Degenerating K\"ahler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric

TL;DR

This paper studies the limits of Kähler-Einstein metrics with cone singularities as the cone angle shrinks, revealing convergence to locally symmetric metrics and Tian-Yau metrics in specific geometric settings.

Contribution

It proves convergence results for Kähler-Einstein cone metrics in locally symmetric and Fano cases, confirming conjectures about their limits and asymptotics.

Findings

Convergence of cone metrics to locally symmetric metrics as cone angle approaches zero.

Asymptotic behavior of metrics in the case of ball quotients.

Validation of the Tian-Yau conjecture for Fano manifolds with anticanonical divisors.

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 K\"ahler-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.