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

Abstract

In this article we study the Quillen norm on the determinant line bundle associated with a family of complex curves with cusps, which admit singular fibers. More precisely, we fix a family of complex curves $\pi : X \to S$, which admit at most double-point singularities. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $X$. Let $\sigma_1, \ldots, \sigma_m : S \to X$ be disjoint holomorphic sections. We denote the divisor $D_{X/S} := \rm{Im}(\sigma_1) + \cdots + \rm{Im}(\sigma_m)$, and endow the relative canonical line bundle $\omega_{X/S}$ with a Hermitian norm such that its restriction at each fiber of $\pi$ induces K\"ahler metric with hyperbolic cusps. This Hermitian norm induces the Hermitian norm on the twisted relative canonical line bundle $\omega_{X/S}(D) := \omega_{X/S} \otimes \mathscr{O}_{X}(D_{X/S})$. For $n \leq 0$, we study the determinant line bundle $\lambda(j^*(\xi \otimes \omega_{X/S}(D)^n))$. We endow it with the Quillen norm by using the analytic torsion from the first paper of this series. Then we study the regularity of this Quillen norm and its asymptotics near the locus of singular curves. The singular terms of the asymptotics turn out to be reasonable enough, so that the curvature of $\lambda(j^*(\xi \otimes \omega_{X/S}(D)^n))$ is well-defined as a current over $S$. 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 formulas of Takhtajan-Zograf and Bismut-Bost. As a consequence of our study, we also get some regularity results on the Weil-Petersson form over the moduli space of pointed curves, which are enough to conclude the well-known fact, originally due to Wolpert, that the Weil-Petersson volume of the moduli space of pointed stable curves is a rational multiple of a power of $\pi$.