrics nor T-balanced metrics. Instead we use simpler Bergman metrics constructed directly from ω. Using asymptotic expansion of Bergman kernel, we show that Bergman metrics are almost balanced in an asymptotical sense (Proposition 2.4), and they can help us to prove the theorem. In this way, we don t need the restriction on the automorphism group; moreover, our argument is more direct. As can be seen from the following, the argument follows [7] closely. The idea of using almost balanced metrics is inspired by works of Mabuchi. In particular, the proof of Lemma 3.3 is inspired by the argument of [5], page 3. See Remark 3.. As the above work shows, the convexity of various functionals is the essential property behind the argument. 2 Notations and Preliminaries 2. Maps between K k and H k We will use some definitions and notations from [7]. The set K defined in () depends on reference Kähler metric ω. However in the following, we will omit writing down this dependence, because it s clear that K is also the set of metrics h on L whose curvature form c (L, h) := 2π log h

4 264 C. Li is a positive (,) form on. LetK k denote the set of Hermitian metrics on L k with positive curvature form, then K k K = K.Let = dim H 0 (, L k ), V = c (L) n.we have maps between K k and H k. Definition 2.. Hilb : K k H k h k s 2 Hilb(h k ) = N k Vk n FS : H k K k H k s 2 FS(H k ) = s 2 Nk = s, s 2 Lk. s 2 h k c (L k, h k ) n, s H 0 (, L k ) In the above definition, {s ; } is an orthonormal basis of the Hermitian complex vector space (H 0 (, L k ), H k ). 2.2 Bergman metrics, expansions of Bergman kernels For any fixed Kähler metric ω c (L), take a Hermitian metric h on L such that c (L, h) = ω, thekth Bergman metric of h is h k = FS(Hilb(h k )) K k. Let {s, } be an orthonormal basis of Hilb(h k ). Define the kth (suitably normalized) Bergman kernel of ω ρ k (ω) = n! Vk n = s 2 h k. Note that h is determined by ω up to a constant, but ρ k (ω) doesn t depend on the chosen h. The following proposition is now well known. Proposition 2.2 ([3, 9, 9, 20, 22]). (i) For fixed ω, there is an asymptotic expansion as k + ρ k (ω) = A 0 (ω) + A (ω)k +... where A i (ω) are smooth functions on defined locally by ω.

5 Kähler Metric Obtains the Minimum of K-energy 265 (ii) In particular A 0 (ω) =, A (ω) = 2 S(ω). (iii) The expansion holds in C in that for any r, N 0 N ρ k(ω) A i (ω)k i C K r,n,ω k N i=0 r () for some constants K r,n,ω. Moreover the expansion is uniform in that for any r, N, there is an integer s such that if ω runs over a set of metrics, which are bounded in C s,andwithω bounded below, the constants K r,n,ω are bounded by some K r,n independent of ω. Remark 2.. We choose a particular normalization of Bergman kernel, so that the expansion starts with order 0, other than order n as it appeared in [6, Proposition 6]. The following approximation result is a corollary of Proposition 2.2.(i) (ii). Corollary 2.3 ([20]). Using the notation at the beginning of this subsection, we have, as k +, (h k ) /k h, and(/k)c (L k, h k ) ω, the convergence in C sense. More precisely, for any r > 0, there exists a constant C r,ω such that log h k k h C r C r,ω k 2, k c (L k, h k ) ω C r C r,ω k 2. (2) Proof. It s easy to see that (h k ) k = h ( s 2 h k ) k =: he φ k.

6 266 C. Li Note that by the expansion in Proposition 2.2.(i) (ii), we have s 2 h k = (n!/vk n ) = s 2 h k n!/vk n = + 2 S(ω)k + O(k 2 ) + 2 Sk + O(k 2 ) = + O(k ). So φ k = k log ( s 2 h k ) = O(k 2 ). The error term is in C sense. So the first inequality in (2) holds. The second inequality in (2) follows because k c (L k, h k ) ω = 2π φ k. Now assume we have a Kähler metric of constant scalar curvature ω in the Kähler class c (L). Take a h K such that ω = c (L, h ). We will make extensive use of the kth Bergman metric of h and its associated objects, so for the rest of this note, we denote H k = Hilb(h k ), h k = FS(H k ) = FS(Hilb(h k )), ω k = c (L k, h k ) = 2π log τ 2 = ( ) Hereafter, we fix an orthonormal basis {τ, } of Hk = Hilb(h k ). The next proposition says we can improve the convergence rate in corollary 2.3 for h. This will be important for us. (Compare [5], (3.8) (3.0)). Proposition 2.4. For any r > 0, there exists some constant C r,ω such that τ 2 h k = C r C r,ω k 2 (3)

