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

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 $\mathbb C^*\times\mathbb C^*$.