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 X _t of genus g > 1, parameterized by the Schottky space G-fraktur sign_g, we define a natural basis of H⁰(X_t},ω_Xtⁿ) 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 G-fraktur sign_g which generalizes the classical product ∏_m=1^∞ (1-q^m)² for n =1 and g =1. We prove the holomorphic factorization formula det'Δ_n\det N_n = c_g,n exp {- 6n ² - 6n + 1/12π S}|F(n)|²} where det^'Δ _n is the zeta-function regularized determinant of the Laplace operator Δ _n in the hyperbolic metric acting on n-differentials, N _n is the Gram matrix of the natural basis with respect to the inner product given by the hyperbolic metric, S is the classical Liouville action -a Kähler potential of the Weil-Petersson metric on G-fraktur sign_g - and c _g,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.