Abstract. We construct a new 20-dimensional family of algebraic hyper-Kähler fourfolds and prove that they are deformationequivalent to the second punctual Hilbert scheme of a K3 surface of degree 22.
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
We study, after Logachev, the geometry of smooth complex Fano threefolds X X with Picard number 1 1 , index 1 1 , and degree 10 10 , and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general such X X has no nontrival automorphisms. By a simple deformation argument and a parameter count, we show that X X is not birational to a quartic double solid, disproving a conjecture of Tyurin. Through a detailed study of the variety of conics contained in X X , a smooth projective irreducible surface of general type with globally generated cotangent bundle, we construct two smooth projective two-dimensional components of the fiber of the period map through a general X X : one is isomorphic to the variety of conics in X X , modulo an involution, another is birationally isomorphic to a moduli space of semistable rank- 2 2 torsion-free sheaves on X X , modulo an involution. The threefolds corresponding to points of these components are obtained from X X via conic and line (birational) transformations. The general fiber of the period map is the disjoint union of an even number of smooth projective surfaces of this type.
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