Symplectic Automorphisms and the Picard Group of a K3 Surface

Let X be a K3 surface, and let G be a finite group acting on X by automorphisms. The action of G on X induces an action on the cohomology of X . We assume G acts symplectically: that is, G acts as the identity on H(X). In this case, the minimum resolution Y of the quotient X/G is itself a K3 surface. Nikulin classified the finite abelian groups which act symplectically on K3 surfaces by analyzing the relationship between X and Y . In the abelian case, Nikulin also described moduli spaces of K3 surfaces with G actions; these topological spaces are subspaces of the moduli space of marked K3 surfaces. [N80a] Mukai showed that any finite group G with a symplectic action on a K3 surface is a subgroup of a member of a list of eleven groups, and gave an example of a symplectic action of each of these maximal groups. [M88] Xiao gave an alternate proof of the classification by listing the possible types of singularities, and Kondō showed that the action of G on the K3 lattice extends to an action on a Niemeier lattice. [X96, K98] The Picard group of X has a primitive sublattice SG determined by the action of G. The rank of SG varies from 8 to 19, depending on G. Thus, K3 surfaces which admit symplectic group actions provide a rich source of examples of families of K3 surfaces with high-rank Picard groups. The monodromy and mirror symmetry properties of algebraic K3 surfaces which admit a sublattice SG of rank 18, and therefore have a Picard group of rank 19, have been extensively studied. (cf. [N01, S07, DK08]) Conversely, if the structure of Pic(X) is known, one may examine its sublattices to detect symplectic group actions on X . Morrison used the structure of SG for G = Z/2Z to study K3 surfaces which admit Shioda-Inose structures. [M84] Recently, Garbagnati and Sarti have computed SG for all possible abelian groups with symplectic action, correcting an earlier computation of Nikulin’s; Garbagnati has also studied SG for dihedral groups, and Hashimoto calculated the invariants of SG for the permutation group G = S5. [GS08, G08a, G08b, G09, H09] In Section 2, we discuss the relationship between the lattice SG and the singularities of X/G for any symplectic G-action, and show how to compute the rank and discriminant of SG. In Section 3, we show that the maps between X , Y , and X/G can be generalized to the realm of moduli spaces, and describe moduli spaces of K3 surfaces with symplectic G-action. Our proof extends the discussion in [N80a] to the case that G is not abelian. The key observation is that we may work backwards from a K3 surface Y endowed with a set of exceptional curves to the K3 surface X .