Order 3 symplectic automorphisms on K3 surfaces

TL;DR

This paper extends the understanding of symplectic automorphisms on K3 surfaces from involutions to automorphisms of order 3, detailing their action on cohomology and geometric structures.

Contribution

It explicitly describes the action of order 3 symplectic automorphisms on the K3 lattice and relates the geometry of quotient surfaces to known structures, generalizing previous involution results.

Findings

Explicit description of automorphism action on K3 lattice

Relations between Néron–Severi groups of X and Y

Generalization of Shioda–Inose structures for order 3 automorphisms

Abstract

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms.