Levi's problem for complex homogeneous manifolds

TL;DR

This paper investigates the structure of complex homogeneous manifolds, proving that pseudoconvexity implies the holomorphic reduction is Stein and the fiber has no non-constant holomorphic functions.

Contribution

It establishes a link between pseudoconvexity and the Stein property of the holomorphic reduction for complex homogeneous manifolds.

Findings

If $G/H$ is pseudoconvex, then $G/J$ is Stein.

The fiber $J/H$ admits no non-constant holomorphic functions.

The holomorphic reduction simplifies the structure of pseudoconvex homogeneous manifolds.

Abstract

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.