On the behavior of analytic torsion for twisted canonical bundles under degenerations

TL;DR

This paper studies how the analytic torsion behaves in degenerating families of algebraic manifolds with group actions, revealing asymptotic expansions and explicit formulas for key coefficients.

Contribution

It establishes the asymptotic expansion of equivariant analytic torsion near degenerations and provides formulas for leading terms using characteristic classes.

Findings

Logarithmic and loglog-type singularities in torsion asymptotics

Explicit formula for the leading coefficient in non-equivariant case

Existence of asymptotic expansions for Quillen and $L^{2}$-metrics

Abstract

Consider a degeneration of projective algebraic manifolds equipped with a compact group action over a curve. Suppose that the total space carries a Nakano semi-positive vector bundle, which is equivariant with respect to this action. We consider the relative canonical bundle twisted by this bundle. Under this setting, we prove that the logarithm of the equivariant analytic torsion of the regular fibers for this coefficient admits an asymptotic expansion near the discriminant locus. The leading term is given by a logarithmic singularity, while the subdominant term is given by a loglog-type singularity. In the non-equivariant case, we provide a formula for the coefficient of the leading term in terms of an integral of characteristic classes associated with the semi-stable reduction of the family. To establish these results, we prove the existence of an asymptotic expansion for both the equivariant Quillen metrics and the $L^{2}$-metrics in the above setting. We also calculate the leading term of the fiber integral of the Bott-Chern classes associated with the degeneration.