In these notes, we provide a summary of recent results on the cohomological properties of compact complex manifolds not endowed with a K hler structure.
On the one hand, the large number of developed analytic techniques makes it possible to prove strong cohomological properties for compact K hler manifolds. On the other, in order to further investigate any of these properties, it is natural to look for manifolds that do not have any K hler structure.
We focus in particular on studying Bott-Chern and Aeppli cohomologies of compact complex manifolds. Several results concerning the computations of Dolbeault and Bott-Chern cohomologies on nilmanifolds are summarized, allowing readers to study explicit examples. Manifolds endowed with almost-complex structures, or with other special structures (such as, for example, symplectic, generalized-complex, etc.), are also considered.