Symplectic automorphisms and the Picard group of a K3 surface

TL;DR

This paper studies how finite groups acting symplectically on K3 surfaces influence their Picard groups, providing methods to compute lattice invariants and classify possible group actions.

Contribution

It introduces techniques to compute the Picard lattice substructure and extends classification results of symplectic group actions on K3 surfaces.

Findings

Computed rank and discriminant of the Picard lattice substructure.

Described moduli spaces of K3 surfaces with symplectic G-action.

Developed classification techniques for symplectic group actions.

Abstract

We consider the symplectic action of a finite group G on a K3 surface. The Picard group of the K3 surface has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then describe moduli spaces of K3 surfaces with symplectic G-action, extending results of Nikulin in the abelian case. We use our moduli spaces to develop techniques for classifying all possible symplectic actions of a group G.