Skip to main content
SpringerLink
Log in

Holomorphic deformations of the Ricci-flat \(\partial \overline \partial \)-manifolds

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat \(\partial \overline \partial \)-manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat \(\partial \overline \partial \)-manifolds.

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. Ahlfors, L.: Lectures on quasiconformal mappings, 2nd edn. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard, University Lecture Series, vol. 38, American Mathematical Society, Providence, 2006

    Book  MATH  Google Scholar 

  2. Clemens, H.: Geometry of formal kuranishi theory. Adv. Math., 198, 311–365 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deligne, P., Griffiths, P., Morgan, J., et al.: Real homotopy theory of Kähler manifolds. Invent. Math., 29 3, 245–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Demailly, J.-P.: Complex analytic and algebraic geometry. http://www-fourier.ujf-grenoble.fr/demailly /books/html (2012)

    MATH  Google Scholar 

  5. Fujiki, A.: On automorphism groups of compact Kähler manifolds. Invent. Math., 44 3, 225–258 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Griffiths, P.: Periods of integrals on algebraic manifolds. ii. local study of the period mapping. Amer. J. Math., 90, 805–865 (1968)

    MATH  Google Scholar 

  7. Liu, K.-F., Rao, S., Yang, X.-K.: Quasi-isometry and deformations of Calabi–Yau manifolds. Invent. Math., 199 2, 423–453 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Morrow, J., Kodaira, K.: Complex manifolds. Holt, Rinehart and Winston Inc., New York, 1971

    MATH  Google Scholar 

  9. Nijenhuis, A., Woolf, W.-B.: Some integration problems in almost-complex and complex manifolds. Annals of Math., 424–489 (1963)

  10. Popovici, D.: Holomorphic deformations of balanced Calabi–Yau -manifolds, arXiv: 1304.0331[math.AG]

  11. Siu, Y.-T.: Curvature of the Weil–Petersson metric in the moduli space of compact Kähler–Einstein manifolds of negative first Chern class. Contributions to several complex variables. In: Aspects Mathematics, vol. E9, Vieweg, Braunschweig, 1986, 261–298

    Google Scholar 

  12. Tian, G.: Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric. Mathematical aspects of string theory (San Diego, California, 1986). In: Advanced Series in Mathematical Physics, vol. 1, World Scientific Publishing, Singapore, 1987, 629–646

    Google Scholar 

  13. Todorov, A.: The Weil–Petersson geometry of the moduli space of SU(n = 3) (Calibi–Yau) manifolds I. Commun. Math. Phys., 126 2, 325–346 (1989)

    Article  MATH  Google Scholar 

  14. Tosatti, V.: Non-Kähler Calabi–Yau manifolds. Analysis, Complex Geometry, and Mathematical Physics: In Honor of Duong H. Phong 644, 261–277 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Varouchas, J.: Käler spaces and proper open morphisms. Math. Ann., 283 1, 13–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wu, C.-C.: On the Geometry of Superstrings with Torsion-thesis, Department of Mathematics, Harvard University, Cambridge MA 02138, April 2006

    Google Scholar 

  17. Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann., 321, 817–884 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, P. R. China

    Hai Sheng Liu

Authors
  1. Hai Sheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hai Sheng Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H. Holomorphic deformations of the Ricci-flat \(\partial \overline \partial \)-manifolds. Acta. Math. Sin.-English Ser. 32, 643–658 (2016). https://doi.org/10.1007/s10114-016-5201-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-016-5201-4

Keywords

MR(2010) Subject Classification

Search

Navigation