Finite symplectic actions on the K3 lattice

Abstract

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