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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \boldmath$2\pi$ and completion of the main proof
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by Xiuxiong Chen, Simon Donaldson and Song Sun PDF
J. Amer. Math. Soc. 28 (2015), 235-278 Request permission

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.
References
  • Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 1074481, DOI 10.1007/BF01233434
  • 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 433520
  • Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
  • R. Berman, K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. arXiv: 1205.6214.
  • R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, Kähler-Ricci flow and Ricci iteration on log-Fano varieties. arXiv:1111.7158.
  • B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. arXiv: 1103.0923
  • 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
  • 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
  • Eugenio Calabi, Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95–114. MR 780039
  • Xiuxiong Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), no. 2, 189–234. MR 1863016
  • X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics and stability. arXiv: 1210.7494. To appear in Int. Math. Res. Not (2013).
  • X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metric on Fano manifolds, I: approximation of metrics with cone singularities. arXiv:1211.4566.
  • 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.
  • Xiuxiong Chen and Bing Wang, Space of Ricci flows I, Comm. Pure Appl. Math. 65 (2012), no. 10, 1399–1457. MR 2957704, DOI 10.1002/cpa.21414
  • 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, DOI 10.1016/S0012-9593(01)01069-2
  • Wei Yue Ding, Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann. 282 (1988), no. 3, 463–471. MR 967024, DOI 10.1007/BF01460045
  • Wei Yue Ding and Gang Tian, Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), no. 2, 315–335. MR 1185586, DOI 10.1007/BF01231335
  • S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349. MR 1988506, DOI 10.4310/jdg/1090950195
  • 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, DOI 10.4310/SDG.2012.v17.n1.a5
  • 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, DOI 10.1007/978-3-642-28821-0_{4}
  • S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. arXiv:1206.2609.
  • Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Singular Kähler-Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. MR 2505296, DOI 10.1090/S0894-0347-09-00629-8
  • A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), no. 3, 437–443. MR 718940, DOI 10.1007/BF01388438
  • 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, DOI 10.1007/978-3-642-61798-0
  • 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
  • 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
  • T. Jeffres, R. Mazzeo, and Y. Rubinstein, Kähler-Einstein metrics with edge singularities. arXiv:1105.5216
  • 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, DOI 10.1007/978-3-642-18808-4
  • C. Li and S. Sun, Conical Kähler-Einstein metric revisited. arXiv:1207.5011
  • Long Li, A note on general Bando-Mabuchi uniqueness theorem. preprint.
  • 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
  • Peng Lu, A local curvature bound in Ricci flow, Geom. Topol. 14 (2010), no. 2, 1095–1110. MR 2629901, DOI 10.2140/gt.2010.14.1095
  • Yozô Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J. 11 (1957), 145–150 (French). MR 94478, DOI 10.1017/S0027763000002026
  • Y. Odaka and S. Sun, Testing log K-stability by blowing up formalism. arXiv: 1112.1353
  • G. Perelman, Ricci flow with surgery on three-manifolds. arXiv: 0303109.
  • 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, DOI 10.1007/978-3-0348-7486-1
  • J. Song and X. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.483
  • Jacopo Stoppa, K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), no. 4, 1397–1408. MR 2518643, DOI 10.1016/j.aim.2009.02.013
  • Song Sun, Note on K-stability of pairs, Math. Ann. 355 (2013), no. 1, 259–272. MR 3004583, DOI 10.1007/s00208-012-0788-y
  • G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172. MR 1055713, DOI 10.1007/BF01231499
  • Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37. MR 1471884, DOI 10.1007/s002220050176
  • G. Tian and B. Wang, On the structure of almost Einstein manifolds. arXiv: 1202.2912.
  • 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
  • Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356, DOI 10.1007/BF01947069
  • B. Wang, Ricci flow on orbifold. arXiv:1003.0151.
  • 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, DOI 10.1002/cpa.3160310304
  • 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
  • MR Author ID: 632654
  • Email: xiu@math.sunysb.edu
  • Simon Donaldson
  • Affiliation: Department of Mathematics, Imperial College London, London, U.K.
  • Email: s.donaldson@imperial.ac.uk
  • Song Sun
  • Affiliation: Department of Mathematics, Imperial College London, London, U.K.
  • MR Author ID: 879901
  • Email: s.sun@imperial.ac.uk
  • 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.
  • © Copyright 2014 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 28 (2015), 235-278
  • MSC (2010): Primary 53C55
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00801-8
  • MathSciNet review: 3264768