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

Let (X, x(0)) 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 x(0), 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 M-X is known to be independent of the complex structure of X. The isomorphism class of the variety M-X is independent of the point x(0) is an element of X. A similar result is proved for the moduli space parametrizing logarithmic GL(r, C)-connections, singular over x(0), 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. (c) 2007 Elsevier Ltd. All rights reserved.