Moduli spaces of simple sheaves on K3 surfaces

Under general assumptions, moduli spaces of coherent sheaves may exist only as Artin stacks. By adding the assumption of simpleness for coherent sheaves on a K3 surface, Mukai proved that the resulting moduli space is a smooth algebraic space admitting a closed, nowhere-degenerate holomorphic 2-form. In this talk I will describe Mukai's proof and provide examples of moduli spaces of sheaves and their connections to holomorphic symplectic varieties. We will mainly follow the papers "On the moduli space of bundles on a K3 surface, I" and "Symplectic structure of the moduli space of sheaves on an abelian or K3 surface", by S. Mukai.