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Cohomology of Complex Manifolds

  • Daniele Angella
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2095)

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.

Keywords

Exact Sequence Spectral Sequence Complex Manifold Cohomological Property Hodge Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Daniele Angella
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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