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Bott–Chern cohomology of solvmanifolds

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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.

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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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  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Via Morgagni 67/A, 50134, Firenze, Italy

    Daniele Angella

  2. Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Japan

    Hisashi Kasuya

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  1. Daniele Angella
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  2. Hisashi Kasuya
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Correspondence to Daniele Angella.

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During the preparation of this work, the first author has been granted with a research fellowship by Istituto Nazionale di Alta Matematica INdAM, and he is supported by the Project PRIN “Varietá reali e complesse: geometria, topologia e analisi armonica,” by the Project FIRB “Geometria Differenziale e Teoria Geometrica delle Funzioni,” by the Project SIR2014 “Analytic aspects in complex and hypercomplex geometry” code RBSI14DYEB, and by GNSAGA of INdAM. The second author is supported by JSPS Research Fellowships for Young Scientists.

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Angella, D., Kasuya, H. Bott–Chern cohomology of solvmanifolds. Ann Glob Anal Geom 52, 363–411 (2017). https://doi.org/10.1007/s10455-017-9560-6

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