Classification of non-symplectic automorphisms of order 3 on $K3$ surfaces

TL;DR

This paper classifies non-symplectic automorphisms of order 3 on algebraic K3 surfaces, focusing on their fixed loci and lattice invariants, advancing understanding of symmetries in complex algebraic geometry.

Contribution

It characterizes fixed loci of order 3 automorphisms on K3 surfaces using invariants of 3-elementary lattices, providing new classification results.

Findings

Fixed loci are characterized by invariants of 3-elementary lattices

Automorphisms act trivially on the Néron-Severi lattice

Provides a classification framework for these automorphisms

Abstract

In this paper, we study non-symplectic automorphisms of order 3 on algebraic $K3$ surface over $\mathbb{C}$ which act trivially on the N\'{e}ron-Severi lattice. In particular we shall characterize their fixed locus in terms of the invariants of 3-elementary lattices.