On the Classification of Stein spaces with Bergman-Einstein metrics

2026/07/16 by Soumya Ganguly, Siddhartha Sahi
#math.CV #math.GR

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Abstract

For every N≥ 2, we prove that the Bergman metric on the regular locus of a finite ball quotient \mathbbB^N/Γ, where Γ⊂ U(N) is finite and fixed-point-free, is Kähler-Einstein if and only if Γ is trivial. Consequently, if Ω is an N-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on Ω_reg is Kähler-Einstein if and only if Ω is biholomorphic to \mathbbB^N. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension N≥ 2.

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