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Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, 100048, China

    Kefeng Liu

  2. Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, 90095-1555, USA

    Kefeng Liu

  3. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Wei Xia

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  1. Kefeng Liu
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  2. Wei Xia
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Correspondence to Wei Xia.

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Liu, K., Xia, W. Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds. Sci. China Math. 62, 171–184 (2019). https://doi.org/10.1007/s11425-016-9241-9

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