José A. Santisteban

COHOMOLOGICALLY KÄHLER MANIFOLDS WITH NO KÄHLER METRICS

We show some examples of compact symplectic solvmanifolds, of dimension greater than four, which are cohomologically Kahler and do not admit Kahler metric since their fundamental groups cannot be the fundamental group of any compact Kahler manifold. Some of the examples that we study were considered by Benson and Gordon (1990). However, whether such manifolds have Kahler metrics was an open question. The formality and the hard Lefschetz property are studied for the symplectic submanifolds constructed by Auroux (1997) and some consequences are discussed.

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting

On seven-dimensional quaternionic contact solvable Lie groups

{\mathfrak{g}^*} = {V_1} \oplus {V_2}

SYMPLECTICALLY ASPHERICAL MANIFOLDS WITH NONTRIVIAL π2 AND WITH NO KÄHLER METRICS

, and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.

Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures

We answer in the affirmative a question posed by Ivanov and Vassilev [13] on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.

Explicit Quaternionic Contact Structures and Metrics with Special Holonomy

In a previous paper, the authors show some examples of compact symplectic solvman-ifolds, of dimension six, which are cohomologically KAahler and they do not admit Kahler metrics because their fundamental groups cannot be the fundamental group of any compact Kahler manifold. Here we generalize such manifolds to higher dimension and, by using Au-roux symplectic submanifolds [3], we construct four-dimensional symplectically aspherical manifolds with nontrivial ¼2 and with no Kahler metrics.