Birational self-maps of hyperkähler manifolds of K3[n]-type

Birational self-maps that are not biregular are hard to find on hyperkähler manifolds. For instance, there aren’t any on K3 surfaces. In this joint work with D. Mattei and Y. Prieto we showed that a general projective Hyperkähler manifold that is deformation equivalent to the Hilbert scheme of n-points on a K3 surface (i.e., of K3[n]-type) cannot admit certain birational self-maps of finite order. This prompted us to investigate birational self-maps moduli of sheaves on K3 surfaces which are of K3[n]-type. Using Markman’s theory of hyperkähler lattices and Bayer–Macri’s study of Bridgeland stability on K3 surfaces, we impose explicit numerical constraints on the topological invariants of the sheaves so that certain birational involutions exist on their moduli space.