Revista Matemática Iberoamericana

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Volume 33, Issue 4, 2017, pp. 1309–1350
DOI: 10.4171/RMI/973

Published online: 2017-11-17

Complex structures of splitting type

Daniele Angella[1], Antonio Otal[2], Luis Ugarte[3] and Raquel Villacampa[4]

(1) Università degli Studi di Firenze, Italy
(2) Academia General Militar, Zaragoza, Spain
(3) Universidad de Zaragoza, Spain
(4) Academia General Militar, Zaragoza, Spain

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

Keywords: Complex structure, splitting type, solvmanifold, Hermitian metric, cohomology, $\partial\bar{\partial}$-manifold

Angella Daniele, Otal Antonio, Ugarte Luis, Villacampa Raquel: Complex structures of splitting type. Rev. Mat. Iberoam. 33 (2017), 1309-1350. doi: 10.4171/RMI/973