Gushel-Mukai varieties: linear spaces and periods

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 hyperkähler 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 hyperkähler fourfold is in both cases a smooth double cover of a hypersurface in ${\mathbf{P}}^5$ called an EPW sextic.

44 pages. Lemma 2.1 slightly expanded. Lemma 2.2, Lemma 3.3 (a Lefschetz-type result for cyclic coverings proved by Cornalba), and Lemma 3.7 are new. Other minor corrections

References in corpus (1)

• Derived categories of Gushel-Mukai varieties

Cited by in corpus (6)

• Stability conditions in families
• Derived categories of Gushel-Mukai varieties
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II
• Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
• Algebraic cycles and Gushel-Mukai fivefolds
• New examples of rational Gushel-Mukai fourfolds