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

Abstract

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 |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. If a conjecture on the behaviour of certain linear systems holds this result together with previous results of ours implies that every numerical (K3)[2] is a deformation of (K3) [2] .

Moduli of double EPW-sextics

“…We showed in [26] that there is a non-trivial involution δ : M → M; we will recall the definition. Let…”

“…Let A ∈ LG( 3 V ) be generic: then Y δV (A) is the classical dual Y ∨ A of Y A , see [26]. The map δ V induces a regular involution…”

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

“…[n] of length n subschemes of an algebraic K3 surface S. Such hyper-Kähler varieties admit projective deformations that are not obtained by the same construction, but they are not well understood except in a few cases, namely the four different families of hyper-Kähler fourfolds constructed in the papers [10], [29], [55], [77]. The varieties constructed by Beauville and Donagi are obtained as Fano varieties of lines of smooth cubic fourfolds in P 5 .…”

“…We will not comment on the proof of (2). 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]). …”

Dual Double EPW-sextics and Their Periods

“…Thus the family of double EPW-sextics is similar to the family of Fano varieties of lines on a cubic 4-fold (see [2]), with the following difference: the Plücker ample divisor on the Fano variety of lines has square 6 for the Beauville-Bogomolov quadratic form (see [1,2]) while the natural polarization of a double EPWsextic has square 2 (see [13]). Let Y ⊂ P 5 be a generic EPW-sextic: we proved in [13] that the dual Y ∨ ⊂ (P 5 ) ∨ is another generic EPW-sextic. Thus we may associate to the natural double cover X of Y a "dual"variety X ∨ namely the natural double cover of Y ∨ .…”