Correspondences for hyperkähler varieties with large Picard numbers

In this note, we explore the connection between hyperkähler manifolds with large Picard numbers and abelian varieties. In particular, we are interested in Morrison’s solution to the (modified) Oda’s conjecture: every K3 surface whose Picard group is large enough (in a certain precise sense) must be related via an algebraic correspondence to an abelian surface. We generalize this theorem to the case of known examples of hyperkähler manifolds such as pointed Hilbert schemes on K3 surfaces.