Cone (formal languages)

Algorithm

Engineering

Pure mathematics

Grassmannian

Statistics

Variety (cybernetics)

Hypersurface

Dimension (graph theory)

Hyperplane

Mechanical engineering

Quadric

Mathematics

Geometry

Cover (algebra)

Cohomology

Kyoto journal of mathematics

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 P5 called an Eisenbud–Popescu–Walter sextic.

Gushel–Mukai varieties: Linear spaces and periods