Peak sections and Bergman kernels on Kähler manifolds with complex hyperbolic cusps

2022/11/08 by Shengxuan Zhou, Zhou, Shengxuan
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2211.04091

openalex publication_date 2022/11/08 · openalex created_date 2022/11/14 · openalex updated_date 2026/07/28

Abstract

By revisiting Tian's peak section method, we obtain a localization principle of the Bergman kernels on Kähler manifolds with complex hyperbolic cusps, which is a generalization of Auvray-Ma-Marinescu's localization result Bergman kernels on punctured Riemann surfaces [Auvray-Ma-Marinescu, Math. Ann., 2021]. Then we give some further estimates when the metric on the complex hyperbolic cusp is a Kähler-Einstein metric or when the manifold is a quotient of the complex ball. By applying our method directly to Poincaré type cusps, we also get a partial localization result.

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