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

Let (X,x0)(X,x0) be any one-pointed compact connected Riemann surface of genus gg, with g≥3g≥3. Fix two mutually coprime integers r>1r>1 and dd. Let MXMX denote the moduli space parametrizing all logarithmic SL(r,C)-connections, singular over x0x0, on vector bundles over XX of degree dd. We prove that the isomorphism class of the variety MXMX determines the Riemann surface XX uniquely up to an isomorphism, although the biholomorphism class of MXMX is known to be independent of the complex structure of XX. The isomorphism class of the variety MXMX is independent of the point x0∈Xx0∈X. A similar result is proved for the moduli space parametrizing logarithmic GL(r,C)-connections, singular over x0x0, on vector bundles over XX of degree dd. The assumption r>1r>1 is necessary for the moduli space of logarithmic GL(r,C)-connections to determine the isomorphism class of XX uniquely.