A classification of homogeneous K\"{a}hler manifolds with discrete isotropy and top nonvanishing homology in codimension two

TL;DR

This paper classifies certain homogeneous Kähler manifolds with discrete isotropy, showing they are essentially products involving Cousin groups and complex lines or tori, under specific homology conditions.

Contribution

It proves that such manifolds have solvable Lie groups and are covered by products of Cousin groups and simple complex groups, extending understanding of their structure.

Findings

G is solvable.

A finite cover of X is biholomorphic to a product C×A.

A is one of {e}, C, C*, or C*×C*}.

Abstract

Suppose $G$ is a connected complex Lie group and $\Gamma$ is a discrete subgroup such that $X := G/\Gamma$ is K\"ahler and the codimension of the top non--vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $G$ is solvable and a finite covering of $X$ is biholomorphic to a product $C\times A$, where $C$ is a Cousin group and $A$ is $\{e \}$, $\mathbb C$, $\mathbb C^*$, or $\mathbb C^*\times\mathbb C^*$.