The monodromy of moduli spaces of sheaves on K3 surfaces

There are very few known deformation classes of irreducible symplectic manifolds, i. e. compact, connected Kahler manifolds that are simply connected and carry a holomorphic symplectic form spanning the space of closed holomorphic 2-forms. Among the tools that are used to study the birational geometry of these manifolds, the monodromy group is of the utmost important, and has been calculated for all the known deformations classes by Markman, Mongardi, Rapagnetta and Onorati. As soon as we allows singularities, we get into the world of irreducible symplectic varieties, for which we know many more deformations classes, and the monodromy group is still defined and plays a major role in the study of their birational geometry. In a joint work with Onorati and Rapagnetta we calculate the monodromy group of the examples of irreducible symplectic varieties given by moduli spaces of semistable sheaves on K3 surfaces.