Chern-Weil and Hilbert-Samuel formulae for singular hermitian line bundles

TL;DR

This paper extends classical Chern-Weil and Hilbert-Samuel formulas to singular hermitian line bundles on complex varieties, utilizing b-divisors and multiplier ideal volumes, with applications to Siegel-Jacobi forms.

Contribution

It introduces a generalized framework for singular metrics on line bundles, expanding the scope of classical formulas using advanced tools like b-divisors and multiplier ideal volumes.

Findings

Established a Chern-Weil type statement for singular metrics.

Derived a Hilbert-Samuel formula for singular hermitian line bundles.

Applied results to Siegel-Jacobi forms over universal abelian varieties.

Abstract

We show a Chern-Weil type statement and a Hilbert-Samuel formula for a large class of singular plurisubharmonic metrics on a line bundle over a smooth projective complex variety. For this we use the theory of b-divisors and the so-called multiplier ideal volume function. We apply our results to the line bundle of Siegel-Jacobi forms over the universal abelian variety endowed with its canonical invariant metric. This generalizes a result of the second author for the universal family of elliptic curves to higher degrees.