Moduli of sheaves on hyper-Kähler manifolds

Hyper-Kähler manifolds are one of the building blocks of compact Ricci flat Kähler manifolds. In many ways they can be seen as higher dimensional analogues of K3 surfaces, with the general theory sharing many similarities to that of K3 surfaces. On the other hand, at the moment few examples are known, and all the constructions are related to moduli spaces of semistable sheaves on K3 surfaces. In the 90’s it was observed that moduli spaces of certain sheaves on higher dimensional hyper-Kähler manifolds are good candidates to be (possibly new examples of) hyper-Kähler themselves, but this approach was considered too difficult to pursue until very recently. After surveying the theory of moduli spaces of sheaves on K3 surfaces, I will outline the recent developments for higher dimensional hyper-Kähler manifolds, focusing on a concrete example of a 10-dimensional “exceptional” hyper-Kähler manifold realized as a moduli space of sheaves on a hyper-Kähler fourfold.