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

Abstract

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta \to 0$ of Kähler-Einstein metrics ${\omega }_{\beta }$ with a cone singularity of angle $2\pi \beta$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that ${\omega }_{\beta }$ converges to the locally symmetric metric and further give asymptotics of ${\omega }_{\beta }$ when $X\setminus D$ 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 ${\omega }_{\beta }$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,{\omega }_{\beta })$ converges to an interval in the Gromov-Hausdorff sense.

51 pages, v2: exposition improved following the referee's suggestions, to appear in Invent. Math

References in corpus (3)

• Schauder estimates for equations with cone metrics, I
• Schauder estimates on products of cones
• Polystable log Calabi-Yau varieties and Gravitational instantons

Cited by in corpus (1)

• Limits of conical Kähler-Einstein metrics on rank one horosymmetric spaces