Degenerating Kahler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric

Let X be a complex projectivemanifold and let D subset of X be a smooth divisor. In this article, we are interested in studying limits when ss -> 0 of Kahler-Einstein metrics omega(ss) with a cone singularity of angle 2 pi ss along D. In our first result, we assume that X \ D is a locally symmetric space and we show that omega(ss) converges to the locally symmetric metric and further give asymptotics of omega(ss) when X \ D is a ball quotient. Our second result deals with the case when X is Fano and D is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of omega(ss) is the complete, Ricci flat Tian-Yau metric on X \ D. Furthermore, we prove that (X, omega(ss)) converges to an interval in the Gromov-Hausdorff sense.

Automated summary

This article studies the limits of Kahler-Einstein metrics with a cone singularity on a smooth divisor D in the complex projective manifold X. Under certain conditions, it is shown that the singular metric converges to a locally symmetric metric and its asymptotic behavior is determined when X \ D is a ball quotient. In the case where X is a Fano manifold and D is anticanonical, it is proven that the rescaled limit of the metric is the complete, Ricci flat Tian-Yau metric on X \ D, and the convergence of (X, omega(ss)) is established in the Gromov-Hausdorff sense.