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Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

Authors: Xiuxiong Chen, Simon Donaldson and Song Sun
Journal: J. Amer. Math. Soc. 28 (2015), 183-197
MSC (2010): Primary 53C55
Published electronically: March 28, 2014
MathSciNet review: 3264766
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Abstract: This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.

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  • [1] Eric Bedford and B. A. Taylor, Uniqueness for the complex Monge-Ampère equation for functions of logarithmic growth, Indiana Univ. Math. J. 38 (1989), no. 2, 455-469. MR 997391 (90i:32025),
  • [2] Robert J. Berman, A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics, Adv. Math. 248 (2013), 1254-1297. MR 3107540,
  • [3] B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. arXiv:1103.0923.
  • [4] Zbigniew Błocki, Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 6, 1697-1701. MR 2021054 (2004m:32073),
  • [5] Eugenio Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958), 105-126. MR 0106487 (21 #5219)
  • [6] F. Campana, H. Guenancia, and M. Paun, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. arXiv:1104.4879.
  • [7] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Diff. Geom. 54 (2000), no. 1, 13-35. MR 1815410 (2003a:53043)
  • [8] Xiuxiong Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 12 (2000), 607-623. MR 1772078 (2001f:32042),
  • [9] X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics and stability. arXiv:1210.7494. To appear in Int. Math. Res. Not (2013).
  • [10] Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge-Ampère equation $ {\rm det}(\partial ^{2}u/\partial x_{i}\partial sx_{j})=F(x,u)$, Commun. Pure Appl. Math. 30 (1977), no. 1, 41-68. MR 0437805 (55 #10727)
  • [11] 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,
  • [12] Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math. 35 (1982), no. 3, 333-363. MR 649348 (83g:35038),
  • [13] Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Singular Kähler-Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607-639. MR 2505296 (2010k:32031),
  • [14] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer, 1998.
  • [15] T. D. Jeffres, R. Mazzeo, and Y. Rubinstein, Kähler-Einstein metrics with edge singularities. arXiv:1105.5216.
  • [16] Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69-117. MR 1618325 (99h:32017),
  • [17] Sławomir Kołodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, 667-686. MR 1986892 (2004i:32062),
  • [18] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487-523, 670 (Russian). MR 661144 (84a:35091)
  • [19] C. Li and S. Sun, Conical Kähler-Einstein metric revisited. arXiv:1207.5011.
  • [20] Haozhao Li, On the lower bound of the $ K$-energy and $ F$-functional, Osaka J. Math. 45 (2008), no. 1, 253-264. MR 2416659 (2009a:32036)
  • [21] Yung-chen Lu, Holomorphic mappings of complex manifolds, J. Differential Geometry 2 (1968), 299-312. MR 0250243 (40 #3482)
  • [22] J. Song and X-W. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.4839.
  • [23] Gábor Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011), no. 1, 319-331. MR 2771134 (2011m:32037),
  • [24] G. Tian and Shing-Tung Yau, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc. 3 (1990), no. 3, 579-609. MR 1040196 (91a:53096),
  • [25] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Commun. Pure Appl. Math. 31 (1978), no. 3, 339-411. MR 480350 (81d:53045),

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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.
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