Singularities of metrics on Hodge bundles and their topological invariants

TL;DR

This paper investigates the singularities of various metrics on Hodge bundles during degenerations of Calabi-Yau varieties, linking these singularities to topological invariants like vanishing cycles and log-canonical thresholds.

Contribution

It provides a detailed analysis of the asymptotic behavior of L^2, Quillen, and BCOV metrics near singular fibers, connecting metric singularities to topological invariants.

Findings

Singularities of metrics relate to topological invariants such as limit Hodge structures.

Asymptotic expansions of metrics reveal dominant and subdominant terms linked to singularity invariants.

Results extend to more general degenerating families for the Quillen metric.

Abstract

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.