Abstract
LetM=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\(\mathcal{C}\left( \mathfrak{g} \right)\) of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification\(\mathfrak{g}^\mathbb{C} \) of the Lie algebra ofG. 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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Research partially supported by MURST and CNR of Italy.
Research partially supported by MURST and CNR of Italy.
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Console, S., Fino, A. Dolbeault cohomology of compact nilmanifolds. Transformation Groups 6, 111–124 (2001). https://doi.org/10.1007/BF01597131
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DOI: https://doi.org/10.1007/BF01597131