Symplectic involutions on deformations of K3[2]

Abstract

Let X be a Hyperkähler manifold deformation equivalent to the Hilbert square of a K3 surface and let ϕ be an involution preserving the symplectic form. We prove that the fixed locus of ϕ consists of 28 isolated points and 1 K3 surface, moreover the anti-invariant lattice of the induced involution on H 2 (X, Z) is isomorphic to E8(−2). Finally we prove that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.

Smith theory and irreducible holomorphic symplectic manifolds

“…The following lemma is a direct generalization of Nikulin's results [35], very close to those stated in Mongardi [32].…”

“…As explained in Mongardi [31], ϕ induces a symplectic automorphism of order 11 on the Fano variety of lines X of the cubic fourfold C, with 5 isolated fixed points. Using our main formula given in Theorem 5.15, one finds that there is only one possibility for the parameters a G (X), m G (X), that is:…”

“…If X is deformation equivalent to S [n] and ρ(X) = 1 then the existence of non-symplectic automorphisms of order p = 23 is not excluded. For a symplectic action on a deformation of S [2] , Mongardi [31] shows that p ≤ 11 so higher order automorphisms on deformations of S [2] are always non-symplectic. The following lemma is a direct generalisation of Nikulin's results [35], very close to those stated in Mongardi [32].…”

“…Note that (g, g) X = 6. By Mongardi [31], for the special choice of the cubic C with a symplectic automorphism of order 11 constructed above, one has…”

“…discriminant for T(X) is the choice of the matrix B (compare with Mongardi [31] for a different argument).…”