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. LetMX 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 MX determines the Riemann surface X uniquely up to an isomorphism, although the biholomorphism class of MX is known to be independent of the complex structure of X. The isomorphism class of the variety MX is independent of the point x0 2 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.