Modular sheaves with many moduli

We exhibit moduli spaces of slope stable vector bundles on general polarized HK varieties (X, h) of type K3^2 which have an irreducible component of dimension 2a²+2, with a an arbitrary integer greater than 1. This is done by studying the case X=S^2 where S is an elliptic K3 surface. We show that in this case there is an irreducible component of the moduli space of stable vector bundles on S^2 which is birational to a moduli space of sheaves on S. We expect that if the moduli space of sheaves on S is a smooth HK variety (necessarily of type K3^a²+1) then the following more precise version holds: the closure of the moduli space of slope stable vector bundles on (X, h) in the moduli space of Gieseker-Maruyama semistable sheaves with its GIT polarization is a general polarized HK variety of type K3^a²+1.