Pseudoconvex domains spread over complex homogeneous manifolds

TL;DR

This paper generalizes conditions under which pseudoconvex domains over complex homogeneous manifolds are Stein, using inner integral curves, and applies this to analyze the holomorphic reduction of such manifolds, especially when G is solvable or reductive.

Contribution

It extends previous results by incorporating inner integral curves to characterize pseudoconvex domains over complex homogeneous manifolds and studies their holomorphic reduction.

Findings

Pseudoconvex domains over complex homogeneous manifolds can be characterized as Stein under certain conditions.

When G is solvable or reductive, the manifold is a G-equivariant fiber bundle over a Stein base.

Holomorphic functions on the fiber are constant, indicating trivial fiber structure in the reduction.

Abstract

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.