Hyperkähler varieties as Brill-Noether loci on curves

Abstract

Consider the moduli space MC (r; KC ) of stable rank r vector bundles on a curve C with canonical determinant, and let h be the maximum number of linearly independent global sections of these bundles. If C embeds in a K3 surface X as a generator of Pic(X) and the genus of C is sufficiently high, we show the Brill-Noether locus BNC ⊂ MC (r; KC ) of bundles with h global sections is a smooth projective Hyperkähler manifold of dimension 2g − 2r⌊ g r ⌋, isomorphic to a moduli space of stable vector bundles on X.