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Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \(2\pi\) and completion of the main proof. (English) Zbl 1311.53059
This is the third (and last) 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., “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-Ampere type PDE using a continuity-type method by constructing metrics which are “bent” along a divisor and then letting the “bending” angle tending to \(2\pi\).
In this paper, the authors study what happens to a sequence of conical Kähler-Einstein manifolds when the cone angle approaches \(2\pi\). Using Gromov-Hausdorff convergence they conclude that such a sequence (up to a subsequence) converges to a limit. Applying the Cheeger-Colding-Tian theory to such a limit they deduce that it is a union of regular points and singular points of “small” Hausdorff dimension. Then they prove that (Theorem 3) the regular set is open and indeed the limiting metric is a smooth Kähler-Einstein metric on it. Two proofs are given for this result. Using this result they proceed to prove their Theorem 2, which (roughly speaking) states that the limiting metric space is actually a Q-Fano variety with a “weak” Kähler-Einstein metric and that this convergence takes place inside some large projective space. Given Theorems 2 and 3, they proceed to complete the proof of the main result (theorem 1) of the paper. Along the way, they establish an Evans-Krylov type estimate.

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