Irreducible symplectic varieties from moduli spaces of sheaves on K3 and Abelian surfaces

We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
Comment: 61 pages, major revisions (the title has changed from the previos version, some of the proofs have been entirely reviewed, the main results have been generalized to a slightly more general notion of general polarization), to appear in Algebraic Geometry