On birational Torelli theorems

Abstract

Let G be a simple simply-connected connected linear algebraic group over \({\mathbb {C}}\). We prove a 2-birational Torelli theorem for the moduli space of semistable principal G-bundles over a smooth curve of genus \(\ge 3\), which says that if two such moduli spaces are 2-birational, then the curves are isomorphic. We also prove a 3-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus \(\ge 4\).