Antisymplectic involutions of holomorphic symplectic manifolds

Abstract: Let X be a holomorphic symplectic manifold, of dimension divisible by 4, and s an antisymplectic involution of X . The fixed locus F of s is a Lagrangian submanifold of X ; we show that its ^A-genus is 1. As an application, we determine all possibilities for the Chern numbers of F when X is a deformation of the Hilbert square of a K3 surface.