Cohomologies and deformations of solvmanifolds and closedness of some properties. (English) Zbl 1384.32018

Angella, Daniele; Kasuya, Hisashi

North-West. Eur. J. Math. 3, 75-105 (2017).

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.

Reviewer: V. V. Gorbatsevich (Moskva)

MSC:

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

Keywords:

solvmanifolds; nilmanifolds; Dolbeault cohomology; Bott-Chern cohomology; deformation; \(\partial \bar{\partial}\)-lemma