Cohomology of Complex Manifolds

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• 作者：Daniele Angella (13)
  
• 刊名：Lecture Notes in Mathematics
• 出版年：2014
• 出版时间：2014
• 年：2014
• 卷：2095
• 期：1
• 页码：95-150
• 全文大小：385KB
• 作者单位：Daniele Angella (13)
  
  13. Dipartimento di Matematica, Universitdi Pisa, Pisa, Italy
  
• ISSN：1617-9692

文摘

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):24574, 1975). In Sect. 2.2, we provide an inequality la Frlicher 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 (Propri cohomologiques dne classe de vari analytiques complexes compactes, Sinaire dnalyse P. Lelong-P. Dolbeault-H. Skoda, anns 1983/1984, Lecture Notes in Math., vol. 1198, Springer, Berlin, 1986, pp. 23343). Finally, in Appendix: Cohomological Properties of Generalized Complex Manifolds, we consider how to extend such results to the symplectic and generalized complex contexts.