Antisymplectic involutions of holomorphic symplectic manifolds

Let X be a holomorphic symplectic manifold, of dimension divisible by four, and sigma be an antisymplectic involution of X. The fixed locus F of sigma is a Lagrangian submanifold of X; we show that its (A) over cap -genus is one. 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.