ec 1 99 5 Non - symplectic involutions of a K 3 surface

Let S be a smooth minimal K3 surface defined over C , G a finite group acting on S. The induced linear action of G on H 0 (ω S) ∼ = C leads to an exact sequence 1 −→ K −→ G −→ N −→ 1 , where the non-symplectic part N is a cyclic group Z m , which acts on the intermediate quotient S/K which is also K3. It is well-known that the Euler number ϕ(m) of m must divide 22 − ρ(S) ([N], Corollary 3.3), in particular ϕ(m) ≤ 21 , hence m ≤ 66. It is also known that if H is non-trivial, then S is algebraic. In this case the quotient of S by the action of G is either an Enriques surface or a rational surface. An example of m = 66 has been constructed in [K], where Kondo also gets the uniqueness of the K3 surface with a non-symplectic action of N ∼ = Z 66 , under the extra condition that N acts trivially on the Néron-Severi group of the surface. (Note that the computation in [K] contains an error, so that the case m = 44 is missing in his final result; the existence of this case is shown in our computation which follows.) The purpose of present article is to determine the K3 surfaces admitting a non-symplectic group N of high order. More precisely, we look at the cases Theorem. 1. There exists no K3 surface admitting a non-symplectic N of order 60. 2. For each of the other 6 cases of m as above, there is exactly one K3 surface S with N ∼ = Z m. The action of N is also unique (up to isomorphisms of S) except in the case of m = 38 , in which case there are 2 different actions.