Cohomologies of deformations of solvmanifolds and closedness of some properties

TL;DR

This paper develops new methods using finite-dimensional complexes to analyze the cohomologies of deformed solvmanifolds, revealing that certain geometric properties are not preserved under deformation.

Contribution

It introduces techniques for computing Dolbeault and Bott-Chern cohomologies of solvmanifolds and provides explicit examples demonstrating the non-closedness of specific properties under deformation.

Findings

The $ ext{∂∂̄}$-Lemma is not stable under deformations.

Hodge and Frölicher spectral sequence degenerations are not preserved.

Explicit examples illustrate the non-closedness of these properties.

Abstract

We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particular that either the $\partial\overline{\partial}$-Lemma or the property that the Hodge and Fr\"olicher spectral sequence degenerates at the first level are not closed under deformations.