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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.

##### MSC:
 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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