Large deviations of empirical measures of zeros on riemann surfaces

We determine a large deviations principle (LDP) for the empirical measure of zeros of random holomorphic sections s of random line bundles L→X over a Riemann surface X of genus g≥1. Zeitouni and Zelditch [27] proved such an LDP in the g=0 case of ℂℙ¹ using an explicit formula for the joint probability current (JPC) of zeros of Gaussian random polynomials. The main purpose of this article is to define Gaussian-type measures on the "vortex moduli space" of all holomorphic sections of all line bundles of degree N and to calculate its JPC as a volume form on the configuration space X^(N) of N points of X.