On a theorem of Wiegerinck

TL;DR

This paper generalizes Wiegerinck's theorem, showing that for certain holomorphic vector bundles over the complex projective line, the space of L2 sections over a domain is either trivial, finite-dimensional, or infinite-dimensional.

Contribution

It extends Wiegerinck's theorem from scalar Bergman spaces to holomorphic L2 sections of vector bundles over the complex projective line.

Findings

The space of holomorphic L2 sections is either equal to the global sections or infinite dimensional.

The result applies to hermitian, holomorphic vector bundles over  with arbitrary domains.

Provides a broader understanding of the structure of L2 sections in complex geometry.

Abstract

A theorem of Wiegerinck says that the Bergman space over any domain in $\mathbb C$ is either trivial or infinite dimensional. We generalize this theorem in the following form. Let E be a hermitian, holomorphic vector bundle over $\mathbb P^1$, the later equipped with a volume form and $D$ an arbitrary domain in $\mathbb P^1$. Then the space of holomorphic L2 sections of $E$ over $D$ is either equal to $H^0(M,E)$ or it has infinite dimension.