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

Abstract

In this paper we describe the fixed locus of a symplectic involution on a hyperk\"ahler manifold of type $K3^{[n]}$ or of Kummer $n$ type. We prove that the fixed locus consists of finitely many copies of Hilbert schemes of $K3$ surfaces of lower dimensions and isolated fixed points.