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The Kähler-Ricci flow and optimal degenerations. (English) Zbl 1447.53081
Summary: We prove that on Fano manifolds, the Kähler-Ricci flow produces a “most destabilising” degeneration, with respect to a new stability notion related to the \(H\)-functional. This answers questions of X. Chen et al. [Geom. Topol. 22, No. 6, 3145–3173 (2018; Zbl 1404.53058)] and W. He [Asian J. Math. 20, No. 4, 645–664 (2016; Zbl 1370.53047)].
We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman’s \(\mu\)-functional on Fano manifolds, resolving a conjecture of G. Tian et al. [Trans. Am. Math. Soc. 365, No. 12, 6669–6695 (2013; Zbl 1298.53037)] as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian et al. [loc. cit.], where either the manifold was assumed to admit a Kähler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.

53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
14J45 Fano varieties
32Q15 Kähler manifolds
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[1] R. H. Bamler. Convergence of Ricci flows with bounded scalar curvature.ArXiv e-prints, March 2016. · Zbl 1410.53063
[2] Shigetoshi Bando. TheK-energy map, almost Einstein K¨ahler metrics and an inequality of the Miyaoka-Yau type.Tohoku Math. J. (2), 39(2):231-235, 1987, MR0887939, Zbl 0678.53061. · Zbl 0678.53061
[3] R. J. Berman and D. Witt Nystrom. Complex optimal transport and the pluripotential theory of K¨ahler-Ricci solitons.ArXiv e-prints, January 2014.
[4] Robert J. Berman. K-polystability ofQ-Fano varieties admitting K¨ahlerEinstein metrics.Invent. Math., 203(3):973-1025, 2016, MR3461370, Zbl 1353.14051.
[5] B. Berndtsson. Probability measures related to geodesics in the space of K¨ahler metrics.ArXiv e-prints, July 2009.
[6] B. Berndtsson. A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in K¨ahler geometry.Invent. Math., 200(1):149-200, 2015, MR3323577, Zbl 1318.53077.
[7] S. Boucksom, T. Hisamoto, and M. Jonsson. Uniform K-stability, DuistermaatHeckman measures and singularities of pairs.Ann. Inst. Fourier, 67(2), 743-841, 2017, MR3669511, Zbl 06821960. · Zbl 1391.14090
[8] L. Bruasse and A. Teleman. Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry.Ann. Inst. Fourier (Grenoble), 55(3):1017- 1053, 2005, MR2149409, Zbl 1093.32009. · Zbl 1093.32009
[9] X. Chen, S. Sun, and B. Wang. K¨ahler-Ricci flow, K¨ahler-Einstein metric, and K-stability.ArXiv e-prints, August 2015.
[10] X. Chen and B. Wang. Space of Ricci flows (II).ArXiv e-prints, May 2014.
[11] Xiuxiong Chen, Simon Donaldson, and Song Sun. K¨ahler-Einstein metrics on Fano manifolds. I, II, III.J. Amer. Math. Soc., 28(1):183-197, 199-234, 235-278, 2015, MR3264766, MR3264767, MR3264768, Zbl 1312.53096, Zbl 1312.53097, Zbl 1311.53059. · Zbl 1312.53096
[12] T. C. Collins and G. Sz´ekelyhidi. K-semistability for irregular Sasakian manifolds.J. Differential Geom.109(1), 81-109, 2018, MR3798716, Zbl 06868031. · Zbl 1403.53039
[13] T. C. Collins and G. Sz´ekelyhidi. The twisted K¨ahler-Ricci flow.Journal f¨ur die Reine und Angewandte Mathematik. [Crelle’s Journal], 716:179-205, 2016, MR3518375, Zbl 1357.53076.
[14] Jean-Pierre Demailly. Mesures de Monge-Amp‘ere et caract´erisation g´eom´etrique des vari´et´es alg´ebriques affines.M´em. Soc. Math. France (N.S.), (19):124, 1985, MR0813252, Zbl 0579.32012.
[15] R. Dervan. Relative K-stability for K¨ahler manifolds.Math Annalen, to appear. · Zbl 1407.53079
[16] W. Ding and G. Tian. K¨ahler-Einstein metrics and the generalized Futaki invariant.Invent. Math., 110:315-335, 1992, MR1185586, Zbl 0779.53044.
[17] W. Ding and G. Tian. The generalized Moser-Trudinger inequality.Nonlinear Analysis and Microlocal Analysis: Proceedings of the International Conference at Nankai Institute of Mathematics. World Scientific, 57-70, 1992. · Zbl 1049.53507
[18] S. Donaldson. The Ding functional, Berndtsson convexity and moment maps. ArXiv e-prints, March 2015. · Zbl 1432.53093
[19] S. K. Donaldson. Scalar curvature and stability of toric varieties.J. Differential Geom., 62(2):289-349, 2002, MR1988506, Zbl 1074.53059.