7 Kähler Metric Obtains the Minimum of K-energy 267 So in particular, 2π log τ 2 = k ω = O(k 2 ) (4) Proof. Since S(ω ) S, by proposition 2.2, we have τ = ( n!/vk n ) Nk 2 h = = O(k 2 ) τ 2 h k = n!/vk n = + 2 S(ω )k + O(k 2 ) + 2 Sk + O(k 2 ) (4) follows because the left hand side of it is equal to 2.3 Aubin Yau functional and Chow norm ( 2π Nk log = We define the Aubin Yau functional with respect to (L k, h k ) by I k (h k e φ ) = 0 dφ(t) dt c (L k, h k dt e φ(t) ) n. Here φ K k,thatis,(/k)φ K. φ(t) is a path connecting 0 and φ in K k. Under the orthonormal basis {τ, } of H k, H 0 (, L k ) = C and P(H 0 (, L k ) ) = CP. τ 2 h k For any H k H k, take an orthonormal basis {s, } of H k.letdeth k denote the determinant of matrix (H k ) β = (H k (s, s β )). {s } determines a projective embedding into CP Nk. (Note that the fixed isomorphism P(H 0 (, L k ) ) = CP Nk is determined by the basis {τ }.) The image of this embedding is denoted by k (H k ) CP and has degree d k = Vk n. k (H k ) has a Chow point [6, 23] ) ˆ k (H k ) W k := H 0 (Gr( n 2, P ), O(d k )) such that the corresponding divisor Zero( ˆ k (H k )) ={L Gr( n 2, P ); L k (H k ) = }.

8 268 C. Li Proposition 2.5 ([6, 23]). have W k has a Chow norm CH(H k ), such that for all H k H k we log det H k Vk n I k(fs(h k )) = Vk n log ˆ k (H k ) 2 CH(Hk ) SL(, C) acts on H k and W k.notethat k (σ Hk ) = σ k(hk ). Define ) f k (σ ) = log ( ˆ k (σ Hk ) 2 CH(H k ) σ SL(, C) It s easy to see that f k (σ σ ) = f k (σ ) for any σ SU ( ),so f k is a function on the symmetric space SL(, C)/SU ( ).Wehave Proposition 2.6 ([7, 8, 8, 23]). f k (σ ) is convex on SL(, C)/SU ( ). To relate K k and H k, following Donaldson [7], we change FS(H k ) in the above formula into general h k K k and define: Definition 2.7. For all h k K k and H k H k, Remark 2.2. P k (h k, H k ) = log det H k Vk n I k(h k ). Note that, for any c R, I k (e c h k ) = cvk n + I k (h k ),so P k (e c h k, e c H k ) = P k (h k, H k ). (5) This definition differs from Donaldson s definition by omitting two extra terms, since we find no use for these terms in the following argument. 3 Proof of Theorem.2 Lemma 3.. For any h k, h k K k,withh k = h ke φ,wehave φ c (L k, h k ) n I k (h k ) I k(h k ) φ c (L k, h k )n. This is [7, Lemma ].

9 Proof. Kähler Metric Obtains the Minimum of K-energy 269 This lemma just says I k is a convex function on K k, regarded as an open subset of C (). We only need to calculate its second derivative along the path h k (t) = h k e tφ : d 2 dt 2 I k(h k ) = φ t φc (L k, h k (t)) n = t φ 2 c (L k, h k (t)) n 0 t and t are the Laplace and gradient operators of Kähler metric c (L k, h k (t)). From now on, fix a ω c (L), take a Hermitian metric h K such that ω = c (L, h). We have the k-th Bergman metric h k = FS(Hilb(h k )) and corresponding Kähler metric ω k = c (L k, h k ). By Corollary 2.3 Lemma 3.2. (h k ) k h, k ω k ω, in C. P k (h k, Hilb(h k )) P k (FS(Hilb(h k )), Hilb(h k )). This is a corollary of [7, Lemma 4]. Since the definition of P is a little different from that in [7], we give a direct proof here. Proof. Let log( Let h k = FS(Hilb(h k)). Then P k (h k, Hilb(h k )) P k (FS(Hilb(h k )), Hilb(h k )) = Vk n(i (h k ) I (h k)). {s, } be an orthonormal basis of Hilb(h k ). Then log(h k /h k) = = s 2 h k ). By Lemma 3. and concavity of the function log, Vk n(i (h k ) I (h k)) Vk n ( log log Vk n ( = log ( ) s 2 h k c (L k, h k ) n s 2 Hilb(h k ) s 2 h k c (L k, h k ) n ) ) = 0.

