On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

TL;DR

This paper characterizes when the Bergman metric on finite ball quotients is Kähler--Einstein, showing it only occurs for the unit ball, and uses this to identify the unit ball among certain Stein spaces.

Contribution

It proves that the Bergman metric on finite abelian ball quotients is Kähler--Einstein only for the trivial case, providing a new characterization of the unit ball among Stein spaces.

Findings

Bergman metric is Kähler--Einstein only for the unit ball.

Characterization of the unit ball among Stein spaces with isolated singularities.

Finite abelian group actions lead to non-Einstein Bergman metrics unless trivial.

Abstract

In this paper, we study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma$, where $\Gamma \subseteq \mathrm{Aut}(\mathbb{B}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is K\"ahler--Einstein if and only if $\Gamma$ is trivial, i.e., when the ball quotient $\mathbb{B}^n/\Gamma$ is the unit ball $\mathbb{B}^n$ itself. As a consequence, we establish a characterization of the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman-Einstein metric.