On cohomological properties of non-Kaehler manifolds

A compact Kaehler manifold always admits a decomposition of the real de
Rham cohomology in subgroups related to the bigraded representatives;
we study the class of almost-complex manifolds having a similar
decomposition: for example, a nice result by T. Draghici, T.-J. Li
and W. Zhang states that such a decomposition always exists on a compact
almost-complex $4$-manifold. This problem is connected to the study of the
relations between the tamed and compatible symplectic cones on a compact
almost-complex manifold. In particular, we will present some results,
obtained in a joint work with Adriano Tomassini, about the relation
between the balanced and strongly-Gauduchon cones and the deformation
properties of the manifolds admitting such a decomposition.