Complex structures of splitting type

TL;DR

This paper classifies six-dimensional solvmanifolds with splitting type complex structures, constructs new non-$ ext{}ar{ ext{}}$-lemma manifolds, and investigates special Hermitian metrics on these structures.

Contribution

It provides a classification of complex structures of splitting type on six-dimensional solvmanifolds and constructs new examples with specific deformation properties.

Findings

Existence of many splitting-type complex structures on Nakamura manifold

Construction of a countable family of non-$ ext{}ar{ ext{}}$-lemma manifolds

Analysis of Hermitian metrics on these solvmanifolds

Abstract

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-$\partial\overline\partial$ manifolds $X_k$, $k\in\mathbb{Z}$, that admit a small holomorphic deformation $\{(X_{k})_{t}\}_{t\in\Delta_k}$ satisfying the $\partial\overline\partial$-Lemma for any $t\in\Delta_k$ 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.