# zbMATH — the first resource for mathematics

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $$2\pi$$. (English) Zbl 1312.53097
This is the second paper in a landmark series of papers that prove the result “K-stability implies the existence of a Kähler-Einstein metric on Fano manifolds”. The other direction, i.e., that the existence of such a metric implies the algebro-geometric condition of K-stability was proven by Tian. The strategy of the proof is to solve a Monge-Ampère-type PDE using a continuity-type method by constructing metrics which are “bent” along a divisor and then tending the “bending” angle to $$2\pi$$. This paper deals with the situation when one has a sequence of conical Kähler-Einstein manifolds $$(X_i, D_i, \omega _i)$$ having a fixed Hilbert polynomial along divisors in a particular linear system ($$-\lambda K_{X_i}$$) with cone angles converging to $$\beta_{\infty} < 2\pi$$. The main theorem (Theorem 1) asserts that there is a Q-Fano variety $$W$$ with a real Weil divisor having KLT singularities such that it has a “weak conical KE metric” and all the $$X_i$$ converge to $$X$$ as projective varieties in a large-dimensional projective space. The proof of this theorem is extremely involved. It makes use of the approximation by smooth metrics proved in the first paper [the authors, ibid. 28, No. 1, 183–197 (2015; Zbl 1312.53096)], and Gromov-Hausdorff as well as Cheeger-Colding theory of convergence. In order to embed these in a large projective space, a Hörmander-type technique is used. In addition, Theorem 2 says that if the limiting Fano variety and the divisor are smooth, then the limiting metric is a smooth conical Kähler-Einstein metric.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14J45 Fano varieties 32Q20 Kähler-Einstein manifolds
Zbl 1312.53096
Full Text:
##### References:
 [1] Anderson, Michael T., Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102, 2, 429-445 (1990) · Zbl 0711.53038 [2] [kn:Brendle] S. Brendle, Ricci flat Kahler metrics with edge singularities. arXiv:1103. 5454. · Zbl 1293.32029 [3] [kn:CZ] S. Calamai and K. Zheng, Geodesics in the space of K\"ahler cone metrics. arXiv:1205.0056 · Zbl 1334.58006 [4] Cheeger, Jeff, Integral bounds on curvature elliptic estimates and rectifiability of singular sets, Geom. Funct. Anal., 13, 1, 20-72 (2003) · Zbl 1086.53051 [5] Cheeger, Jeff; Colding, Tobias H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math. (2), 144, 1, 189-237 (1996) · Zbl 0865.53037 [6] Cheeger, Jeff; Colding, Tobias H., On the structure of spaces with Ricci curvature bounded below. I, J. Diff. Geom., 46, 3, 406-480 (1997) · Zbl 0902.53034 [7] Cheeger, Jeff; Colding, Tobias H., On the structure of spaces with Ricci curvature bounded below. III, J. Diff. Geom., 54, 1, 37-74 (2000) · Zbl 1027.53043 [8] [kn:CDS0] 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 [9] [kn:CDS1] X-X. Chen, S. Donaldson, and S. Sun, K\"ahler-Einstein metric on Fano manifolds, I: approximation of metrics with cone singularities. arXiv:1211.4566 · Zbl 1312.53096 [10] 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 [11] [kn:DS] S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, arXiv:1206.2609. · Zbl 1318.53037 [12] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed, Singular K\"ahler-Einstein metrics, J. Amer. Math. Soc., 22, 3, 607-639 (2009) · Zbl 1215.32017 [13] [kn:JMR] T. Jeffres, R. Mazzeo, and T. Rubinstein, K\"ahler-Einstein metrics with edge singularities. arXiv:1105.5216 · Zbl 1337.32037
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.