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 P 5 called an EPW sextic.We work over the field of complex numbers. A smooth Gushel-Mukai variety is ([DK1, Definition 2.1]) a smooth dimensionally transverse intersectionof the cone over the Grassmannian Gr(2, V 5 ) of 2-dimensional subspaces in a fixed 5-dimensional vector space V 5 , with a linear subspace P(W ) and a quadric Q. This class of varieties includes all smooth prime Fano varieties X of dimension n ≥ 3, coindex 3, and degree 10 (i.e., such that there is an ample class H with Pic(X) = ZH, K X = −(n − 2)H, and H n = 10; see [DK1, Theorem 2.16]).One can naturally associate with any smooth GM variety of dimension n a triple (V 6 , V 5 , A), called a Lagrangian data, where V 6 is a 6-dimensional vector space containing V 5 as a hyperplane and the subspace A ⊂ 3 V 6 is Lagrangian with respect to the symplectic structure on 3 V 6 given by wedge product. Moreover, P(A) ∩ Gr(3, V 6 ) = ∅ in P( 3 V 6 ) when n ≥ 3 (we say that A has no decomposable vectors).Conversely, given a Lagrangian data (V 6 , V 5 , A) with no decomposable vectors in A, one can construct two smooth GM varieties of respective dimensions n = 5−ℓ and n = 6−ℓ (where ℓ := dim(A ∩ 3 V 5 ) ≤ 3), with associated Lagrangian data (V 6 , V 5 , A) ([DK1, Theorem 3.10 and Proposition 3.13]; see Section 2.1 for more details).Given a Lagrangian subspace A ⊂ 3 V 6 , we define three chains of subschemes

New examples of rational Gushel-Mukai fourfolds

“…In [DIM15] (see also [DK18a,DK16,DK18b]), similarly to Hassett's analysis of cubic fourfolds (see [Has99,Has00]), the authors studied Gushel-Mukai fourfolds via Hodge theory and via the period map. In particular, they showed that inside X 10 there is a countable union d GM d of (not necessarily irreducible) hypersurfaces parametrizing special Gushel-Mukai fourfolds, that is fourfolds [X] ∈ X 10 that contain a surface S whose cohomology class does not come from the Grassmannian G(1, 4).…”

Gushel–Mukai varieties: Moduli

“…Theorem 4.5 ( [DK2,DK5]). Let X be a smooth GM variety of dimension n ∈ {3, 4, 5, 6}, with associated Lagrangian A.…”

Double covers of quadratic degeneracy and Lagrangian intersection loci

“…The Hilbert schemes F d−1 (Q 1 (X)/P(V 5 )) were identified in [DK2,Proposition 4.1] with some irreducible components of the Hilbert schemes F d−1 (X) of (d − 1)-dimensional linear spaces on X. The connected fibers of its Stein factorization over P(V 5 ) were described in [DK2,Theorems 4.2,4.3,and 4.7].…”

“…give double EPW sextics and EPW cubes as special cases. These double covers also appear in the theory of Gushel-Mukai varieties ( [DK1,DK2]) and this was the original motivation for this work. In the relative situation (for the universal family of EPW varieties), a similar double cover is the base for a generalized root stack construction in terms of which the moduli stack of Gushel-Mukai varieties is described in [DK3].…”