Gushel–Mukai varieties: Linear spaces and periods

Abstract

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-Kähler $4$-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 $\mathsf{Gr}(2,5)$, a quadric, and two hyperplanes (resp., of the cone over $\mathsf{Gr}(2,5)$ and a quadric). The associated hyper-Kähler $4$-fold is in both cases a smooth double cover of a hypersurface in ${\mathbf{P}}^5$ called an Eisenbud–Popescu–Walter sextic.