Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities
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- by Xiuxiong Chen, Simon Donaldson and Song Sun
- J. Amer. Math. Soc. 28 (2015), 183-197
- DOI: https://doi.org/10.1090/S0894-0347-2014-00799-2
- Published electronically: March 28, 2014
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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.References
- 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, DOI 10.1512/iumj.1989.38.38021
- 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, DOI 10.1016/j.aim.2013.08.024
- B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. arXiv:1103.0923.
- 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, DOI 10.1512/iumj.2003.52.2346
- 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 106487
- F. Campana, H. Guenancia, and M. Paun, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. arXiv:1104.4879.
- 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
- Xiuxiong Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 12 (2000), 607–623. MR 1772078, DOI 10.1155/S1073792800000337
- X-X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics and stability. arXiv:1210.7494. To appear in Int. Math. Res. Not (2013).
- Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge-Ampère equation $\textrm {det}(\partial ^{2}u/\partial x_{i}\partial sx_{j})=F(x,u)$, Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68. MR 437805, DOI 10.1002/cpa.3160300104
- 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}
- Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
- 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
- D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer, 1998.
- T. D. Jeffres, R. Mazzeo, and Y. Rubinstein, Kähler-Einstein metrics with edge singularities. arXiv:1105.5216.
- Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69–117. MR 1618325, DOI 10.1007/BF02392879
- Sławomir Kołodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), no. 3, 667–686. MR 1986892, DOI 10.1512/iumj.2003.52.2220
- 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
- C. Li and S. Sun, Conical Kähler-Einstein metric revisited. arXiv:1207.5011.
- Haozhao Li, On the lower bound of the $K$-energy and $F$-functional, Osaka J. Math. 45 (2008), no. 1, 253–264. MR 2416659
- Yung-chen Lu, Holomorphic mappings of complex manifolds, J. Differential Geometry 2 (1968), 299–312. MR 250243
- J. Song and X-W. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.4839.
- Gábor Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011), no. 1, 319–331. MR 2771134, DOI 10.1112/S0010437X10004938
- 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, DOI 10.1090/S0894-0347-1990-1040196-6
- 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
Bibliographic 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 2104 American Mathematical Society
- Journal: J. Amer. Math. Soc. 28 (2015), 183-197
- MSC (2010): Primary 53C55
- DOI: https://doi.org/10.1090/S0894-0347-2014-00799-2
- MathSciNet review: 3264766