A classification of homogeneous Kähler 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 Kähler 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 $\{e\}$, $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, or ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$.

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