The Torelli theorem for the moduli spaces of connections on a Riemann surface

Abstract

Let (X,x₀) be any one-pointed compact connected Riemann surface of genus g, with g≥ 3. Fix two mutually coprime integers r>1 and d. Let M_x denote the moduli space parametrizing all logarithmic SL(r,C) -connections, singular over x₀, on vector bundles over X of degree d. We prove that the isomorphism class of the variety M_x determines the Riemann surface X uniquely up to an isomorphism, although the biholomorphism class of is known to be independent of the complex structure of X. The isomorphism class of the variety M_x is independent of the point x₀∈X. A similar result is proved for the moduli space parametrizing logarithmic GL(r,C)-connections, singular over x₀, on vector bundles over X of degree d. The assumption r>1 is necessary for the moduli space of logarithmic GL(r,C)-connections to determine the isomorphism class of X uniquely.