Moduli spaces of bundles over nonprojective K3 surfaces

We study moduli spaces of sheaves over nonprojective K3 surfaces. More precisely, let omega be a Kahler class on a K3 surface S, let rgeq 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 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.