Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula

Details Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula

2004-10-12

arxiv.org/abs/math/0410294v2

Mathematics

Complex Variables

LaTex, 31 pages, no figures; minor typos corrected, final version to appear in Geometric and Functional Analysis

Scientific paper

For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the Schottky space S_g, we define a natural basis for the holomorphic n-differentials on X_t 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 S_g which generalizes the classical product \prod(1-q^m)^2 appearing in the Dedekind eta function for n=1 and g=1. We prove a holomorphic factorization formula expressing the regularized determinant of the Laplacian as a product of |F(n)|^2, a holomorphic anomaly depending on the classical Liouville action (a Kahler potential of S_g), and the determinant of the Gram matrix of the natural basis. 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.

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