Induced Automorphisms on Irreducible Symplectic Manifolds

TL;DR

This paper introduces a criterion for identifying automorphisms on $K3^{[n]}$ type manifolds induced by $K3$ surface automorphisms, classifies certain automorphisms, and explores their geometric properties.

Contribution

It develops a new criterion for induced automorphisms, applies it to classify non-symplectic automorphisms on $K3^{[2]}$ manifolds, and discusses variations for other types of irreducible symplectic manifolds.

Findings

Most non-symplectic prime order automorphisms on $K3^{[2]}$ type manifolds are classified.

A criterion for automorphisms induced by $K3$ surface automorphisms is established.

Descriptions of Picard lattices for manifolds with lagrangian fibrations are provided.

Abstract

We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of $K3^{[n]}$ type is, in fact, induced by an automorphism of a $K3$ surface and the manifold is a moduli space of stable objects on the $K3$. This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of $K3^{[2]}$ type and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore we provide a description of the Picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.