Bott-Chern cohomology of solvmanifolds

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.