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Chen, Xiuxiong; Donaldson, Simon; Sun, Song

Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \(2\pi\) and completion of the main proof. (English) Zbl 1311.53059

J. Am. Math. Soc. 28, No. 1, 235-278 (2015).
This is the third (and last) paper in a landmark series of papers that prove the result “K-stability implies the existence of a Kähler-Einstein metric on Fano manifolds”. The other direction, i.e., “the existence of such a metric implies the algebro-geometric condition of K-stability” was proven by Tian. The strategy of the proof is to solve a Monge-Ampere type PDE using a continuity-type method by constructing metrics which are “bent” along a divisor and then letting the “bending” angle tending to \(2\pi\).
In this paper, the authors study what happens to a sequence of conical Kähler-Einstein manifolds when the cone angle approaches \(2\pi\). Using Gromov-Hausdorff convergence they conclude that such a sequence (up to a subsequence) converges to a limit. Applying the Cheeger-Colding-Tian theory to such a limit they deduce that it is a union of regular points and singular points of “small” Hausdorff dimension. Then they prove that (Theorem 3) the regular set is open and indeed the limiting metric is a smooth Kähler-Einstein metric on it. Two proofs are given for this result. Using this result they proceed to prove their Theorem 2, which (roughly speaking) states that the limiting metric space is actually a Q-Fano variety with a “weak” Kähler-Einstein metric and that this convergence takes place inside some large projective space. Given Theorems 2 and 3, they proceed to complete the proof of the main result (theorem 1) of the paper. Along the way, they establish an Evans-Krylov type estimate.
Reviewer: Vamsi Pritham Pingali (Baltimore)

Cited in 15 Reviews
Cited in 214 Documents

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J45 Fano varieties
32Q20 Kähler-Einstein manifolds

Keywords:

Kähler-Einstein metric; Fano manifolds; cone angle; K-stability
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References:

