Science China Mathematics

, Volume 62, Issue 1, pp 171–184 | Cite as

Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds

  • Kefeng Liu
  • Wei XiaEmail author


For a one parameter family of Calabi-Yau threefolds, Green et al. (2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV (Bershadsky-Cecotti-Ooguri-Vafa) invariants, studied by Fang et al. (2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.


BCOV invariants Calabi-Yau manifolds singularities Arakelov inequalities 


55R55 14D06 14J32 14D07 14C40 



This work was supported by National Natural Science Foundation of China (Grant No. 11531012). The second author thanks Professors Carlos Simpson and Matt Kerr for helpful discussions. The authors thank the referees for carefully reading the manuscript and their valuable comments.


  1. 1.
    Bershadsky M, Cecotti S, Ooguri H, et al. Holomorphic anomalies in topological field theories. Nuclear Phys B, 1993, 405: 279–304MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bershadsky M, Cecotti S, Ooguri H, et al. Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Comm Math Phys, 1994, 165: 311–427MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bismut J-M, Bost J-B. Fibrés déterminants, métriques de Quillen et dégénérescence des courbes. Acta Math, 1990, 165: 1–103MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bismut J-M, Gillet H, Soulé C. Analytic torsion and holomorphic determinant bundles I: Bott-Chern forms and analytic torsion. Comm Math Phys, 1988, 115: 49–78MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bismut J-M, Gillet H, Soulé C. Analytic torsion and holomorphic determinant bundles II: Direct images and Bott-Chern forms. Comm Math Phys, 1988, 115: 79–126MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bismut J-M, Gillet H, Soulé C. Analytic torsion and holomorphic determinant bundles III: Quillen metrics on holomorphic determinants. Comm Math Phys, 1988, 115: 301–351MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cattani E, Kaplan A, Schmid W. Degeneration of Hodge structures. Ann of Math (2), 1986, 123: 457–535MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheeger J. Analytic torsion and the heat equation. Ann of Math (2), 1979, 109: 1–21MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eriksson D, Freixas G, Mourougane C. Singularities of metrics on Hodge bundles and their topological invariants. J Algebraic Geom, 2018, in pressGoogle Scholar
  10. 10.
    Fang H, Lu Z. Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli. J Reine Angew Math, 2005, 588: 49–69MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fang H, Lu Z, Yoshikawa K-I. Analytic torsion for Calabi-Yau threefolds. J Differential Geom, 2008, 80: 175–259MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Green M, Griffths P, Kerr M. Some enumerative global properties of variations of Hodge structures. Mosc Math J, 2009, 9: 469–530MathSciNetzbMATHGoogle Scholar
  13. 13.
    Griffths P. Periods of integrals on algebraic manifolds. II: Local study of the period mapping. Amer J Math, 1968, 90: 805–865Google Scholar
  14. 14.
    Griffths P. Topics in Transcendental Algebraic Geometry. Princeton: Princeton University Press, 1984Google Scholar
  15. 15.
    Griffths P, Schmid W. Locally homogeneous complex manifolds. Acta Math, 1969, 123: 253–302MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hunt B. A bound on the Euler number for certain Calabi-Yau 3-folds. J Reine Angew Math, 1990, 411: 137–170MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kulikov V S, Kurchanov P F. Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians. New York: Springer, 1998CrossRefGoogle Scholar
  18. 18.
    Landman A. On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities. Trans Amer Math Soc, 1973, 181: 89–126MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Liu K. Geometric height inequalities. Math Res Lett, 1996, 3: 693–702MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu K. Remarks on the geometry of moduli spaces. Proc Amer Math Soc, 1996, 124: 689–695MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lu Z. On the Hodge metric of the universal deformation space of Calabi-Yau threefolds. J Geom Anal, 2001, 11: 103–118MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lu Z, Sun X. Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds. J Inst Math Jussieu, 2004, 3: 185–229MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Müller W. Analytic torsion and R-torsion of Riemannian manifolds. Adv Math, 1978, 28: 233–305MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Peters C. Arakelov-type inequalities for Hodge bundles. ArXiv:0007102v1, 2000Google Scholar
  25. 25.
    Ray D B, Singer I M. R-torsion and the Laplacian on Riemannian manifolds. Adv Math, 1971, 7: 145–210MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ray D B, Singer I M. Analytic torsion for complex manifolds. Ann of Math (2), 1973, 98: 154–177MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schmid W. Variation of Hodge structure: The singularities of the period mapping. Invent Math, 1973, 22: 211–319MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tian G. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric. In: Mathematical Aspects of String Theory. Advanced Series in Mathematical Physics, vol. 1. Singapore: World Scientific, 1987, 629–646CrossRefGoogle Scholar
  29. 29.
    Todorov A N. The Weil-Petersson geometry of the moduli space of SU(≥ 3) (Calabi-Yau) manifolds i. Comm Math Phys, 1989, 126: 325–346MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tosatti V, Zhang Y. Triviality of fibered Calabi-Yau manifolds without singular fibers. Math Res Lett, 2013, 21: 905–918MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Viehweg E. Quasi-Projective Moduli for Polarized Manifolds. Berlin: Springer, 1995CrossRefzbMATHGoogle Scholar
  32. 32.
    Viehweg E, Zuo K. Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds. J Algebraic Geom, 2005, 15: 771–791MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang C. Curvature properties of the Calabi-Yau moduli. Doc Math, 2003, 8: 577–590MathSciNetzbMATHGoogle Scholar
  34. 34.
    Yau S-T. A general Schwarz lemma for Kähler manifolds. Amer J Math, 1978, 100: 197–203MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yau S-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation, I. Comm Pure Appl Math, 1978, 31: 339–411MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yoshikawa K-I. Analytic torsion for Borcea-Voisin threefolds. ArXiv:1410.0212v2, 2014zbMATHGoogle Scholar
  37. 37.
    Yoshikawa K-I. Degenerations of Calabi-Yau threefolds and BCOV invariants. Internat J Math, 2014, 26: 217–250MathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCapital Normal UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA
  3. 3.Center of Mathematical SciencesZhejiang UniversityHangzhouChina

Personalised recommendations