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

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.Comment: 21 pages, no figure

Non-Kaehler Heterotic String Solutions with non-zero fluxes and non-constant dilaton

Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kaehler nilmanifold are presented. In the conformally compact case the dilaton is determined by the real slices of the elliptic Weierstrass function. The dilaton of non-compact complete solutions is given by the fundamental solution of the Laplacian on $R^4$.Comment: LaTeX 2e, 17 page

A six-dimensional compact symplectic solvmanifold without Kahler structures

International audienceThe purpose of this paper is to construct a compact symplectic (non-nilpotent) solvmanifold $M^{6} = \Gamma / G$ of dimension $6$ which does not admit Kähler structures. We show that the minimal model of $M^{6}$ is not formal by proving that there are non-trivial (quadruple) Massey products, however we remark that all the (triple) Massey products of $M^{6}$ vanish

Symplectically aspherical manifolds with nontrivial π2 and with no Kähler metrics.

In a previous paper, the authors show some examples of compact symplectic solvman-ifolds, of dimension six, which are cohomologically KÄahler 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