Gushel-Mukai varieties: linear spaces and periods

TL;DR

This paper extends Beauville and Donagi's classical results to Gushel-Mukai varieties, establishing isomorphisms between their primitive cohomologies and those of associated hyperk"ahler fourfolds, which are double covers of EPW sextics.

Contribution

It proves new isomorphisms of Hodge structures for Gushel-Mukai varieties, generalizing known results for cubic fourfolds and their Fano varieties of lines.

Findings

Primitive cohomology of Gushel-Mukai varieties is isomorphic to that of associated hyperk"ahler fourfolds.

Hyperk"ahler fourfolds are double covers of EPW sextics.

Results unify the understanding of cohomological properties across different classes of varieties.

Abstract

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperk\"ahler fourfold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., 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 hyperk\"ahler fourfold is in both cases a smooth double cover of a hypersurface in ${\bf P}^5$ called an EPW sextic.