Bergman spaces of natural G-manifolds

TL;DR

This paper investigates the properties of Bergman spaces on certain complex manifolds constructed from unimodular Lie groups and principal bundles, showing conditions under which these spaces are large.

Contribution

It establishes a new criterion involving tangent spaces for the size of Bergman spaces on G-manifolds with boundary.

Findings

Bergman space is large under specific tangent space conditions.

Includes examples like G-complexifications of known complex manifolds.

Provides new insights into the structure of Bergman spaces on G-manifolds.

Abstract

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G--> M-->X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if there exists a point p in bM such that T_p(G) is contained in the complex tangent space of bM at p, then the Bergman space of M is large. Natural examples include the gauged G-complexifications of Heinzner, Huckleberry, and Kutzschebauch.