Degenerating Riemann Surfaces and the Quillen Metric

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.