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Cohomologies and deformations of solvmanifolds and closedness of some properties. (English) Zbl 1384.32018
The Dolbeault and Bott-Chern cohomologies and deformations of solvmanifolds by means of finite-dimensional complexes are studied. These cohomologies for some classes of complex nilmanifolds and solvmanifolds \(G/\Gamma\) (of splitting types and holomorphically parallelizable solvmanifolds) are calculated. The finite-dimensional complexes for Dolbeault and Bott-Chern cohomologies and deformations are calculated explicitly for Nakamura 3-dimensional solvmanifold (G=\(\mathbf C \cdot \mathbf C^2\)) with a lot of details and for Sawai-Yamada generalized manifolds (\(G=\mathbf C \cdot N\) for some nilpotent Lie group \(N\)). Explicit examples show that the property of \(E_1\)-degeneration of the Hodge and Frölicher spectral sequences (here the result is known) and the property of satisfying the \(\partial \bar{\partial}\)-Lemma are not closed under holomorphic deformations.

32M10 Homogeneous complex manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32G05 Deformations of complex structures
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