K3 surfaces with a symplectic action of $(\mathbb Z/2\mathbb Z)^2$

TL;DR

This paper investigates the symplectic actions of the group (Z/2Z)^2 on K3 surfaces, analyzing their effects on cohomology, quotient maps, and moduli space components, revealing new phenomena in non-cyclic group actions.

Contribution

It provides a lattice-theoretic characterization of quotient singularities and describes the moduli space correspondence for non-cyclic group actions on K3 surfaces.

Findings

Describes the action of (Z/2Z)^2 on H^2(X,Z)

Provides a lattice-theoretic characterization of quotient singularities

Establishes a correspondence between moduli space components

Abstract

We study the symplectic action of the group (Z/2Z)^2 on a K3 surface X: we describe its action on H^2(X,Z) and the maps induced in cohomology by the rational quotient maps; we give a lattice-theoretic characterization of the resolution of singularities of the quotient X/i, where i is any of the involutions in (Z/2Z)^2. Assuming X is projective, we describe the correspondence between irreducible components of its moduli space, and those of the resolution of singularities of its quotients: this being the first description of this correspondence for a non-cyclic action, we see new phenomena, of which we provide explicit examples assuming $X$ has a polarization of degree 4.