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

Abstract

Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic $\text{SL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree $d$. We prove that the isomorphism class of the variety ${\mathcal M}_X$ determines the Riemann surface $X$ uniquely up to an isomorphism, although the biholomorphism class of ${\mathcal M}_X$ is known to be independent of the complex structure of $X$. The isomorphism class of the variety ${\mathcal M}_X$ is independent of the point $x_0 \in X$. A similar result is proved for the moduli space parametrizing logarithmic $\text{GL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree $d$.