Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Abstract

Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is an irreducible symplectic 4-fold numerically equivalent to the Hilbert square of a K3-surface, that H is an ample divisor on X of square 2 for Beauville's quadratic form and that the map associated to |H| is the composition of the quotient map $X\to Y$ for an anti-symplectic involution on X followed by an immersion of Y; then Y is an EPW-sextic and $X\to Y$ is the natural double cover.