Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

By Kieran G. O’Grady. 2005

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Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in P 5 as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3 surface (K3) [2] and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is a numerical (K3) [2] , that H is an ample divisor on X of square 2 for Beauville’s quadratic form and that the map X 99K |H| _ is the composition of the quotient X ! Y for an anti-symplectic involution on X followed by an immersion Y ,! |H| _ ; then Y is an EPW-sextic and X ! Y is the natural double cover.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b15bf8ce6142d645d94da665ffa079067df2a4c2

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Citations in the corpus

Enriques manifolds (2010, background): “O’Grady [29] constructed hyperkähler 4-folds as double covers of certain sextic hyperplane in P(5).”
Double covers of EPW-sextics (2010, background): “Here we will present a direct proof (we will not appeal to [16] nor to [14]).”
Derived categories of Gushel–Mukai varieties (2016, background): “O’Grady [41, 42, 43, 44, 45, 46] extensively investigated the geometry of EPW sextics, and proved in particular that (see also [7, Theorem B.”
Symplectic involutions of holomorphic symplectic four‐folds (2010, background): “ Moreover, they show that there is an isomorphism of Hodge structures α : H4(X,Z) ∼= H2(F,Z). The double cover of an EPW sextic. This example has been introduced and intensively studied by O’Grady in [14] and many other papers. Starting from a 6-dimensional vector space V and from a general enough L”
Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$-elementary lattices (2014, background): “Such a hypersurface is not smooth, but for a general A it has a smooth double coverXA which is an IHS−K3 [2] (see [32]).”
Holomorphic symplectic geometry: a problem list (2010, background): “no variety of lines contained in a cubic fourfold ([7]; g =8) 2. The “variety of sum of powers” associated to a cubic fourfo ld ([19]; g =8) 3. The double cover of certain sextic hypersurfaces in P5 ([31]; g =6) 4. The subspace of the Grassmannian G(6,V)consisting of 6-planesL such that σ|L =0, wher”
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, background): “O’Grady studied double covers of Eisenbud-Popescu-Walter sextics in [101].”
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, background): “Let us just say that the result in (2) was partially extended by Ferretti in [40] to the case of O’Grady fourfolds (see [77]).”
Symplectic involutions on deformations of K3[2] (2011, background): “The papers [19, 20] by O’Grady study DoubleEPW sextics which are naturally endowed with an antisymplectic involution.”