Finite Symplectic Actions on the K3 Lattice

TL;DR

This paper investigates finite symplectic group actions on K3 surfaces, demonstrating that such actions on the second cohomology lattice are uniquely determined by the group, with only five exceptions.

Contribution

It establishes the classification of symplectic actions on the K3 lattice, revealing that they depend solely on the group structure, except for five specific groups.

Findings

Most symplectic actions are uniquely determined by the group

Five groups are exceptions with non-unique actions

Provides a classification framework for symplectic actions on K3 surfaces

Abstract

In this paper, we study finite symplectic actions on K3 surfaces X, i.e. actions of finite groups G on X which act on H^{2,0}(X) trivially. We show that the action on the K3 lattice H^2(X,Z) induced by a symplectic action of G on X depends only on G up to isomorphism, except for five groups.