Dolbeault Cohomology of compact Nilmanifolds

S. Console; A. Fino

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arXiv · math/9803135

Dolbeault Cohomology of compact Nilmanifolds

S. Console·A. Fino

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Abstract

Let $M= G/Γ$ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the Dolbeault cohomology of $M$ is isomorphic to the one of the differential bigraded algebra associated to the complexification $\cg^\C$ of the Lie algebra of $G$. To obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.

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BibTeX RIS

S. Console, A. Fino. 2001-01-23. Dolbeault Cohomology of compact Nilmanifolds. https://arxiv.org/abs/math/9803135

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