Torelli theorem for moduli stacks of vector bundles and principal G-bundles

Abstract

Given any irreducible smooth complex projective curve $X$, of genus at least $2$, consider the moduli stack of vector bundles on $X$ of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve $X$ and the rank of the vector bundles. The case of trivial determinant, rank $2$ and genus $2$ is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is $\mathbb{P}^3$ thus independently of the curve). We also prove a Torelli theorem for moduli stacks of principal $G$-bundles on a curve of genus at least $3$, where $G$ is any non-abelian reductive group.