Torelli theorems for moduli of logarithmic connections and parabolic bundles

Abstract

Let Z be a finite subset of a compact connected Riemann Surface X. Let \({\fancyscript{M}_X^{lc}}\) denote the moduli space of pairs (L, D) where L is a line bundle on X and D is a logarithmic connection on L singular along Z. Then \({\fancyscript{M}_X^{lc}}\) has a natural symplectic structure [ωX]. We show that the pair \({(\fancyscript{M}_X^{lc},[\omega_X])}\) determines X and there are no nonconstant algebraic functions on \({\fancyscript{M}_X^{lc}}\). We also prove a Torelli type theorem for the moduli space of parabolic bundles.