Torelli theorem for moduli spaces of SL(r,C) -connections on a compact Riemann surface.

Abstract

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let denote the moduli space of holomorphic SL(r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL(r,ℂ)-connections on X.