Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi\). (English) Zbl 1312.53097

Chen, Xiuxiong; Donaldson, Simon; Sun, Song

J. Am. Math. Soc. 28, No. 1, 199-234 (2015).

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.

Reviewer: Vamsi Pritham Pingali (Baltimore)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
14J45	Fano varieties
32Q20	Kähler-Einstein manifolds

Keywords:

Kähler-Einstein metrics; Fano manifolds; Cone angle less than \(2\pi\); Q-Fano

Citations:

Zbl 1312.53096

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