Irreducible symplectic varieties from moduli spaces of sheaves on K3 surfaces

In a recent paper Hoering and Peternell completed the proof of the Bogomolov decomposition in the singular projective setting: every normal projective variety which is smooth in codimension 2 and has canonical singularities and numerically trivial canonical bundles, admits a quasi-étale cover which is a product of complex tori, Calabi-Yau varieties and irreducible symplectic varieties. These last are the singular analogue of hyperkaehler manifolds, and share many features with them. In a joint work with A. Rapagnetta we show that all moduli spaces of semistable sheaves over projective K3 surfaces (with respect to generic polarizations) are irreducible symplectic varieties, with the only exception of symmetric products of K3 surfaces. Moreover, we describe their second integral cohomology, their Beauville form and their Fujiki constant.