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Motivic integration and birational invariance of BCOV invariants

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Abstract

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi–Yau manifolds, which is now called the BCOV torsion. Based on it, a metric-independent invariant, called BCOV invariant, was constructed by Fang–Lu–Yoshikawa and Eriksson–Freixas i Montplet–Mourougane. The BCOV invariant is conjecturally related to the Gromov–Witten theory via mirror symmetry. Based upon previous work of the second author, we prove the conjecture that birational Calabi–Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi–Yau varieties with Kawamata log terminal singularities, and prove its birational invariance for Calabi–Yau varieties with canonical singularities. We provide an interpretation of our construction using the theory of motivic integration.

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Notes

  1. Here we use the definition of [31], which differs from the one in [34] by an explicit power of \(2\pi \).

  2. i.e., their bounded derived categories of coherent sheaves are equivalent as \({\mathbb {C}}\)-linear triangulated categories. Note that the derived equivalence of birational Calabi–Yau threefolds was proved by Bridgeland [15, Theorem 1.1] and there are derived equivalent Calabi–Yau threefolds that are not birationally equivalent [14, 27, 56].

  3. The terminology refers to the fact that if \(\chi (X)\ne 0\), then \(\frac{\phi ({\mathbb {P}}(V))}{\chi ({\mathbb {P}}(V))}=\frac{\phi (X)}{\chi (X)}+\frac{\phi ({\mathbb {C}}\textrm{P}^{r-1})}{\chi ({\mathbb {C}}\textrm{P}^{r-1})}\).

  4. It is usually denoted by \(Z(X, {\mathcal {I}}_{D}; T)\), where \({\mathcal {I}}_{D}={\mathcal {O}}_{X}(-D)\) is the ideal sheaf of D.

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Acknowledgements

We are indebted to Professor Ken-Ichi Yoshikawa for his guidance throughout the project. We would like to thank Professor Xianzhe Dai, Professor Dennis Eriksson, Professor Gerard Freixas i Montplet, Professor Chen Jiang, Professor Xiaonan Ma, and Professor Vincent Maillot for helpful discussions. Lie Fu is supported by the Radboud Excellence Initiative from the Radboud University, by the project FanoHK (ANR-20-CE40-0023) of Agence Nationale de la Recherche in France, and by the University of Strasbourg Institute for Advanced Study (USIAS). Yeping Zhang is supported by KIAS individual Grant MG077401 at Korea Institute for Advanced Study.

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    Lie Fu

  2. Shanghai, China

    Yeping Zhang

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Fu, L., Zhang, Y. Motivic integration and birational invariance of BCOV invariants. Sel. Math. New Ser. 29, 25 (2023). https://doi.org/10.1007/s00029-023-00832-3

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