Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem

Abstract

We study the Quillen metric on the determinant line bundle associated with a family of complex singular curves with hyperbolic cusp singularities.More precisely, we fix a family of complex curves, which admit at most double-point singularities. We endow the fibers of this family with Kähler metrics, which are defined away from a finite set of points, a divisor on the total space of the family, in the vicinity of which the metric has Poincaré-type singularities. We fix a holomorphic vector bundle over the total space of the family and endow it with a Hermitian metric with at most logarithmic singularities on the divisor, coming from the power of the relative canonical line bundle twisted by the divisor. An important example is the family of stable surfaces with marked points endowed with constant scalar curvature metric.The associated determinant line bundle is naturally endowed with the Quillen metric defined using the analytic torsion from the first paper of this series, generalizing the classical definition of Ray-Singer. We study the regularity of this Quillen metric near the locus of singular curves. The singularities turn out to be reasonable enough, so that the curvature of the Chern connection of the determinant line bundle endowed with the Quillen norm is well-defined as a current. We derive the explicit formula for this current, which gives a refinement of Riemann-Roch-Grothendieck theorem at the level of currents. This generalizes the curvature theorems of Takhtajan-Zograf and Bismut-Bost.