[Paper Review] How to determine a K3 surface from a finite automorphism

Simon Brandhorst|arXiv (Cornell University)|Apr 29, 2016

Algebraic Geometry and Number Theory33 references3 citations

TL;DR

This paper establishes that a finite non-symplectic automorphism with a small transcendental lattice determines a K3 surface up to isomorphism via an n-th root of unity and an ideal in ℤ[ζₙ]. It generalizes Vorontsov's theorem, classifies high-order purely non-symplectic automorphisms, and proves the existence of infinitely many K3 surfaces with commuting symplectic and non-symplectic automorphisms of order 5, uniquely determined by their Néron-Severi lattice and explicit equations.

ABSTRACT

In this article we pursue the question when an automorphism determines a (complex) K3 surface up to isomorphism. We prove that if the automorphism is finite non-symplectic and the transcendental lattice small, then the isomorphism class of the K3 surface is determined by an n-th root of unity and an ideal in $\mathbb{Z}[\zeta_n]$. As application we give a generalization of Vorontsov's theorem and the classification of purely non-symplectic automorphisms of high order. Furthermore, we prove that there exist infinitely many K3 surfaces with a symplectic and a non-symplectic automorphism of order $5$. If they commute such a K3 surface is unique. We give a description of its Neron-Severi lattice as well as equations of its generators.

Motivation & Objective

• To determine when a finite non-symplectic automorphism uniquely determines a complex K3 surface up to isomorphism.
• To generalize Vorontsov's theorem on K3 surfaces with finite automorphism groups.
• To classify purely non-symplectic automorphisms of high order on K3 surfaces.
• To investigate the existence and structure of K3 surfaces admitting both symplectic and non-symplectic automorphisms of order 5.
• To describe the Néron-Severi lattice and provide explicit equations for generators of such K3 surfaces when automorphisms commute.

Proposed method

• Use of the transcendental lattice to constrain the isomorphism class of the K3 surface.
• Construction of invariants via n-th roots of unity and ideals in the ring of integers ℤ[ζₙ] of cyclotomic fields.
• Application of the theory of finite non-symplectic automorphisms on K3 surfaces to classify their action on cohomology.
• Leveraging the Néron-Severi lattice to characterize the algebraic cycles and determine uniqueness.
• Explicit computation of generators and equations for K3 surfaces with commuting automorphisms of order 5.
• Use of the classification of automorphism groups on K3 surfaces to extend results to higher-order cases.

Experimental results

Research questions

• RQ1 Under what conditions does a finite non-symplectic automorphism determine a K3 surface up to isomorphism?
• RQ2 How can the isomorphism class of a K3 surface be encoded using roots of unity and ideals in ℤ[ζₙ]?
• RQ3 Can Vorontsov's theorem be generalized to include K3 surfaces with small transcendental lattices?
• RQ4 Do there exist infinitely many K3 surfaces admitting both symplectic and non-symplectic automorphisms of order 5?
• RQ5 What is the structure of the Néron-Severi lattice and explicit equations for such K3 surfaces when the automorphisms commute?

Key findings

• A finite non-symplectic automorphism with a small transcendental lattice determines the isomorphism class of a K3 surface via an n-th root of unity and an ideal in ℤ[ζₙ].
• The paper generalizes Vorontsov's theorem to K3 surfaces with finite non-symplectic automorphisms and small transcendental lattices.
• The classification of purely non-symplectic automorphisms of high order is extended using the same invariant framework.
• Infinitely many K3 surfaces exist that admit both a symplectic and a non-symplectic automorphism of order 5.
• When such automorphisms commute, the K3 surface is uniquely determined, with its Néron-Severi lattice fully described and explicit equations for its generators provided.