# Hyperkahler varieties as Brill-Noether loci on curves

## Location

Consider the moduli space $M_C(r; K_C)$ 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, I will show the Brill-Noether locus $BN_C \subset M_C(r; K_C)$ of bundles with $h$ global sections is a smooth projective Hyperkahler manifold, isomorphic to a moduli space of stable vector bundles on $X$. The main technique is to apply wall-crossing with respect to Bridgeland stability conditions on K3 surfaces.

The synchronous discussion for Soheyla Feyzbakhsh’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2022-05-27-sf (and will be deleted after ~3-7 days).