A birational Torelli theorem with a Brauer class

Abstract

Let ${\text{M}}_C(2,{\mathcal{O}}_C)\cong {\mathbb{P}}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let ${\hat{I}}_C^2$ denote the natural Brauer class over the stable locus. We prove that if $f^{\ast }({\hat{I}}_{C^{\prime }}^2)={\hat{I}}_C^2$ for some birational map $f$ from ${\text{M}}_C(2,{\mathcal{O}}_C)$ to ${\text{M}}_{C^{\prime }}(2,{\mathcal{O}}_{C^{\prime }})$, then the Jacobians of $C$ and of $C^{\prime }$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C^{\prime }$ are isomorphic. Similar statements hold for Kummer surfaces in ${\mathbb{P}}^3$ and for quadratic line complexes.

15 pages