Bott-Chern cohomology of solvmanifolds

TL;DR

This paper investigates methods to compute Bott-Chern cohomology for specific solvmanifolds, providing explicit complexes and calculations that reveal new insights into complex structure deformations.

Contribution

It introduces explicit finite-dimensional double complexes for computing Bott-Chern cohomology of certain solvmanifolds, including Nakamura manifolds, and examines their deformation properties.

Findings

Computed Bott-Chern cohomology for Nakamura manifolds.

Showed the  extbackslash d extbackslash d extbackslash d extbackslash d extbackslash d extbackslash d-lemma is not closed under deformations.

Developed explicit complexes for solvmanifolds of splitting type.

Abstract

We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott-Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type $\mathbb{C}^n\ltimes_\varphi N$ where $N$ is nilpotent. As an application, we compute the Bott-Chern cohomology of the complex parallelizable Nakamura manifold and of the completely-solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the $\partial\overline\partial$-Lemma is not strongly-closed under deformations of the complex structure.