Kähler structures on compact solvmanifolds

In a previous paper, the authors proved that the only compact nilmanifolds $\Gamma \backslash G$ which admit Kähler structures are tori. Here we consider a more general class of homogeneous spaces $\Gamma \backslash G$, where $G$ is a completely solvable Lie group and $\Gamma$ is a cocompact discrete subgroup. Necessary conditions for the existence of a Kähler structure are given in terms of the structure of $G$ and a homogeneous representative $\omega$ of the Kähler class in ${H^2}(\Gamma \backslash G;\mathbb{R})$. These conditions are not sufficient to imply the existence of a Kähler structure. On the other hand, we present examples of such solvmanifolds that have the same cohomology ring as a compact Kähler manifold. We do not know whether some of these solvmanifolds admit Kähler structures.