Moduli spaces of bundles over nonprojective K3 surfaces

Abstract

We study moduli spaces of sheaves over nonprojective K3 surfaces. More precisely, let $\omega$ be a Kähler class on a K3 surface $S$, let $r\ge 2$ be an integer, and let $v=(r,\xi ,a)$ be a Mukai vector on $S$. We show that if the moduli space $M$ of ${\mu }_{\omega }$-stable vector bundles with associated Mukai vector $v$ is compact, then $M$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. Moreover, we show that there is a Hodge isometry between $v^{\perp }$ and $H^2(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective.