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 ∈ 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.