Antisymplectic involutions of holomorphic symplectic manifolds

Let X be a holomorphic symplectic manifold, of dimension divisible by four, and σ be an antisymplectic involution of X. The fixed locus F of σ is a Lagrangian submanifold of X; we show that its Â-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.