Cohomology of Nilmanifolds

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


Nilmanifolds and solvmanifolds appear as “toy-examples” in non-Kähler geometry: indeed, on the one hand, non-tori nilmanifolds admit no Kähler structure, (Benson and Gordon, Topology 27(4):513–518, 1988; Lupton and Oprea, J. Pure Appl. Algebra 91(1–3):193–207, 1994), and, more in general, solvmanifolds admitting a Kähler structure are characterized, (Hasegawa, Proc. Am. Math. Soc. 106(1):65–71, 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 à 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’s 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.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

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