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

Abstract

We study the behavior of the Quillen metric for families of Riemann surfaces with hyperbolic cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we’ve shown that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously over the locus of singular curves. Here we show that modulo some explicit universal constant, this continuous extension coincides with the Quillen metric of the normalization of singular curves. As a consequence, we get an explicit relation in terms of the Bott–Chern classes between the Quillen metric associated with a metric with cusps and the Quillen metric associated with a metric on the compactified Riemann surface. We also prove compatibility between our version of the analytic torsion and the version of Takhtajan–Zograf, defined through lengths of closed geodesics.