Degenerating Riemann surfaces and the Quillen metric

TL;DR

This paper offers a geometric interpretation of the degeneration of the Quillen metric on Riemann surfaces, connecting it with Deligne's Riemann-Roch theorem and monodromy eigenvalues, extending previous results.

Contribution

It introduces a geometric perspective on Quillen metric degeneration using Deligne's Riemann-Roch theorem, generalizing earlier work by Bismut-Bost and Yoshikawa.

Findings

Interpretation of the singular part as a discriminant

Continuous part as a degeneration of Deligne product metrics

Asymptotic development involving monodromy eigenvalues

Abstract

The degeneration of the Quillen metric for a one-parameter family of Riemann surfaces has been studied by Bismut-Bost and Yoshikawa. In this article we propose a more geometric point of view using Deligne's Riemann-Roch theorem. We obtain an interpretation of the singular part of the metric as a discriminant and the continuous part as a degeneration of the metric on Deligne products, which gives an asymptotic development involving the monodromy eigenvalues. This generalizes the results of Bismut-Bost and is a version of Yoshikawa's results on the degeneration of the Quillen metric for general degenerations with isolated singularities in the central fiber.