Cohomology of Nilmanifolds

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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：151-232
• 全文大小：653KB
• 作者单位：Daniele Angella (13)
  
  13. Dipartimento di Matematica, Universitdi Pisa, Pisa, Italy
  
• ISSN：1617-9692

文摘

Nilmanifolds and solvmanifolds appear as oy-examplesin non-Khler geometry: indeed, on the one hand, non-tori nilmanifolds admit no Khler structure, (Benson and Gordon, Topology 27(4):51318, 1988; Lupton and Oprea, J. Pure Appl. Algebra 91(1):19307, 1994), and, more in general, solvmanifolds admitting a Khler structure are characterized, (Hasegawa, Proc. Am. Math. Soc. 106(1):651, 1989); on the other hand, the geometry and cohomology of solvmanifolds can be often reduced to study left-invariant geometry.In Sect. 3.1, it is shown that, for certain classes of complex structures on nilmanifolds (that is, compact quotients of connected simply-connected nilpotent Lie groups by co-compact discrete subgroups), the de Rham, Dolbeault, Bott-Chern, and Aeppli cohomologies are completely determined by the associated Lie algebra endowed with the induced linear complex structure, Theorem 3.6, giving a sort of result la Nomizu for the Bott-Chern cohomology. This will allow us to explicitly study the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, in Sect. 3.2, and of the complex structures on six-dimensional nilmanifolds in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa classification, (Ceballos et al., Classification of complex structures on 6-dimensional nilpotent Lie algebras, arXiv:1111.5873v3 [math.DG], 2011), in Sect. 3.3. Finally, in Appendix: Cohomology of Solvmanifolds, we recall some facts concerning cohomologies of solvmanifolds.