In these notes, we provide a summary of recent results on the cohomological properties of compact complex manifolds not endowed with a KAchler structure. On the one hand, the large number of developed analytic techniques makes it possible to prove strong cohomological properties for compact KAchler 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 KAchler 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.For an arbitrary G-left-invariant complex structure on a nilmanifold X D nG, it is not known whether iWH ... or rational, one can construct a series of exact sequences of Lie algebras being compatible with both the rational and complex structures.
|Title||:||Cohomological Aspects in Complex Non-Kähler Geometry|
|Publisher||:||Springer - 2013-11-22|