On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics
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Publication:5043822
DOI10.1093/IMRN/RNAB120zbMath1504.32067arXiv2009.07416OpenAlexW3176508281MaRDI QIDQ5043822
Ming Xiao, Peter Ebenfelt, Hang Xu
Publication date: 6 October 2022
Published in: Unnamed Author (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/2009.07416
Kähler-Einstein manifolds (32Q20) Integral representations; canonical kernels (Szeg?, Bergman, etc.) (32A25)
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