Moduli spaces of bundles over non-projective K3 surfaces

Abstract

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,ξ,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $ξ$ and $ω$ is a "generic" Kähler class on $S$, we show that the moduli space $M$ of $μ_ω-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. If $M$ parametrizes only locally free sheaves, it is moreover hyperkähler. Finally, we show that there is an isometry between $v^{\perp}$ and $H^{2}(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective.