Complex structures of splitting type

We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold $X$, and they allow us to construct a countable family of compact complex non-$\mathrm{\partial }\bar{\mathrm{\partial }}$ manifolds $X_k$, $k\in \mathbb{Z}$, that admit a small holomorphic deformation $\{(X_k{)}_t{\}}_{t\in {\mathrm{\Delta }}_k}$ satisfying the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma for any $t\in {\mathrm{\Delta }}_k$ except for the central fibre. Moreover, a study of the existence of special Hermitian metrics is also carried out on six-dimensional solvmanifolds with splitting-type complex structures.