Hermitian structures on six dimensional nilmanifolds

Let (J,g) be a Hermitian structure on a compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We give classifications of 6-dimensional nilmanifolds M admitting strong K\"ahler with torsion, balanced or locally conformal K\"ahler structures (J,g).Comment: LaTeX, 24 pages, 1 figur

Symplectic harmonicity and generalized coeffective cohomologies

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by C.-J. Tsai, L.-S. Tseng and S.-T. Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms $\omega_t$ such that the dimensions of these symplectic cohomology groups vary with respect to $t$. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifolds, and concrete examples with special cohomological properties are obtained on an $8$-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.Comment: 25 pages; revised version, new Theorem 5.7 and Section 8 added, references update

On the geometry underlying supersymmetric flux vacua with intermediate SU(2) structure

We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in certain sense near to the balanced condition. Using this result we show that compact simply connected manifolds do not admit type IIB solutions. New solutions of the SUSY equations IIA and IIB are constructed from hyperk\"ahler 4-manifolds, special hypo 5-manifolds and 6-dimensional solvmanifolds.Comment: 21 pages; to be published in Classical Quantum Gravit

On the Strominger system and holomorphic deformations

We show that the property of existence of solution to the Strominger system in dimension six is neither open nor closed under holomorphic deformations of the complex structure. These results are obtained both in the case of positive slope parameter as well as in the case of negative slope parameter in the anomaly cancellation equation

Weakly Lefschetz symplectic manifolds

The harmonic cohomology of a Donaldson symplectic submanifold and of an Auroux symplectic submanifold are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the $s$-Lefschetz propery. In particular, we consider the symplectic blow-ups of the complex projective space along weakly Lefschetz symplectic submanifolds. As an application we construct, for each even integer $s\geq 2$, compact symplectic manifolds which are $s$-Lefschetz but not $(s+1)$-Lefschetz.Comment: 22 pages; many improvements from previous versio

On the real homotopy type of generalized complex nilmanifolds

We prove that for any $n\geq 4$ there are infinitely many real homotopy types of $2n$-dimensional nilmanifolds admitting generalized complex structures of every type $k$, for $0 \leq k \leq n$. This is in deep contrast to the $6$-dimensional case.Comment: 11 pages; observation added concerning higher dimensions; change of title and abstrac