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

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 $L^2$-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.