Complex and Kaehler structures on compact homogeneous manifolds - their existence, classification and moduli problem

TL;DR

This paper surveys recent advances in understanding the existence, classification, and moduli of complex and Kaehler structures on compact homogeneous manifolds, highlighting their geometric and topological implications.

Contribution

It provides a comprehensive overview of current research on complex and Kaehler structures specifically on compact homogeneous manifolds.

Findings

Conditions for existence of complex structures

Classification results for Kaehler structures

Open problems in moduli space of these structures

Abstract

The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classification and moduli problems of such structures on compact homogeneous manifolds.