Skip to main content
SpringerLink
Log in

Cohomologies of Generalized Complex Manifolds and Nilmanifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fiber bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angella, D.: The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angella, D., Calamai, S., Latorre, A.: On cohomological decomposition of generalized-complex structures. J. Geom. Phys. 98, 227–241 (2015)

  3. Angella, D., Kasuya, H.: Cohomologies of deformations of solvmanifoldsand closedness of some properties, arXiv:1305.6709v2 [math.CV], to appear in Mathematica Universalis

  4. Angella, D., Kasuya, H.: Symplectic Bott–Chern cohomology of solvmanifolds, arXiv:1308.4258 [math.SG]

  5. Cavalcanti, G.R.: New aspects of the \(dd^c\)-lemma, Oxford University D. Phil thesis, arXiv:math/0501406v1 [math.DG]

  6. Cavalcanti, G.R.: The decomposition of forms and cohomology of generalized complex manifolds. J. Geom. Phys. 57(1), 121–132 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cavalcanti, G.R., Gualtieri, M.: Generalized complex structures on nilmanifolds. J. Symplectic Geom. 2(3), 393–410 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Console, S., Fino, A.: Dolbeault cohomology of compact nilmanifolds. Transform. Groups 6(2), 111–124 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Console, S.: Dolbeault cohomology and deformations of nilmanifolds. Rev. Un. Mat. Argentina 47(1), 51–60 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Gualtieri, M.: Generalized complex geometry. Ann. Math. (2) 174(1), 75–123 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hattori, A.: Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. 1960(8), 289–331 (1960)

    MathSciNet  MATH  Google Scholar 

  12. Hitchin, N.J.: Generalized Calabi–Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kasuya, H.: Differential Gerstenhaber algebras and generalized deformations of solvmanifolds. arXiv:math/1211.4188v3 [math.DG]

  14. McCleary, J.: A user’s guide to spectral sequences, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 58. Cambridge University Press, Cambridge (2001)

  15. Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. 59(3), 531–538 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  16. Raghunathan, M.S.: Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, New York (1972)

    Book  Google Scholar 

  17. Rollenske, S.: Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds. J. Lond. Math. Soc. 79(2), 346–362 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tsai, C.-J., Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge Theory on Symplectic Manifolds: III, arXiv:1402.0427v2 [math.SG]

  19. Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: I. J. Differ. Geom. 91(3), 383–416 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: II. J. Differ. Geom. 91(3), 417–443 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Tseng, L.-S., Yau, S.-T.: Generalized cohomologies and supersymmetry. Commun. Math. Phys. 326(3), 875–885 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Gil R. Cavalcanti, Sönke Rollenske, Giovanni Bazzoni, and Adela Latorre for useful discussions and for helpful comments on preliminary versions. During the preparation of the work, Daniele Angella has been granted by a research fellowship by Istituto Nazionale di Alta Matematica INdAM and by a Junior Visiting Position at Centro di Ricerca “Ennio de Giorgi”; he is also supported by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, by the Project FIRB “Geometria Differenziale e Teoria Geometrica delle Funzioni”, by SNS GR14 grant “Geometry of non-Kähler manifolds”, and by GNSAGA of INdAM. Simone Calamai is supported by GNSAGA of INdAM, by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, and by SNS GR14 grant “Geometry of non-Kähler manifolds”. Hisashi Kasuya is supported by JSPS Research Fellowships for Young Scientists.

Author information

Author notes

  1. Daniele Angella

    Present address: Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134, Firenze, Italy

Authors and Affiliations

  1. Centro di Ricerca Matematica “Ennio de Giorgi”, Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126, Pisa, Italy

    Daniele Angella

  2. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134, Firenze, Italy

    Simone Calamai

  3. Department of Mathematics, Tokyo Institute of Technology, 1-12-1-H-7, O-okayama, Meguro, Tokyo, 152-8551, Japan

    Hisashi Kasuya

Authors
  1. Daniele Angella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simone Calamai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hisashi Kasuya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Angella.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Angella, D., Calamai, S. & Kasuya, H. Cohomologies of Generalized Complex Manifolds and Nilmanifolds. J Geom Anal 27, 142–161 (2017). https://doi.org/10.1007/s12220-016-9677-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-016-9677-z

Keywords

Mathematics Subject Classification

Search

Navigation