Chern-Weil theory for line bundles with the family Arakelov metric

TL;DR

This paper extends Chern-Weil theory to canonically metrized line bundles in one-parameter families of smooth complex curves, generalizing previous results related to Jacobi forms and modular curves using b-divisors.

Contribution

It introduces a generalized Chern-Weil type result for line bundles with the family Arakelov metric, broadening the scope of previous specific cases.

Findings

Proves a Chern-Weil type formula for canonically metrized line bundles.

Generalizes previous results on Jacobi forms and modular curves.

Utilizes the notion of b-divisors as a key tool.

Abstract

We prove a result of Chern-Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J.I. Burgos Gil, J. Kramer and U. K\"uhn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full level structure, equipped with the Petersson metric. Our main tool, as in the work by Burgos Gil, Kramer and K\"uhn, is the notion of a b-divisor.