Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem

TL;DR

This paper investigates how the Quillen metric behaves under degeneration of cusps in Riemann surfaces, establishing its compatibility with singularities, moduli space morphisms, and relating it explicitly to compactified metrics.

Contribution

It proves that the Quillen metric extends continuously over singular curves and coincides with the normalized curve's Quillen metric, refining previous results with explicit constants.

Findings

Quillen metric extends continuously over singular curves.

Compatibility of Quillen metric with clutching morphisms in moduli space.

Explicit relation between cusped and compactified Quillen metrics via Bott-Chern forms.

Abstract

We study the behavior of the Quillen metric for the family of Riemann surfaces with cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we've seen that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously over the locus of singular curves. The main result of this article shows that, modulo some explicit universal constant, this continuous extension coincides with the Quillen metric of the normalization of singular curves. This result shows that the Quillen metric is compatible with the adjunction of cusps. When this theorem is applied directly to the moduli space of curves, we obtain the compatibility of the Quillen metric with clutching morphisms in the moduli space of pointed stable curves. As one application, we obtain the compatibility between our definition of the analytic torsion and the definition of Takhtajan-Zograf using lengths of closed geodesics. As a consequence of the proof of the main theorem, we get an explicit relation in terms of Bott-Chern forms between the Quillen metric associated with a cusped metric and the Quillen metric associated with a metric on the compactified Riemann surface. This refines relative compact perturbation theorem we obtained before by pinning down the universal constant.