Motivic integration and the birational invariance of BCOV invariants

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 the 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 the 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.

arXiv commentFinal version to appear in Selecta Mathematica

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Cited by in corpus (2)

• On genus one mirror symmetry in higher dimensions and the BCOV conjectures
• Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $t{t}^{\ast }$ Structures