Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms

TL;DR

This paper introduces a Lie algebraic method to analyze compact complex homogeneous manifolds with transitive automorphism groups, providing new proofs and a canonical abelian Lie algebra construction.

Contribution

It develops a Lie algebraic approach to study complex homogeneous manifolds, extending previous work and simplifying proofs of key theorems.

Findings

New proofs of classical results on complex homogeneous manifolds

A canonical abelian Lie algebra associated with complex structures

Extension of earlier work by Samelson and Pittie

Abstract

The purpose of this paper is to develop a Lie algebraic approach to obtain new proofs of important results of H.-C. Wang, Tits and Wolf-Wang-Ziller on compact complex homogeneous manifolds emphasizing only those that admit a transitive compact group of biholomorphic transformations. The method only uses some standard results in Lie theory. The new approach provides a method of associating a canonical abelian Lie algebra with a given integrable complex structure on a compact Lie algebra which extends the earlier work of Samelson and Pittie.