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 $g\ge 3$. Fix two mutually coprime integers $r>1$ and $d$. Let $M_X$ denote the moduli space parametrizing all logarithmic $\mathrm{SL}(r,C)$-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 $x_0\in X$. A similar result is proved for the moduli space parametrizing logarithmic $\mathrm{GL}(r,C)$-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 $\mathrm{GL}(r,C)$-connections to determine the isomorphism class of $X$ uniquely.