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Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. (English) Zbl 1312.53096
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.

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