Hyperkähler manifolds and sheaves

Moduli spaces of hyperkähler manifolds or of sheaves on them are often nonseparated. We will discuss results where this phenomenon reflects interesting geometric aspects, e.g. deformation equivalence of birational hyperkähler manifolds or cohomological properties of derived autoequivalences. In these considerations the Ricci-
flat structure often plays a crucial role via the associated twistor space providing global deformations of manifolds and bundles.