Peak sections and Bergman kernels on K\"ahler manifolds with complex hyperbolic cusps

TL;DR

This paper extends localization principles for Bergman kernels on K"ahler manifolds with complex hyperbolic cusps, providing new estimates and applying the method to Poincaré type cusps, generalizing previous results.

Contribution

It revisits Tian's peak section method to generalize localization results of Bergman kernels on manifolds with complex hyperbolic cusps and derives new estimates for specific metrics.

Findings

Localization principle for Bergman kernels on hyperbolic cusp manifolds

New estimates for K"ahler-Einstein metrics on cusps

Partial localization results for Poincaré type cusps

Abstract

By revisiting Tian's peak section method, we obtain a localization principle of the Bergman kernels on K\"ahler 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 K\"ahler-Einstein metric or when the manifold is a quotient of the complex ball. By applying our method directly to Poincar\'e type cusps, we also get a partial localization result.