[20] S. K. Donaldson. Lower bounds on the Calabi functional.J. Differential Geom., 70(3):453-472, 2005, MR2192937, Zbl 1149.53042. · Zbl 1149.53042
[21] W. He. K¨ahler-Ricci soliton and H-functional.Asian J. of Math, 20(4), 2016, 645-663, MR3570456, Zbl 1370.53047.
[22] Weiyong He. On the convergence of the Calabi flow.Proc. Amer. Math. Soc., 143(3):1273-1281, 2015, MR3293741, Zbl 1314.53129. · Zbl 1314.53129
[23] Tomoyuki Hisamoto. On the limit of spectral measures associated to a test configuration of a polarized K¨ahler manifold.J. Reine Angew. Math., 713:129- 148, 2016, MR3483627, Zbl 1343.32017. · Zbl 1343.32017
[24] George R. Kempf. Instability in invariant theory.Ann. of Math. (2), 108(2):299- 316, 1978, MR0506989, Zbl 0406.14031. · Zbl 0406.14031
[25] B. Kleiner and J. Lott. Notes on Perelman’s papers.Geom. Topol.12(5):2587- 2855, 2008, MR2460872, Zbl 1204.53033. · Zbl 1204.53033
[26] N. Pali. Characterization of Einstein-Fano manifolds via the K¨ahler-Ricci flow. Indiana Univ. Math. J., 57(7):3241-3274, 2008, MR2492232, Zbl 1165.53050. · Zbl 1165.53050
[27] G. Perelman. The entropy formula for the Ricci flow and its geometric applications.ArXiv Mathematics e-prints, November 2002.
[28] D. H. Phong and J. Song and J. Sturm and X. Wang. The Ricci flow on the sphere with marked points.arXiv:1407.1118 · Zbl 1431.53104
[29] D. H. Phong and J. Song and J. Sturm and X. Wang. Convergence of the conical Ricci flow onS2to a soliton.arXiv:1503.04488
[30] D. H. Phong and J. Song and J. Sturm and B. Weinkove. The K¨ahler-Ricci flow and the ¯∂operator on vector fields.J. Differential Geometry, 81:631-647, 2009, MR2487603, Zbl 1162.32014. · Zbl 1162.32014
[31] D. H. Phong and J. Sturm. Regularity of geodesic rays and Monge-Amp‘ere equations.Proc. Amer. Math. Soc., 138(10):3637-3650, 2010, MR2661562, Zbl 1205.31004. · Zbl 1205.31004
[32] J. Ross and R. P. Thomas. A study of the Hilbert-Mumford criterion for the stability of projective varieties.J. Algebraic Geom., 16(2):201-255, 2007, MR2274514, Zbl 1200.14095. · Zbl 1200.14095
[33] O. S. Rothaus. Logarithmic Sobolev inequalities and the spectrum of Schr¨odinger operators.J. Funct. Anal., 42:110-120, 1981, MR0620582, Zbl 0471.58025.
[34] N. Sesum and G. Tian. Bounding scalar curvature and diameter along the K¨ahler-Ricci flow (after Perelman).J. Inst. Math. Jussieu, 7(3):575-587, 2008, MR2427424, Zbl 1147.53056. · Zbl 1147.53056
[35] Gang Tian. K¨ahler-Einstein metrics with positive scalar curvature.Invent. Math., 130(1):1-37, 1997, MR1471884, Zbl 0892.53027.
[36] Gang Tian, Shijin Zhang, Zhenlei Zhang, and Xiaohua Zhu. Perelman’s entropy and K¨ahler-Ricci flow on a Fano manifold.Trans. Amer. Math. Soc., 365(12):6669-6695, 2013, MR3105766, Zbl 1298.53037. · Zbl 1298.53037
[37] Gang Tian and Xiaohua Zhu. Convergence of K¨ahler-Ricci flow.J. Amer. Math. Soc., 20(3):675-699, 2007, MR2291916, Zbl 1185.53078. · Zbl 1185.53078
[38] Gang Tian and Xiaohua Zhu. Convergence of the K¨ahler-Ricci flow on Fano manifolds.J. Reine Angew. Math., 678:223-245, 2013, MR3056108, Zbl 1276.14061.
[39] David Witt Nystr¨om. Test configurations and Okounkov bodies.Compos. Math., 148(6):1736-1756, 2012, MR2999302, Zbl 1276.14017.
[40] S.-T. Yau. Open problems in geometry.Proc. Symposia Pure Math., 54:1-28, 1993, MR1216573, Zbl 0801.53001.
[41] R. Ye. The logarithmic Sobolev inequality along the Ricci flow.Commun. Math. Stat., 3(1), 1-36, 2015, MR3333694, Zbl 1319.53078.
[42] Q. S. Zhang. A uniform Sobolev inequality under Ricci flow.IMRN, 1-12, 2007, MR2354801, Zbl 1141.53064.
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