Cohomology of Complex Manifolds

Abstract

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 $$\partial \overline{\partial }$$ -Lemma, referring to Deligne et al. (Invent. Math. 29(3):245–274, 1975). In Sect. 2.2, we provide an inequality à la Frölicher for the Bott-Chern cohomology, Theorem 2.13, which also allows to characterize the validity of the $$\partial \overline{\partial }$$ -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 (Propriétés cohomologiques d’une classe de variétés analytiques complexes compactes, Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 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.