[Paper Review] Extensions of maximal symplectic actions on K3 surfaces

Simon Brandhorst, Kenji Hashimoto|arXiv (Cornell University)|Oct 14, 2019

Advanced Algebra and Geometry26 references4 citations

TL;DR

This paper classifies finite groups $G$ of automorphisms on complex K3 surfaces where the symplectic subgroup $G_s$ is one of Mukai's 11 maximal symplectic groups and $G_s \lneq G$. Using lattice-theoretic classification of orthogonal subgroups and the Torelli theorem, it proves the existence and uniqueness of 42 such pairs $(X,G)$, all of which are rigid and correspond to singular K3 surfaces of Picard number 20.

ABSTRACT

We classify pairs $(X,G)$ consisting of a complex K3 surface $X$ and a finite group $G \leq Aut(X)$ such that the subgroup $G_s \lneq G$ consisting of symplectic automorphisms is among the $11$ maximal symplectic ones as classified by Mukai.

Motivation & Objective

• To classify all finite groups $G$ of automorphisms on complex K3 surfaces where the symplectic part $G_s$ is one of Mukai's 11 maximal symplectic groups.
• To determine the structure of $G$ when $G_s \lneq G$, particularly focusing on the extension by a cyclic group $\mu_n$.
• To establish that such K3 surfaces must be singular (Picard number 20) and rigid, with no moduli.
• To provide explicit projective models for 25 of the 42 pairs and list the remaining 17 as open problems.
• To derive the full automorphism group structure and projective equations for the K3 surfaces with $G \cong M_{20} \rtimes \mu_4$.

Proposed method

• Classify finite subgroups of the orthogonal group $O(\Lambda)$ of the K3 lattice $\Lambda \cong H^2(X,\mathbb{Z})$ up to conjugacy, preserving the symplectic action.
• Use the strong Torelli theorem for K3 surfaces to ensure that the lattice polarization determines the surface up to isomorphism.
• Apply the surjectivity of the period map to guarantee the existence of such K3 surfaces with the given lattice and group action.
• Construct explicit projective models via complete linear systems of ample line bundles $l$ of degree 40 on the K3 surface.
• Use Cremona transformations and invariant theory to lift automorphisms from a quotient $Y$ to the K3 surface $X$ via resolution of singularities.
• Verify automorphism actions via computational algebra using Singular, particularly on the defining ideal of the non-normal model $\overline{X} \subset \mathbb{P}^5$.

Experimental results

Research questions

• RQ1 Which finite groups $G$ extend a given maximal symplectic group $G_s$ (among Mukai's 11) as a proper subgroup in $\operatorname{Aut}(X)$ for a K3 surface $X$?
• RQ2 What are the geometric and arithmetic properties of K3 surfaces admitting such extensions, particularly regarding their Picard number and moduli?
• RQ3 Can explicit projective equations be constructed for the K3 surfaces and their automorphisms in the extended group action?
• RQ4 How does the action of the non-symplectic part $G/G_s \cong \mu_n$ affect the lattice polarization and the geometry of the surface?
• RQ5 What is the structure of the full automorphism group $\operatorname{Aut}(X)$ for these singular K3 surfaces?

Key findings

• There are exactly 42 isomorphism classes of pairs $(X,G)$ where $G_s$ is one of the 11 maximal symplectic groups and $G_s \lneq G$.
• All such K3 surfaces $X$ are singular, i.e., have Picard number 20, and are rigid, meaning they admit no nontrivial deformations.
• The non-symplectic part $G/G_s$ is always cyclic of even order, and the pair $(X,G)$ is determined up to isomorphism by $G_s$ and any involution in $G/G_s$.
• For 25 of the 42 pairs, the paper provides explicit projective equations in $\mathbb{P}^{21}$ via a complete linear system of degree 40.
• The K3 surface with $G \cong M_{20} \rtimes \mu_4$ is realized as a resolution of a complete intersection of type $(2,2,2)$ in $\mathbb{P}^5$, with an explicit Cremona transformation lifting the automorphism.
• The full automorphism group of the $M_{20} \rtimes \mu_4$-surface is generated by the symplectic group $M_{20}$ and a non-symplectic automorphism of order 4, with the action on cohomology realized via lattice isometries.