1 9 O ct 1 99 8 Ω - Admissible Theory II : New metrics on determinant of cohomology And Their applications to moduli spaces of punctured Riemann surfaces

For singular metrics, Ray and Singer’s analytic torsion formalism cannot be applied. Hence we do not have the so-called Quillen metric on determinant of cohomology with respect to a singular metric. In this paper, we introduce a new metric on determinant of cohomology by adapting a totally different approach. More precisely, by strengthening results in the first paper of this series, we develop an admissible theory for compact Riemann surfaces with respect to singular volume forms, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we prove the Mumford type fundamental relations for metrized determinant line bundles over moduli spaces of punctured Riemann surfaces. Moreover, using an idea of D’Hoker-Phong and Sarnak, we introduce a natural admssible metric associated to a punctured Riemann surface via the Arakelov-Poincaré volume, a new invariant for a punctured Riemann surface. With this admissible metric, we make an intensive yet natural study on two Kähler forms on the moduli space of punctured Riemann surfaces associated to the Weil-Petersson metric and the Takhtajan-Zograf metric (defined by using Eisenstein series. Among others, we, together with Fujiki, show that the Takhtajan-Zograf Kähler form is indeed the first Chern form of a certain metrized line bundle). All this finally leads to a more geometric interpretation of our new determinant metrics in terms of special values of Selberg zeta functions. We end this paper by proposing an arithmetic factorization in terms of Weil-Petersson metrics, cuspidal metrics and Selberg zeta functions, which then serves as the most global picture for viewing Riemann surfaces.