Finite subgroups of automorphisms of K3 surfaces

TL;DR

This paper provides a comprehensive classification of finite automorphism subgroups of K3 surfaces using Hodge theory, lattice theory, and computational methods, linking algebraic and geometric perspectives.

Contribution

It offers the first complete classification of finite automorphism groups of K3 surfaces up to deformation, utilizing Hodge theoretic data and computational techniques.

Findings

Complete classification of finite automorphism subgroups of K3 surfaces.

Connection between automorphism groups and Hodge theoretic data.

Use of computer-aided methods involving hermitian lattices over number fields.

Abstract

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves hermitian lattices over number fields.