Bergman kernels on punctured Riemann surfaces

TL;DR

This paper studies the Bergman kernel on punctured Riemann surfaces with Poincaré metrics, showing it localizes near singularities and providing optimal estimates for its growth, advancing understanding of complex geometric analysis.

Contribution

It demonstrates the localization of the Bergman kernel near punctures and derives optimal uniform estimates involving fractional growth, extending previous results to punctured surfaces.

Findings

Bergman kernel localizes around punctures

Local model is the Bergman kernel of the punctured unit disc

Provides optimal uniform estimates with fractional growth order

Abstract

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincar\'e metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power.