Higher Rank Bergman Kernels on Compact Riemann Surfaces

TL;DR

This paper establishes the existence of a global asymptotic expansion for higher rank Bergman kernels on compact Riemann surfaces with Griffiths-positive curvature, advancing understanding in complex geometry and vector bundle analysis.

Contribution

It proves the existence of a global asymptotic expansion of Bergman kernels for symmetric powers of vector bundles with Griffiths-positive curvature on compact Riemann surfaces.

Findings

Asymptotic expansion of Bergman kernels established

Results apply to vector bundles with Griffiths-positive curvature

Advances in complex geometric analysis

Abstract

Let X be a compact Riemann surface equipped with a real-analytic K\"ahler form $\omega$ and let E be a holomorphic vector bundle over $X$ equipped with a real-analytic Hermitian metric $h$. Suppose that the curvature of $h$ is Griffiths-positive. We prove the existence of a global asymptotic expansion in powers of $k$ of the Bergman kernel associated to $(S^k E, S^k h)$ and $\omega$.