Finiteness and quasi-simplicity for symmetric K3 -surfaces

Abstract

We compare the smooth and deformation equivalence of actions of finite groups on $K3$-surfaces by holomorphic and antiholomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an action is determined by the induced action in the homology. On the other hand, we construct two examples to show first that, in general, the homological type of an action does not even determine its topological type, and second that $K3$-surfaces $X$ and $\overline{X}$ with the same Klein action do not need to be equivariantly deformation equivalent even if the induced action on $H^{2,0}(X)$ is real, that is, reduces to multiplication by $\pm 1$.