On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces

TL;DR

This paper investigates the dimension of bundle-valued Bergman spaces on compact Riemann surfaces, showing they are either finite-dimensional (matching global sections) or infinite-dimensional, with the latter characterized by potential theory.

Contribution

It establishes a dichotomy for the dimension of Bergman spaces of holomorphic sections of vector bundles on compact Riemann surfaces, linking infinite-dimensionality to potential-theoretic properties.

Findings

Bergman space of restricted bundle sections is either finite or infinite dimensional.

Infinite dimensionality characterized by potential-theoretic properties of the domain.

The result generalizes classical function theory to bundle-valued settings.

Abstract

Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$.