Skip to main content
SpringerLink
Log in

On the Weil-Petersson Volume and the First Chern Class of the Moduli Space of Calabi-Yau Manifolds

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weli-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.

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. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165(2), 311–427 (1994)

    MathSciNet  Google Scholar 

  2. Doran, C., Morgan, J.: Integral monodromy and Calabi-Yau Moduli. Preprint

  3. Fang, H., Lu, Z.: Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli. To appear Journal fur die reine und angewandte Mathematik, http://arxiv.org/list/math.DG/031007, 2003

  4. Griffiths, P., ed.: Topics in transcendental algebraic geometry, Volume 106 of Annals of Mathematics Studies, Princeton, NJ: Princeton University Press, 1984

  5. Jost, J., Yau, S.-T.: Harmonic mappings and algebraic varieties over function fields. Amer. J. Math. 115(6), 1197–1227 (1993)

    MathSciNet  Google Scholar 

  6. Kawamata, Y.: Characterization of abelian varieties. Compositio Math. 43(2), 253–276 (1981)

    MathSciNet  Google Scholar 

  7. Li, P.: Curvature and function theory on Riemannian manifolds. In: Surveys in differential geometry, VII, Somerville, MA: Int. Press, 2000, pp. 375–432

  8. Lu, Z.: On the geometry of classifying spaces and horizontal slices. Amer. J. Math. 121(1), 177–198 (1999)

    MathSciNet  Google Scholar 

  9. Lu, Z.: On the Hodge metric of the universal deformation space of Calabi-Yau threefolds. J. Geom. Anal. 11(1), 103–118 (2001)

    MathSciNet  Google Scholar 

  10. Lu, Z., Sun, X.: Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds. J. Inst. Math. Jussieu 3(2), 185–229 (2004)

    MathSciNet  Google Scholar 

  11. Mumford, D.: Hirzebruch's proportionality theorem in the noncompact case. Invent. Math. 42, 239–272 (1977)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Schumacher, G.: The curvature of the Petersson-Weil metric on the moduli space of Kähler-Einstein manifolds. In: Complex analysis and geometry, Univ. Ser. Math., New York: Plenum, 1993, pp. 339–354

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

  15. Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: Mathematical aspects of string theory (San Diego, Calif., 1986), Volume 1 of Adv. Ser. Math. Phys., Singapore: World Sci. Publishing, 1987, pp. 629–646

  16. Todorov, A.: Weil-petersson volumes of the moduli spaces of CY manifolds pp. 629–646. http://arxiv.org/list/hep-th/0408033, 2004

  17. Todorov, A.N.: Introduction to Weil-Petersson Geometry of the moduli space of CY manifolds. Preprint

  18. Todorov, A.N.: The Weil-Petersson geometry of the moduli space of SU(n≥ 3) (Calabi-Yau) manifolds. I. Commun. Math. Phys. 126(2), 325–346 (1989)

    Google Scholar 

  19. Trapani, S.: On the determinant of the bundle of meromorphic quadratic differentials on the Deligne-Mumford compactification of the moduli space of Riemann surfaces. Math. Ann. 293(4), 681–705 (1992)

    MathSciNet  Google Scholar 

  20. Viehweg, E.: Quasi-projective moduli for polarized manifolds. Volume 30 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Berlin: Springer-Verlag 1995

  21. Wang, C.-L.: On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds. Math. Res. Lett. 4(1), 157–171 (1997)

    ADS  MathSciNet  Google Scholar 

  22. Yau, S.T.: A general Schwarz lemma for Kähler manifolds. Amer. J. Math. 100(1), 197–203 (1978)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Zhiqin Lu & Xiaofeng Sun

Authors
  1. Zhiqin Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofeng Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiqin Lu.

Additional information

Communicated by P. Sarnak

The first author is supported by NSF CareerAward DMS 0347033 and an Alfred P. Sloan Fellowship.

The second author is supported by NSF grant DMS 0202508.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lu, Z., Sun, X. On the Weil-Petersson Volume and the First Chern Class of the Moduli Space of Calabi-Yau Manifolds. Commun. Math. Phys. 261, 297–322 (2006). https://doi.org/10.1007/s00220-005-1441-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1441-3

Keywords

Search

Navigation