10 270 C. Li Lemma 3.3. There exists a constant C > 0, depending only on h and h, such that P k (FS(Hilb(h k )), Hilb(h k )) P k (FS(H k ), H k ) Ck. Proof. Recall that Hk = Hilb(h k ) and {τ ; } is an orthonormal basis of Hk (see ( )). Let H k = Hilb(h k ) and {s ; } be an orthonormal basis of H k. Transforming by a matrix in SU ( ), we can assume s = e λ τ Evaluating the norm Hilb(h k ) on both sides, we see that e 2λ = N k Vk n τ 2 h k ωk n. (6) Since by Corollary 2.3 we have the following uniform convergence in C : (h k ) k h, k ω k ω. There exists a constant C > 0, C 2 > 0, depending only on h and h, such that C k h k C h k k, C 2 ω k ω k C 2 ω, so we see from (6) that λ Ck. Let λ = (/ ), H k = e2λ H k, ˆλ = λ λ. Then {ŝ β= λ β orthonormal basis of H k.notethatˆλ satisfies the same estimate as λ : = eˆλ τ } is an ˆλ Ck. (7) (e ˆ ) β = eˆλ δ β is a diagonal matrix in SL(, C). By scaling invariance of P k (5) and Proposition 2.5, we have P k (FS(H k ), H k ) = P k (FS(H k ), H k ) = Vk n log ˆ k (H k ) 2 CH(H k ) (8) P k (FS(Hk ), H k ) = Vk n log ˆ k (Hk ) 2 CH(Hk ). (9) As in Section 2.3, let k (s) = e s ˆ k (H k ) f k (s) = log ˆ k (s) 2 CH(H k ).

11 Kähler Metric Obtains the Minimum of K-energy 27 Then k (0) = k (H k ) and k() = k (H k ) = k(h k ). By Proposition 2.6, f k (s) is a convex function of s, so f k () f k (0) f k (0). We can estimate f k (0) by the estimates in Proposition 2.4: f k (0) = = = ˆλ τ 2 h k τ 2 h k ˆλ τ 2 h k + O(k 2 ) O(k 2 )( where the last equality is because of By the estimate for ˆλ = ˆλ (7), we get = τ 2 h k So f k () f k (0) f k (0) Ckn,and ˆλ 2π log τ 2 = ( + O(k 2 ))(kω ) n τ 2 h k (kω ) n = Vkn )(kω ) n = f k (0) Ck 2 k Ck n. ˆλ = 0 Vk n(log ˆ k (H k ) 2 CH Vk n log ˆ k (H k ) 2 CH ) = Vk n( f k() f k (0)) C Vk nkn Ck. n So the lemma follows from identities (8) and (9). Remark 3.. The proof of this lemma is similar to the argument in the beginning part of [5, Section 5] where Mabuchi proved K-semistability of varieties with constant scalar curvature metrics. Roughly speaking, here we consider geodesic segment connecting H k and H k in H k, while Mabuchi [5, Section 5] considered geodesic ray in H k defined by a test configuration. The estimates in Proposition 2.4 show that, to prove the K-semistability as in Mabuchi s argument [5, Section 5], we only need Bergman metrics of h instead of Mabuchi s T-balanced metrics.

12 272 C. Li Remark 3.2. The referee pointed out that the similar argument also appears in the proof of Theorem 2 in [7]. Remark 3.3. In [7, Corollary 2], Hk is taken to be balance metric, that is, H k is a fixed point of the mapping Hilb(FS( )). Then the difference in Lemma 3.3 is nonnegative, instead of bounded below by error term Ck. Lemma 3.4. There exists a constant C > 0, which only depends on h, such that P k (FS(H k ), H k ) P k (h k, Hilb(h k )) Ck 2 Proof. Recall from ( ) that: Hilb(h k ) = H k, h k = FS(H k ) = FS(Hilb(h k )), so it s easy to see that For any section s of L k, s 2 h k = By proposition 2.4. P k (FS(Hk ), H k ) P k (h k, Hilb(h k )) = Vk n(i k(h k ) I k(h k )) log h k h k s 2 h k τ 2 h k h k h k.so = = τ 2 h k = log( + O(k 2 )) =O(k 2 ). So by Lemma 3., we get Vk n(i k(h k ) I k(h k )) Ck 2 Definition 3.5. For any Kähler metric ω φ = c (L, h) + ( /2π) φ [ω], let h k (φ) = h k e kφ. Define L k (ω φ ) = P k (h k (φ), Hilb(h k (φ)))

