Hyperk\"{a}hler varieties as Brill-Noether loci on curves

TL;DR

This paper demonstrates that certain Brill-Noether loci on high-genus curves embedded in K3 surfaces form smooth hyperk"ahler manifolds, linking curve theory with hyperk"ahler geometry.

Contribution

It establishes a new geometric correspondence between Brill-Noether loci on curves and moduli spaces of stable bundles on K3 surfaces, revealing hyperk"ahler structures.

Findings

Brill-Noether loci are smooth hyperk"ahler manifolds.

These loci are isomorphic to moduli spaces on K3 surfaces.

Dimension formula for the hyperk"ahler manifolds.

Abstract

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 $g$ of $C$ is sufficiently high, we show the Brill-Noether locus $BN_C \subset M_C(r; K_C)$ of bundles with $h$ global sections is a smooth projective Hyperk\"{a}hler manifold of dimension $2g -2r \lfloor \frac{g}{r}\rfloor$, isomorphic to a moduli space of stable vector bundles on $X$.