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

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 Kähler--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.

21 pages