Symplectic involutions of K3[n]$K3^{[n]}$ type and Kummer n$n$ type manifolds

Abstract

In this paper, we describe the fixed locus of a symplectic involution on a hyper-Kähler manifold of type K 3 [ n ] $K3^{[n]}$ or of Kummer n $n$ type. We prove that the fixed locus consists of finitely many copies of deformations of Hilbert schemes of K 3 $K3$ surfaces of lower dimensions and isolated fixed points.