K3 surfaces with a symplectic automorphism of order 4

TL;DR

This paper analyzes the action of order 4 symplectic automorphisms on K3 surfaces, describing their cohomological effects, lattice structures, and providing explicit projective models for related surfaces.

Contribution

It offers a detailed lattice-theoretic characterization of quotient surfaces and explicit models for K3 surfaces with order 4 symplectic automorphisms.

Findings

Describes the isometry induced by automorphism on second cohomology

Provides a lattice-theoretic classification of quotient surfaces

Constructs three explicit projective models for the surfaces

Abstract

Given $X$ a K3 surface admitting a symplectic automorphism $\tau$ of order 4, we describe the isometry $\tau^*$ on $H^2(X,\mathbb Z)$. Having called $\tilde Z$ and $\tilde Y$ respectively the minimal resolutions of the quotient surfaces $Z=X/\tau^2$ and $Y=X/\tau$, we also describe the maps induced in cohomology by the rational quotient maps $X\rightarrow\tilde Z,\ X\rightarrow\tilde Y$ and $\tilde Y\rightarrow\tilde Z$: with this knowledge, we are able to give a lattice-theoretic characterization of $\tilde Z$, and find the relation between the N\'eron-Severi lattices of $X,\tilde Z$ and $\tilde Y$ in the projective case. We also produce three different projective models for $X,\tilde Z$ and $\tilde Y$, each associated to a different polarization of degree 4 on $X$.