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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Kähler structures on compact solvmanifolds

Authors: Chal Benson and Carolyn S. Gordon
Journal: Proc. Amer. Math. Soc. 108 (1990), 971-980
MSC: Primary 53C55; Secondary 22E25, 22E40, 32M05, 32M10
MathSciNet review: 993739
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Abstract: 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.

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