Cohomological aspects on complex and symplectic manifolds

In the Thesis we discuss quantitative and qualitative cohomological properties on complex and symplectic manifolds. A very special class of smooth manifolds is represented by Kaehler manifolds which are endowed with a complex structure, a metric structure and a symplectic structure which are compatible to each other. This fact implies many strong results, even at a topological level. Therefore one is led to study obstructions to the existence of a Kaehler metric and to weaken the geometric structures that are involved and/or to weaken their relations. A global tool to study this is provided by cohomology groups which are invariant by the considered geometric structures. In particular, we consider (almost-)complex manifolds, symplectic manifolds and locally-conformally symplectic manifolds. We discuss comparisons on the dimensions of suitable cohomology groups either of different manifolds related by structure-preserving, proper, surjective maps or on the same manifold discussing the relations with the d-delta-lemma (an important property in differential geometry). Moreover, we will study Hodge theory for such cohomology groups focusing also, in the complex setting, on the algebraic structure of the space of harmonic forms associated to the Bott-Chern cohomology.