The Levi Problem On Strongly Pseudoconvex $G$-Bundles

TL;DR

This paper investigates the Levi problem for strongly pseudoconvex G-bundles, showing that under certain conditions, the space of square-integrable holomorphic functions on the total space is infinite-dimensional.

Contribution

It establishes that for strongly pseudoconvex G-bundles with a holomorphic G-action satisfying a local property, the space of square-integrable holomorphic functions is infinite-dimensional.

Findings

The space of square-integrable holomorphic functions on M is infinite G-dimensional.

G acts by holomorphic transformations satisfying a local property.

The setting involves a principal G-bundle over a compact boundary manifold with strong pseudoconvexity.

Abstract

Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ the total space of a principal bundle $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on $M$ is infinite $G$-dimensional.