[1] Anderson, Michael T., Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102, 2, 429-445 (1990) · Zbl 0711.53038 · doi:10.1007/BF01233434
[2] Aubin, Thierry, \'Equations du type Monge-Amp\`“ere sur les vari\'”et\'es k\`“ahleriennes compactes, C. R. Acad. Sci. Paris S\'”er. A-B, 283, 3, Aiii, A119-A121 (1976) · Zbl 0333.53040
[3] Bedford, Eric; Taylor, B. A., A new capacity for plurisubharmonic functions, Acta Math., 149, 1-2, 1-40 (1982) · Zbl 0499.01024 · doi:10.1007/BF02392348
[4] [Ber2] R. Berman, K-polystability of Q-Fano varieties admitting K\"ahler-Einstein metrics. arXiv: 1205.6214.
[5] [BBEGZ] R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, K\"ahler-Ricci flow and Ricci iteration on log-Fano varieties. arXiv:1111.7158. · Zbl 1430.14083
[6] [Bern] B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. arXiv: 1103.0923 · Zbl 1318.53077
[7] Cheeger, Jeff; Colding, Tobias H., On the structure of spaces with Ricci curvature bounded below. II, J. Diff. Geom., 54, 1, 13-35 (2000) · Zbl 1027.53042
[8] Calabi, Eugenio, On K\"ahler manifolds with vanishing canonical class. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, 78-89 (1957), Princeton University Press: Princeton, NJ:Princeton University Press · Zbl 0080.15002
[9] Calabi, Eugenio, Extremal K\"ahler metrics. II. Differential geometry and complex analysis, 95-114 (1985), Springer: Berlin:Springer · Zbl 0574.58006
[10] Chen, Xiuxiong, The space of K\"ahler metrics, J. Diff. Geom., 56, 2, 189-234 (2000) · Zbl 1041.58003
[11] [CDS0] X-X. Chen, S. Donaldson, and S. Sun, K\"ahler-Einstein metrics and stability. arXiv: 1210.7494. To appear in Int. Math. Res. Not (2013). · Zbl 1331.32011
[12] [CDS1] X-X. Chen, S. Donaldson, and S. Sun, K\"ahler-Einstein metric on Fano manifolds, I: approximation of metrics with cone singularities. arXiv:1211.4566. · Zbl 1312.53096
[13] [CDS2] X-X. Chen, S. Donaldson, and S. Sun, K\"ahler-Einstein metric on Fano manifolds, II: limits with cone angle less than \(2\pi \). arXiv:1212.4714.
[14] Chen, Xiuxiong; Wang, Bing, Space of Ricci flows I, Comm. Pure Appl. Math., 65, 10, 1399-1457 (2012) · Zbl 1252.53076 · doi:10.1002/cpa.21414
[15] Demailly, Jean-Pierre; Koll{\'a}r, J{\'a}nos, Semi-continuity of complex singularity exponents and K\`“ahler-Einstein metrics on Fano orbifolds, Ann. Sci. \'”Ecole Norm. Sup. (4), 34, 4, 525-556 (2001) · Zbl 0994.32021 · doi:10.1016/S0012-9593(01)01069-2
[16] Ding, Wei Yue, Remarks on the existence problem of positive K\"ahler-Einstein metrics, Math. Ann., 282, 3, 463-471 (1988) · Zbl 0661.53045 · doi:10.1007/BF01460045
[17] Ding, Wei Yue; Tian, Gang, K\"ahler-Einstein metrics and the generalized Futaki invariant, Invent. Math., 110, 2, 315-335 (1992) · Zbl 0779.53044 · doi:10.1007/BF01231335
[18] Donaldson, S. K., Scalar curvature and stability of toric varieties, J. Diff. Geom., 62, 2, 289-349 (2002) · Zbl 1074.53059
[19] Donaldson, S. K., Stability, birational transformations and the Kahler-Einstein problem. Surveys in differential geometry. Vol. XVII, Surv. Diff. Geom. 17, 203-228 (2012), International Press, Boston, MA · Zbl 1382.32018
[20] Donaldson, S. K., K\"ahler metrics with cone singularities along a divisor. Essays in mathematics and its applications, 49-79 (2012), Springer: Heidelberg:Springer · Zbl 1326.32039 · doi:10.1007/978-3-642-28821-0\_4
[21] [kn:DS] S. Donaldson and S. Sun, Gromov-Hausdorff limits of K\"ahler manifolds and algebraic geometry. arXiv:1206.2609. · Zbl 1388.53074
[22] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed, Singular K\"ahler-Einstein metrics, J. Amer. Math. Soc., 22, 3, 607-639 (2009) · Zbl 1215.32017 · doi:10.1090/S0894-0347-09-00629-8
[23] Futaki, A., An obstruction to the existence of Einstein K\"ahler metrics, Invent. Math., 73, 3, 437-443 (1983) · Zbl 0506.53030 · doi:10.1007/BF01388438
[24] Gilbarg, David; Trudinger, Neil S., Elliptic partial differential equations of second order, Classics in Mathematics, xiv \(+517\) pp. (2001), Springer-Verlag: Berlin:Springer-Verlag · Zbl 1042.35002
[25] Hamilton, Richard S., The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol.II, Cambridge, MA, 1993, 7-136 (1995), International Press, Cambridge, MA · Zbl 0867.53030
[26] H{\"o}rmander, Lars, An introduction to complex analysis in several variables, North-Holland Mathematical Library 7, xii \(+254\) pp. (1990), North-Holland Publishing Co.: Amsterdam:North-Holland Publishing Co. · Zbl 0685.32001
[27] [JMR] T. Jeffres, R. Mazzeo, and Y. Rubinstein, K\"ahler-Einstein metrics with edge singularities. arXiv:1105.5216 · Zbl 1337.32037
[28] Lazarsfeld, Robert, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 49, xviii \(+385\) pp. (2004), Springer-Verlag: Berlin:Springer-Verlag · Zbl 1093.14500 · doi:10.1007/978-3-642-18808-4
[29] [LS] C. Li and S. Sun, Conical K\"ahler-Einstein metric revisited. arXiv:1207.5011
[30] [LL] Long Li, A note on general Bando-Mabuchi uniqueness theorem. preprint.
[31] Li, Peter, Lecture notes on geometric analysis, Lecture Notes Series 6, iv \(+90\) pp. (1993), Seoul National University Research Institute of Mathematics Global Analysis Research Center: Seoul:Seoul National University Research Institute of Mathematics Global Analysis Research Center · Zbl 0822.58001
[32] Lu, Peng, A local curvature bound in Ricci flow, Geom. Topol., 14, 2, 1095-1110 (2010) · Zbl 1194.53056 · doi:10.2140/gt.2010.14.1095
[33] Matsushima, Yoz{\^o}, Sur la structure du groupe d’hom\'eomorphismes analytiques d’une certaine vari\'et\'e k\`“ahl\'”erienne, Nagoya Math. J., 11, 145-150 (1957) · Zbl 0091.34803
[34] [OS] Y. Odaka and S. Sun, Testing log K-stability by blowing up formalism. arXiv: 1112.1353 · Zbl 1326.14096
[35] [perelman02] G. Perelman, Ricci flow with surgery on three-manifolds. arXiv: 0303109. · Zbl 1130.53002
[36] Siu, Yum Tong, Lectures on Hermitian-Einstein metrics for stable bundles and K\`“ahler-Einstein metrics, DMV Seminar 8, 171 pp. (1987), Birkh\'”auser Verlag: Basel:Birkh\'”auser Verlag · Zbl 0631.53004
[37] [SW] J. Song and X. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.483
[38] Stoppa, Jacopo, K-stability of constant scalar curvature K\"ahler manifolds, Adv. Math., 221, 4, 1397-1408 (2009) · Zbl 1181.53060 · doi:10.1016/j.aim.2009.02.013
[39] Sun, Song, Note on K-stability of pairs, Math. Ann., 355, 1, 259-272 (2013) · Zbl 1271.53068 · doi:10.1007/s00208-012-0788-y
[40] Tian, G., On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., 101, 1, 101-172 (1990) · Zbl 0716.32019 · doi:10.1007/BF01231499
[41] Tian, Gang, K\"ahler-Einstein metrics with positive scalar curvature, Invent. Math., 130, 1, 1-37 (1997) · Zbl 0892.53027 · doi:10.1007/s002220050176
[42] [WT12] G. Tian and B. Wang, On the structure of almost Einstein manifolds. arXiv: 1202.2912.
[43] Trudinger, Neil S., Regularity of solutions of fully nonlinear elliptic equations, Boll. Un. Mat. Ital. A (6), 3, 3, 421-430 (1984) · Zbl 0559.35030
[44] Uhlenbeck, Karen K., Connections with \(L^p\) bounds on curvature, Commun. Math. Phys., 83, 1, 31-42 (1982) · Zbl 0499.58019
[45] [wang10] B. Wang, Ricci flow on orbifold. arXiv:1003.0151.
[46] Yau, Shing Tung, On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation. I, Commun. Pure Appl. Math., 31, 3, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[47] Yau, Shing-Tung, Open problems in geometry. Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Symp. Pure Math. 54, 1-28 (1993), Amer. Math. Soc.: Providence, RI:Amer. Math. Soc. · Zbl 0801.53001
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