Symplectic involutions of holomorphic symplectic four-folds

The main goal of this paper is the study of fixed points of a symplectic involution over an irreducible holomorphic symplectic manifold of dimension 4 such that b2=23. We show that there are only three possibilities for the number of fixed points and of fixed K3 surfaces. We conjecture that only one case can actually occur, the one with 28 isolated fixed points and 1 fixed K3 surface, and that such an involution can never fix an abelian surface. We provide evidence for the conjecture by verifying it in some examples, as the Hilbert scheme of a K3 surface, the Fano variety of a cubic in ℙ5 and the double cover of an Eisenbud-Popescu-Walter sextic.