Holomorphic Function Theory on Pseudoconvex Complex Homogeneous Manifolds

Introduction

In this paper, we analyze the holomorphic function theory of pseudoconvex complex homogeneous manifolds. We restrict our attention to complex homogeneous manifolds with connected complex Lie group G. Using the concept of inner integral curves defined by Hirschowitz, we generalize a recent result by Kim.

Main Results

Our main results are the following:

1. We show that a strongly pseudoconvex homogeneous domain in a complex manifold is biholomorphic to the complex unit ball.
2. We use this result to characterize reductive homogeneous K$\ddot{a}$hler manifolds in terms of their isotropy subgroups.
3. We prove a generalization of Kim's result on the existence of inner integral curves for holomorphic vector fields on complex homogeneous manifolds.

Applications

Our results have applications to the study of complex homogeneous manifolds, holomorphic function theory, and K$\ddot{a}$hler geometry.

Keywords

Complex homogeneous manifolds, holomorphic functions, pseudoconvexity, inner integral curves, K$\ddot{a}$hler geometry