Non-symplectic involutions on manifolds of $K3^{[n]}$-type

TL;DR

This paper classifies non-symplectic involutions on manifolds of $K3^{[n]}$-type, describing their lattice structures, and explores families and examples related to these involutions.

Contribution

It provides a classification of invariant and anti-invariant lattices under non-symplectic involutions and describes moduli spaces of such manifolds with explicit examples.

Findings

Classification of discriminant forms for involution lattices

Explicit descriptions of lattices with small invariant rank

Construction of moduli spaces and examples as twisted sheaf moduli

Abstract

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant forms of the invariant and anti-invariant lattice for the action of the involution on cohomology, and explicitly describe the lattices in the cases where the invariant has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geq 19$ and $n\leq 5$, and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.