Gushel-Mukai varieties: Linear spaces and periods

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic 4-fold and the primitive second cohomology of its variety of lines, a smooth hyper-Kahler 4-fold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp., 6), that is, smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2, 5), a quadric, and two hyperplanes (resp., of the cone over Gr(2, 5) and a quadric). The associated hyper-Kahler 4-fold is in both cases a smooth double cover of a hypersurface in P-5 called an Eisenbud-Popescu-Walter sextic.