On the dimension of Bergman spaces on $\mathbb{P}^1$

TL;DR

This paper provides potential-theoretic criteria to determine the dimension of Bergman spaces of holomorphic sections of line bundles over subsets of the complex projective line, extending previous results by Sz"Hoke.

Contribution

It introduces new potential-theoretic characterizations for the dimension of Bergman spaces on subsets of , generalizing prior work by Szke.

Findings

Characterization of Bergman space dimension via potential theory

Extension of Szke's results to line bundle sections

New criteria for holomorphic section spaces on 

Abstract

Inspired by a result by Sz\H{o}ke, we give potential-theoretic characterizations of the dimension of the Bergman space of holomorphic sections of a restriction of a holomorphic line bundle of $\mathbb{P}^1$ to some open set $D\subset\mathbb{P}^1$.