Holomorphic Factorization of Determinants of Laplacians on Riemann Surfaces and a Higher Genus Generalization of Kronecker’s First Limit Formula

For a family of compact Riemann surfaces Xt of genus g > 1, parameterized by the Schottky space Sg, we define a natural basis of H(Xt, ω n Xt ) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F (n) on Sg which generalizes the classical product ∏ ∞ m=1 (1 − q) for n = 1 and g = 1. We prove the holomorphic factorization formula det′∆n detNn = cg,n exp { − 6n − 6n+ 1 12π S } |F (n)| , where det′∆n is the zeta-function regularized determinant of the Laplace operator ∆n in the hyperbolic metric acting on n-differentials, Nn is the Gram matrix of the natural basis with respect to inner product given by the hyperbolic metric, S is the classical Liouville action — a Kähler potential of the Weil-Petersson metric on Sg — and cg,n is a constant depending only on g and n. The factorization formula reduces to Kronecker’s first limit formula when n = 1 and g = 1, and to Zograf’s factorization formula for n = 1 and g > 1.