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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
Articles
  • 19 Downloads

Abstract

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.

Keywords

BCOV invariants Calabi-Yau manifolds singularities Arakelov inequalities 

MSC(2010)

55R55 14D06 14J32 14D07 14C40 

Notes

Acknowledgements

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.

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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

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