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15 February 2021 BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures
Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane
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Duke Math. J. 170(3): 379-454 (15 February 2021). DOI: 10.1215/00127094-2020-0045

Abstract

Calabi–Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky, Cecotti, Ooguri, and Vafa (BCOV), it is expected that genus 1 curve-counting on a Calabi–Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray–Singer holomorphic analytic torsions. To this end, extending work of Fang, Lu, and Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi–Yau manifolds of arbitrary dimension. To determine it, knowledge of its behavior at the boundary of moduli spaces is imperative. To address this problem, we prove general results on degenerations of L2-metrics on Hodge bundles and their determinants, refining the work of Schmid. We express the singularities of these metrics in terms of limiting Hodge structures and derive consequences for the dominant and subdominant singular terms of the BCOV invariant.

Acknowledgments

We want to express our wholehearted gratitude to Ken-Ichi Yoshikawa, for many discussions and criticisms on these topics and for his generous explanations of his own work in this field. Many of these conversations occurred at Kyoto University, which we thank for its ample hospitality. We moreover thank the Centre Emile Borel at the Institute Henri Poincaré for financial support and reception during the Research in Paris program “Secondary Invariants in Mirror Symmetry” in 2018.

The first author is also grateful to IRMAR for financial support in Rennes during a one-month stay in 2018. He also wants to extend his thanks to the Knut and Alice Wallenberg Foundation and the Göran Gustafsson Foundation for their travel support that made several visits possible. The second author was partially supported by Agence Nationale de la Recherche grant ANR-18-CE40-0017 (PERGAMO) and the Partenariat Hubert Curien–Sakura. We finally thank the two anonymous referees, whose constructive comments helped us improve the article.

Citation

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Dennis Eriksson. Gerard Freixas i Montplet. Christophe Mourougane. "BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures." Duke Math. J. 170 (3) 379 - 454, 15 February 2021. https://doi.org/10.1215/00127094-2020-0045

Information

Received: 1 May 2019; Revised: 15 January 2020; Published: 15 February 2021
First available in Project Euclid: 23 December 2020

Digital Object Identifier: 10.1215/00127094-2020-0045

Subjects:
Primary: 14J32
Secondary: 32G20 , 58J52 , 58K55 , 58K65

Keywords: analytic torsion , Calabi-Yau manifolds , degenerations of Hodge structures , mathematical mirror symmetry

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 3 • 15 February 2021
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