Motivic integration and the birational invariance of BCOV invariants

TL;DR

This paper proves the birational invariance of the BCOV invariant for Calabi--Yau manifolds and varieties, extending its construction to singular cases and interpreting it through motivic integration, thus advancing mirror symmetry and birational geometry.

Contribution

It establishes the birational invariance of the BCOV invariant for Calabi--Yau varieties with singularities and connects it to motivic integration, broadening its applicability.

Findings

BCOV invariant is birationally invariant for Calabi--Yau manifolds.

Extended the BCOV invariant to Calabi--Yau varieties with Kawamata log terminal singularities.

Provided a motivic integration interpretation of the BCOV invariant.

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.