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Article

Gushel–Mukai varieties: Linear spaces and periods

Kyoto Journal of Mathematics. 2019. Vol. 4. No. 59. P. 897-953.
Kuznetsov A. G., Debarre O.

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