BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures

TL;DR

This paper extends the BCOV invariant to Calabi--Yau manifolds of any dimension, analyzing its behavior at degenerations of Hodge structures to connect complex geometry with mirror symmetry and enumerative geometry.

Contribution

It introduces the BCOV invariant for higher-dimensional Calabi--Yau manifolds and establishes its asymptotic behavior at degenerations using new results on Hodge bundle metrics.

Findings

Derived precise asymptotics of BCOV invariants at degenerations

Connected BCOV invariants to topological data and intersection theory

Extended Fang--Lu--Yoshikawa's work to arbitrary dimensions

Abstract

Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--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--Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of its behaviour at the boundary of moduli spaces is imperative. We address this problem by proving precise asymptotics along one-parameter degenerations, in terms of topological data and intersection theory. Central to the approach are new results on degenerations of $L^2$ metrics on Hodge bundles, combined with information on the singularities of Quillen metrics in our previous work.