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Annals of Global Analysis and Geometry

, Volume 52, Issue 4, pp 363–411 | Cite as

Bott–Chern cohomology of solvmanifolds

Article

Abstract

We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott–Chern cohomology. We are especially aimed at studying the Bott–Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott–Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type \(\mathbb {C}^n\ltimes _\varphi N\) where N is nilpotent. As an application, we compute the Bott–Chern cohomology of the complex parallelizable Nakamura manifold and of the completely solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the \(\partial \overline{\partial }\)-Lemma is not strongly closed under deformations of the complex structure.

Keywords

Dolbeault cohomology Bott–Chern cohomology Solvmanifolds Invariant complex structure 

Mathematics Subject Classification

22E25 53C30 57T15 53C55 53D05 

Notes

Acknowledgements

The first author would like to warmly thank Adriano Tomassini for his constant support and encouragement, for his several advices, and for many inspiring conversations. The second author would like to express his gratitude to Toshitake Kohno for helpful suggestions and stimulating discussions. The authors would like also to thank Luis Ugarte for suggestions and remarks. Thanks also to Maria Beatrice Pozzetti and to the anonymous Referees, whose suggestions improved the presentation of the paper.

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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “Ulisse Dini”Università di FirenzeFirenzeItaly
  2. 2.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsakaJapan

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