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

Authors: Xiuxiong Chen, Simon Donaldson and Song Sun
Journal: J. Amer. Math. Soc. 28 (2015), 235-278
MSC (2010): Primary 53C55
Published electronically: March 28, 2014
MathSciNet review: 3264768
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Abstract: This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2$ \pi $. We also put all our technical results together to complete the proof of the main theorem.

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  • [1] Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 1074481, 10.1007/BF01233434
  • [2] Thierry Aubin, Équations du type Monge-Ampère sur les variétés kähleriennes compactes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, Aiii, A119–A121. MR 0433520
  • [3] Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, 10.1007/BF02392348
  • [4] R. Berman, K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. arXiv: 1205.6214.
  • [5] R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, Kähler-Ricci flow and Ricci iteration on log-Fano varieties. arXiv:1111.7158.
  • [6] B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. arXiv: 1103.0923
  • [7] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom. 54 (2000), no. 1, 13–35. MR 1815410
  • [8] Eugenio Calabi, On Kähler manifolds with vanishing canonical class, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 78–89. MR 0085583
  • [9] Eugenio Calabi, Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95–114. MR 780039
  • [10] Xiuxiong Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), no. 2, 189–234. MR 1863016
  • [11] X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics and stability. arXiv: 1210.7494. To appear in Int. Math. Res. Not (2013).
  • [12] X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metric on Fano manifolds, I: approximation of metrics with cone singularities. arXiv:1211.4566.
  • [13] X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metric on Fano manifolds, II: limits with cone angle less than $ 2\pi $. arXiv:1212.4714.
  • [14] Xiuxiong Chen and Bing Wang, Space of Ricci flows I, Comm. Pure Appl. Math. 65 (2012), no. 10, 1399–1457. MR 2957704, 10.1002/cpa.21414
  • [15] Jean-Pierre Demailly and János Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525–556 (English, with English and French summaries). MR 1852009, 10.1016/S0012-9593(01)01069-2
  • [16] Wei Yue Ding, Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann. 282 (1988), no. 3, 463–471. MR 967024, 10.1007/BF01460045
  • [17] Wei Yue Ding and Gang Tian, Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), no. 2, 315–335. MR 1185586, 10.1007/BF01231335
  • [18] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349. MR 1988506
  • [19] S. K. Donaldson, Stability, birational transformations and the Kahler-Einstein problem, Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., vol. 17, Int. Press, Boston, MA, 2012, pp. 203–228. MR 3076062, 10.4310/SDG.2012.v17.n1.a5
  • [20] S. K. Donaldson, Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications, Springer, Heidelberg, 2012, pp. 49–79. MR 2975584, 10.1007/978-3-642-28821-0_4
  • [21] S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. arXiv:1206.2609.
  • [22] Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Singular Kähler-Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. MR 2505296, 10.1090/S0894-0347-09-00629-8
  • [23] A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), no. 3, 437–443. MR 718940, 10.1007/BF01388438
  • [24] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • [25] Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
  • [26] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
  • [27] T. Jeffres, R. Mazzeo, and Y. Rubinstein, Kähler-Einstein metrics with edge singularities. arXiv:1105.5216
  • [28] Robert Lazarsfeld, 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], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472
  • [29] C. Li and S. Sun, Conical Kähler-Einstein metric revisited. arXiv:1207.5011
  • [30] Long Li, A note on general Bando-Mabuchi uniqueness theorem. preprint.
  • [31] Peter Li, Lecture notes on geometric analysis, Lecture Notes Series, vol. 6, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 1320504
  • [32] Peng Lu, A local curvature bound in Ricci flow, Geom. Topol. 14 (2010), no. 2, 1095–1110. MR 2629901, 10.2140/gt.2010.14.1095
  • [33] Yozô Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J. 11 (1957), 145–150 (French). MR 0094478
  • [34] Y. Odaka and S. Sun, Testing log K-stability by blowing up formalism. arXiv: 1112.1353
  • [35] G. Perelman, Ricci flow with surgery on three-manifolds. arXiv: 0303109.
  • [36] Yum Tong Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar, vol. 8, Birkhäuser Verlag, Basel, 1987. MR 904673
  • [37] J. Song and X. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.483
  • [38] Jacopo Stoppa, K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), no. 4, 1397–1408. MR 2518643, 10.1016/j.aim.2009.02.013
  • [39] Song Sun, Note on K-stability of pairs, Math. Ann. 355 (2013), no. 1, 259–272. MR 3004583, 10.1007/s00208-012-0788-y
  • [40] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172. MR 1055713, 10.1007/BF01231499
  • [41] Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37. MR 1471884, 10.1007/s002220050176
  • [42] G. Tian and B. Wang, On the structure of almost Einstein manifolds. arXiv: 1202.2912.
  • [43] Neil S. Trudinger, Regularity of solutions of fully nonlinear elliptic equations, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 3, 421–430. MR 769173
  • [44] Karen K. Uhlenbeck, Connections with 𝐿^{𝑝} bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356
  • [45] B. Wang, Ricci flow on orbifold. arXiv:1003.0151.
  • [46] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, 10.1002/cpa.3160310304
  • [47] Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1–28. MR 1216573

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Additional Information

Xiuxiong Chen
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651 – and – School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China

Simon Donaldson
Affiliation: Department of Mathematics, Imperial College London, London, U.K.

Song Sun
Affiliation: Department of Mathematics, Imperial College London, London, U.K.

Received by editor(s): March 8, 2013
Received by editor(s) in revised form: October 4, 2013, and January 13, 2014
Published electronically: March 28, 2014
Additional Notes: The first author was partly supported by National Science Foundation grant No 1211652; the last two authors were partly supported by the European Research Council award No 247331.
Article copyright: © Copyright 2014 American Mathematical Society