13 Kähler Metric Obtains the Minimum of K-energy 273 Lemma 3.6 ([7]). There exist constants μ k, such that L k (ω φ ) + μ k = 2 ν ω(ω φ ) + O(k ). Here O(k ) depends on ω and ω φ. Proof. Let ψ(t) = tφ K connecting 0 and φ, h k (t) = h k e tkφ, ω t = ω + t( /2π) φ, t be the Laplace operator of metric ω t. Plugging in expansions for Bergman kernels ρ k in Proposition 2.2, we get d dt P k (h k (t), Hilb(h k (t))) = n! n! Vk n k n s 2 h k (t) ( kφ + tφ)k n ωt n + Vk n = V k n + 2 S k (ω t ) + t ρ k (ω t ))φω kn t + ( kρ n + k V = (S(ω t ) S)φωt n 2V + O(k ) kφk n ω n t φωt n {ω t, 0 t } have uniformly bounded geometry, so by Proposition 2.2.(3), the expansions above are uniform. So the lemma follows after integrating the above equation. Proof of Theorem.2. Let ω φ = c (L, h) + 2π φ [ω], h k = h k (φ) = h k e kφ By Lemma 3.2, Lemma 3.3, Lemma 3.4 P k (h k, Hilb(h k )) P k (FS(Hilb(h k )), Hilb(h k )) P k (FS(H k ), H k ) + O(k ) = P k (h k, Hilb(h k )) + O(k ) So by Lemma 3.6 ν ω (ω φ ) = 2L k (ω φ ) + 2μ k + O(k ) = 2 P k (h k, Hilb(h k )) + 2μ k + O(k ) 2 P k (h k, Hilb(h k )) + 2μ k + O(k ) = 2L k (ω ) + 2μ k + O(k ) = ν ω (ω ) + O(k ) The Theorem follows by letting k +.

14 274 C. Li Acknowledgements The author would like to thank: Professor Gang Tian for his constant encouragement and help; Yanir Rubinstein for pointing out some inaccuracies in the first version of the paper, and for his friendship and encouragement. The author would also like to thank all the participants, particularly Yalong Shi, in the geometry seminar in Peking University. Their patience and encouragement all help the author to understand the work of Donaldson. The author also thanks the referees for careful reading of the paper. References [] Bando, S. The K-energy map, almost Einstein Kähler metrics and an inequality of the Miyaoka Yau type. Tohuku Mathematical Journal 39 (987): [2] Bando, S., and T. Mabuchi. Uniqueness of Einstein Kähler Metrics Modulo Connected Group Actions. In Algebraic Geometry, Sendai 985, 40. Advanced Studies in Pure Mathematics 0. Tokyo: Kinokuniya; Amsterdam: North-Holland, 987. [3] Catlin, D. The Bergman Kernel and a Theorem of Tian. In Analysis and Geometry in Several Complex Variables, Katata, 997, 23. Boston: Birkhauser, 999. [4] Chen,., and S. Sun. Space of Kähler metrics (V)-Kähler quantization. preprint arxiv: [5] Chen,., and G. Tian. Geometry of Kähler metrics and holomorphic foliation by discs. Publications Mathématiques de l IHÉS 07 (2008): 07. [6] Donaldson, S. K. Scalar curvature and projective embeddings I. Journal of Differential Geometry 59 (200): [7] Donaldson, S. K. Scalar curvature and projective embeddings, II. Quarterly Journal of Mathematics 56 (2005): [8] Kempf, G., and L. Ness. The Length of Vectors in Representation Spaces. In Algebraic Geometry, Lecture Notes in Mathematics 732. Springer, 979. [9] Lu, Z. On the lower order terms of the asymptotic expansion of Tian Yau Zelditch. American Journal of Mathematics 22, no. 2 (2000): [0] Mabuchi, T. K-energy maps integrating Futaki invariants. Tohuku Mathematical Journal 38 (986): [] Mabuchi, T. An obstruction to asymptotic semistability and approximate critical metrics. Osaka Journal of Mathematics 4 (2004): [2] Mabuchi, T. Stability of extremal Kähler manifolds. Osaka Journal of Mathematics 4 (2004): [3] Mabuchi, T. An energy-theoretic approach to the Hitchin Kobayashi correspondence for manifolds I. Inventiones Mathematicae 59 (2005): [4] Mabuchi, T. An energy-theoretic approach to the Hitchin Kobayashi correspondence for manifolds II. Osaka Journal of Mathematics 46 (2009): [5] Mabuchi, T. K-stability of constant scalar curvature polarization. preprint arxiv:

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