Symmetries of order four on K3 surfaces

TL;DR

This paper classifies order four automorphisms on K3 surfaces, focusing on non-symplectic cases and their fixed loci, expanding understanding of their geometric and cohomological properties.

Contribution

It provides a classification of purely non-symplectic automorphisms of order four on K3 surfaces based on fixed locus structure and cohomology action.

Findings

Classification of non-symplectic automorphisms with fixed curves of genus g>0

Analysis of automorphisms fixing rational curves

Partial results on other fixed locus configurations

Abstract

We study automorphisms of order four on K3 surfaces. The symplectic ones have been first studied by Nikulin, they are known to fix six points and their action on the K3 lattice is unique. In this paper we give a classification of the purely non-symplectic automorphisms by relating the structure of their fixed locus to their action on cohomology, in the following cases: the fixed locus contains a curve of genus g>0; the fixed locus contains at least a curve and all the curves fixed by the square of the automorphism are rational. We give partial results in the other cases. Finally, we classify non-symplectic automorphisms of order four with symplectic square.