Symplectic involutions on deformations of K3^[2]

TL;DR

This paper studies symplectic involutions on hyperk"ahler varieties deformation equivalent to Hilbert squares of K3 surfaces, revealing their fixed loci, lattice structures, and deformation properties.

Contribution

It characterizes fixed loci and lattice structures of symplectic involutions on these varieties and shows their deformation equivalence to involutions on Hilbert squares of K3 surfaces.

Findings

Fixed locus consists of 28 points and a K3 surface.

Anti-invariant lattice is isomorphic to E_8(-2).

Any such pair deforms to a Hilbert square with a Nikulin involution.

Abstract

Let X be a Hyperk\"{a}hler variety deformation equivalent to the Hilbert square on a K3 surface and let f be an involution preserving the symplectic form. We prove that the fixed locus of f 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 E_8(-2). Finally we prove that any couple consisting of one such variety 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 Nikulin involution on the K3 surface.