Torelli theorems for moduli of logarithmic connections and parabolic bundles
Let \(X\) be a compact connected Riemann surface. Fix \(n\) points \(x_i\) on \(X\), let \(Z\) be the divisor \( \sum x_i\). For each \(i\), fix \(r_i \in {\mathbb Q}\). For a logarithmic connection \((L,D)\) (here \(L\) is a holomorphic line bundle and \(D:L\to L\otimes \Omega _X\otimes {\mathcal O}_X(D)\) satisfies Leibniz rule, such that \(\mathrm{Res}(L,D,x_i)=r_i\)) one has \(\sum \mathrm{Res}(L,D,x_i)=-\deg(L)\). Let \({\mathcal M}_X^{lc}\) be the moduli space of pairs \((L,D)\) satisfying \(\mathrm{Res}(L,D,x_i)=r_i\). One shows that \({\mathcal M}_X^{lc} \not = \emptyset\) provided \(\sum r_i \in {\mathbb Z}\). \({\mathcal M}_X^{lc}\) has a natural symplectic structure; let its class be \([\omega_X]\). Using among others, the classical Torelli theorem, one shows the following Torelli-type theorem: for two compact connected Riemann surfaces \(X,Y\), if there exists an isomorphism \({(\mathcal M}_X^{lc}, [\omega _X]) \cong ({\mathcal M}_X^{lc},[\omega _Y]) \), then \(X\cong Y\). One proves also a Torelli-type theorem for the moduli space of parabolic bundles, generalizing some results of \textit{V. Balaji, I. Biswas} and \textit{S. del Bano Rollin} [Math. Proc. Camb. Philos. Soc. 130, No. 2, 269--280 (2001; Zbl 1063.14042)] from rank \(2\) to arbitrary rank.
scientific article; zbMATH DE number 5944338
scientific article