Kähler structures on compact solvmanifolds
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- by Chal Benson and Carolyn S. Gordon
- Proc. Amer. Math. Soc. 108 (1990), 971-980
- DOI: https://doi.org/10.1090/S0002-9939-1990-0993739-4
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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.References
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Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 971-980
- MSC: Primary 53C55; Secondary 22E25, 22E40, 32M05, 32M10
- DOI: https://doi.org/10.1090/S0002-9939-1990-0993739-4
- MathSciNet review: 993739