This paper classifies Hermitian structures on 6-dimensional nilmanifolds M = Γ\G for which the fundamental 2-form is ∂∂-closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limaçon-shaped curve in the… (More)
We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang in , in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain… (More)
Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G 2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G 2 structure. By suitably deforming the SU(3)… (More)
On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT manifold at a point or along a complex… (More)
Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in  for 2-step nil-manifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the… (More)
In this paper we provide examples of hypercomplex manifolds which do not carry HKT structure, thus answering a question in . We also prove that the existence of HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure with… (More)
We consider 5-manifolds with a contact form arising from a hypo structure , which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU (3)-structure. For half-flat manifolds with a Killing vector field X preserving the SU (3)-structure we study the geometry of the… (More)
The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups (g, J, Ω) such that the Ricci tensor is J-invariant.
We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized Kähler structure and which does not admit any Kähler metric. This is in contrast with the case of nilmanifolds which cannot admit any invariant generalized Kähler structure unless they are tori.
We show that a 7-dimensional non-compact Ricci-flat Riemannian man-ifold with Riemannian holonomy G2 can admit non-integrable G2 structures of type R ⊕ S 2 0 (R 7) ⊕ R 7 in the sense of Fernández and Gray. This relies on the construction of some G2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family… (More)