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-∂∂¯¯¯ manifolds Xk, k∈Z, that admit a small holomorphic deformation {(Xk)t}t∈Δk satisfying the ∂∂¯¯¯-Lemma for any t∈Δk 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.