Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. (English) Zbl 1312.53096 J. Am. Math. Soc. 28, No. 1, 183-197 (2015). This is the first paper in a landmark series of papers that prove the theorem “K-stability implies the existence of a Kähler-Einstein metric.” In this paper the goal (Theorem 1.1) is to approximate any Kähler-Einstein metric with cone singularities on a divisor (i.e., if the divisor is \(z=0\), then near the divisor the metric is equivalent to the standard cone metric \(\sqrt{-1} \beta ^2 | z | ^{2\beta -2}dz \wedge d\bar{z} + \sqrt{-1} \sum dz^i \wedge d\bar{z} ^i \) and is Einstein away from the divisor) in the Gromov-Hausdorff sense by means of smooth Kähler metrics with positive Ricci curvature and bounded diameter. This is accomplished by first smoothing out the volume form of the given metric, solving the Calabi-Yau equation, then modifying the volume form of the resulting metric so as to have positive Ricci curvature and bounded diameter. A sharper version of this theorem (Theorem 1.2) is also proven which provides a uniform positive lower bound on the Ricci curvature of the smooth metrics. Reviewer: Vamsi Pritham Pingali (Baltimore) Cited in 18 ReviewsCited in 222 Documents MSC: 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14J45 Fano varieties 32Q20 Kähler-Einstein manifolds Keywords:Kähler-Einstein; Fano variety; approximation; cone singularities; Gromov-Hausdorff limit × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bedford, Eric; Taylor, B. A., Uniqueness for the complex Monge-Amp\`ere equation for functions of logarithmic growth, Indiana Univ. Math. J., 38, 2, 455-469 (1989) · Zbl 0677.32002 · doi:10.1512/iumj.1989.38.38021 [2] Berman, Robert J., A thermodynamical formalism for Monge-Amp\`“ere equations, Moser-Trudinger inequalities and K\'”ahler-Einstein metrics, Adv. Math., 248, 1254-1297 (2013) · Zbl 1286.58010 · doi:10.1016/j.aim.2013.08.024 [3] [Berndt11] B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. arXiv:1103.0923. · Zbl 1318.53077 [4] B{\l }ocki, Zbigniew, Uniqueness and stability for the complex Monge-Amp\`“ere equation on compact K\'”ahler manifolds, Indiana Univ. Math. J., 52, 6, 1697-1701 (2003) · Zbl 1054.32024 · doi:10.1512/iumj.2003.52.2346 [5] Calabi, Eugenio, Improper affine hyperspheres of convex type and a generalization of a theorem by K. J\"orgens, Michigan Math. J., 5, 105-126 (1958) · Zbl 0113.30104 [6] [CGP11] F. Campana, H. Guenancia, and M. Paun, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. arXiv:1104.4879. · Zbl 1310.32029 [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] Chen, Xiuxiong, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices, 12, 607-623 (2000) · Zbl 0980.58007 · doi:10.1155/S1073792800000337 [9] [CDS] 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 [10] Cheng, Shiu Yuen; Yau, Shing Tung, On the regularity of the Monge-Amp\`ere equation \({\rm det}(\partial^2u/\partial x_i\partial sx_j)=F(x,u)\), Commun. Pure Appl. Math., 30, 1, 41-68 (1977) · Zbl 0347.35019 [11] 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 [12] Evans, Lawrence C., Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35, 3, 333-363 (1982) · Zbl 0469.35022 · doi:10.1002/cpa.3160350303 [13] 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 [14] [GT] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer, 1998. · Zbl 1042.35002 [15] [JMR] T. D. Jeffres, R. Mazzeo, and Y. Rubinstein, K\"ahler-Einstein metrics with edge singularities. arXiv:1105.5216. · Zbl 1337.32037 [16] Ko{\l }odziej, S{\l }awomir, The complex Monge-Amp\`ere equation, Acta Math., 180, 1, 69-117 (1998) · Zbl 0913.35043 · doi:10.1007/BF02392879 [17] Ko{\l }odziej, S{\l }awomir, The Monge-Amp\`“ere equation on compact K\'”ahler manifolds, Indiana Univ. Math. J., 52, 3, 667-686 (2003) · Zbl 1039.32050 · doi:10.1512/iumj.2003.52.2220 [18] Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat., 46, 3, 487-523, 670 (1982) · Zbl 0511.35002 [19] [LiSun] C. Li and S. Sun, Conical K\"ahler-Einstein metric revisited. arXiv:1207.5011. · Zbl 1296.32008 [20] Li, Haozhao, On the lower bound of the \(K\)-energy and \(F\)-functional, Osaka J. Math., 45, 1, 253-264 (2008) · Zbl 1138.53056 [21] Lu, Yung-chen, Holomorphic mappings of complex manifolds, J. Differential Geometry, 2, 299-312 (1968) · Zbl 0167.36602 [22] [SW12] J. Song and X-W. Wang, The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.4839. [23] Sz{\'e}kelyhidi, G{\'a}bor, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math., 147, 1, 319-331 (2011) · Zbl 1222.32046 · doi:10.1112/S0010437X10004938 [24] Tian, G.; Yau, Shing-Tung, Complete K\"ahler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc., 3, 3, 579-609 (1990) · Zbl 0719.53041 · doi:10.2307/1990928 [25] 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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