Kähler structures on compact solvmanifolds

Abstract

In a previous paper, the authors proved that the only compact nilmanifolds Γ ∖ G \Gamma \backslash G which admit Kähler structures are tori. Here we consider a more general class of homogeneous spaces Γ ∖ G \Gamma \backslash G , where G G is a completely solvable Lie group and Γ \Gamma is a cocompact discrete subgroup. Necessary conditions for the existence of a Kähler structure are given in terms of the structure of G G and a homogeneous representative ω \omega of the Kähler class in H 2 ( Γ ∖ G ; R ) {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.