Nikulin Involutions on K3 Surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on MorrisonNikulin involutions, these are Nikulin involutions which interchange two copies of E8(−1) in the Néron Severi group. In his paper [Ni1] Nikulin started the study of finite groups of automorphisms on K3 surfaces, in particular those leaving the holomorphic two form invariant, these are called symplectic. He proves that when the group G is cyclic and acts symplectically, then G ∼= Z/nZ, 1 ≤ n ≤ 8. Symplectic automorphisms of K3 surfaces of orders three, five and seven are investigated in the paper [GS]. Here we consider the case of G ∼= Z/2Z, generated by a symplectic involution ι. Such involutions are called Nikulin involutions (cf.[Mo, Definition 5.1]). A Nikulin involution on the K3 surface X has eight fixed points, hence the quotient Ȳ = X/ι has eight nodes, by blowing them up one obtains a K3 surface Y . In the paper [Mo] Morrison studies such involutions on algebraic K3 surfaces with Picard number ρ ≥ 17 and in particular on those surfaces whose Néron Severi group contains two copies of E8(−1). These K3 surfaces always admit a Nikulin involution which interchanges the two copies of E8(−1). We call such involutions Morrison-Nikulin involutions. The paper of Morrison motivated us to investigate Nikulin involutions in general. After a study of the maps on the cohomology induced by the quotient map, in the second section we show that an algebraic K3 surface with a Nikulin involution has ρ ≥ 9 and that the Néron Severi group contains a primitive sublattice isomorphic with E8(−2). Moreover if ρ = 9 (the minimal possible) then the following two propositions are the central results in the paper: Proposition 2.2. Let X be a K3 surface with a Nikulin involution ι and assume that the Néron Severi group NS(X) of X has rank nine. Let L be a generator of E8(−2) ⊥ ⊂ NS(X) with L = 2d > 0 and let Λ2d := ZL⊕ E8(−2) (⊂ NS(X)). Then we may assume that L is ample and: (1) in case L ≡ 2 mod 4 we have Λ2d = NS(X); The second author is supported by DFG Research Grant SA 1380/1-1. 2000 Mathematics Subject Classification: 14J28, 14J10.