A Torelli theorem for moduli spaces of principal bundles over a curve
The authors prove a Torelli-type theorem for the moduli space of principal \(G\)-bundles. The classical Torelli Theorem states that two curves must be isomorphic if their principally polarized Jacobians, or polarized moduli spaces of topologically trivial line bundles, are isomorphic. The authors of the present paper generalize this result by replacing the Jacobian with a more general moduli space.NEWLINENEWLINEThey replace the Jacobian by the moduli space \(\text{M}_{G}^{d}(X)\) of polystable principal \(G\)-bundle of fixed topological type \(d\) on a smooth complex projective curve \(X\). Suppose that \(G\) and \(G'\) are two non-abelian connected reductive groups. The main theorem of this paper states that two curves \(X, X'\) of genus at least \(3\) are isomorphic if the modules spaces \(\text{M}_{G}^{d}(X)\) and \(\text{M}_{G'}^{d'}(X')\) are isomorphic.NEWLINENEWLINEThe authors prove the theorem by showing that \(X\) can be reconstructed from \(\text{M}^{d}_{G}(X)\). Most of the paper is devoted to proving this when \(\text{M}^{d}_{G}(X)\) contains a strictly semistable point. When no such point exists, \(G\) must be of a special form: The quotient of \(G\) by its center must be isomorphic to a product \(\text{PGL}_{n_1} \times \dots \times \text{PGL}_{n_r}\) of projective linear groups. In this special case, the authors reconstruct a suitable moduli space of semistable vector bundles and then use a previous Torelli-type theorem of \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math. (2) 101, 391--417 (1975; Zbl 0314.14004)] and \textit{A. N. Tyurin} [Math. USSR, Izv. 4, (1970), 343--370 (1970; Zbl 0225.14011)].NEWLINENEWLINEWhen \(\text{M}^{d}_{G}(X)\) contains a strictly semistable point, the authors reconstruct the Jacobian with its principal polarization and then use the classical Torelli Theorem. To reconstruct the Jacobian, the authors prove several results on the singular locus and the strictly semistable locus of \(\text{M}_{G}^{d}(X)\). For example, they prove the smooth locus of \(\text{M}_{G}^{d}(X)\) is the locus of regularly stable \(G\)-bundles.
scientific article; zbMATH DE number 6064513
scientific article