Cohomology of Complex Manifolds

In this chapter, we study cohomological properties of compact complex manifolds. In particular, we are concerned with studying the Bott-Chern cohomology, which, in a sense, constitutes a bridge between the de Rham cohomology and the Dolbeault cohomology of a complex manifold. In Sect. 2.1, we recall some definitions and results on the Bott-Chern and Aeppli cohomologies, see, e. g., Schweitzer (Autour de la cohomologie de Bott-Chern, arXiv: 0709. 3528 [math. AG], 2007), and on the Lemma, referring to Deligne et al. (Invent. Math. 29(3): 245-274, 1975). In Sect. 2.2, we provide an inequality a la Frolicher for the Bott-Chern cohomology, Theorem 2.13, which also allows to characterize the validity of the Lemma in terms of the dimensions of the Bott-Chern cohomology groups, Theorem 2.14; the proof of such inequality is based on two exact sequences, firstly considered by J. Varouchas in (Proprietes cohomologiques d'une classe de varietes analytiques complexes compactes, Seminaire d'analyse P. Lelong-P. Dolbeault-H. Skoda, annees 1983/1984, Lecture Notes in Math., vol. 1198, Springer, Berlin, 1986, pp. 233-243). Finally, in Appendix: Cohomological Properties of Generalized Complex Manifolds, we consider how to extend such results to the symplectic and generalized complex contexts.