# Finite subgroups of automorphisms of K3 surfaces

@inproceedings{Brandhorst2021FiniteSO, title={Finite subgroups of automorphisms of K3 surfaces}, author={Simon Brandhorst and Tommy Hofmann}, year={2021} }

. 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…

## 6 Citations

### An explicit model of a polarised K3 surface of genus 5 with a symplectic action of order 192

- Mathematics
- 2022

We give a projective model of a complex polarised K3 surface via the knowledge of a ﬁnite group acting on it. This paper presents the theory used to develop an algorithm for this purpose. It relies…

### Classification of Symplectic Birational Involutions of Manifolds of $OG10$ type

- Mathematics
- 2022

. In this paper we give a classiﬁcation of symplectic birational involutions of manifolds of OG 10-type. We approach this classiﬁcation with three techniques - via involutions of the Leech lattice,…

### Non-symplectic automorphisms of order multiple of seven on K3 surfaces

- Mathematics
- 2022

. In this paper we present a complete classiﬁcation of non-symplectic automorphisms of K3 surfaces whose order is a multiple of seven by describing the topological type of their ﬁxed locus. In the…

### Ricci-flat manifolds of generalized ALG asymptotics

- Mathematics
- 2022

In complex dimensions ≥ 3, we provide a geometric existence for generalized ALG complete non-compact Ricci ﬂat K¨ahler manifolds with Schwartz decay i.e. metric decay in any polynomial rate to an ALG…

### An explicit model of a polarised K3 surface of degree 8 with a symplectic action of T 192

- Mathematics
- 2022

The author gives a projective model of a complex polarised K3 surface via the knowledge of a ﬁnite group acting on it. This paper presents the theory used to develop an algorithm for this purpose. It…

### Equivariant derived equivalence and rational points on K3 surfaces

- Mathematics
- 2022

. We study arithmetic properties of derived equivalent K3 surfaces over the ﬁeld of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.

## References

SHOWING 1-10 OF 63 REFERENCES

### Isometries of lattices and automorphisms of K3 surfaces

- Mathematics
- 2021

. The aim of this paper is to give necessary and suﬃcient conditions for an integral polynomial to be the characteristic polynomial of a semi-simple isometry of some even unimodular lattice of given…

### An atlas of K3 surfaces with finite automorphism group

- MathematicsÉpijournal de Géométrie Algébrique
- 2022

We study the geometry of the K3 surfaces $X$ with a finite number
automorphisms and Picard number $\geq 3$. We describe these surfaces classified
by Nikulin and Vinberg as double covers of simpler…

### Connected Components of the Moduli of Elliptic K3 Surfaces

- MathematicsMichigan Mathematical Journal
- 2018

The combinatorial type of an elliptic K3 surface with a zero section is the pair of the ADE -type of singular fibers and the torsion part of the Mordell-Weil group. We determine the set of connected…

### K3 surfaces with non-symplectic automorphisms of prime order

- Mathematics
- 2009

In this paper we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e. we describe the topological structure of their fixed locus and determine their…

### Symmetries of order four on K3 surfaces

- Mathematics
- 2011

In this paper we study automorphisms of order four on K3 surfaces. We give a classification of the non-symplectic ones when either the square of the automorphism is symplectic, or its fixed locus…

### Order eight non-symplectic automorphisms on elliptic K3 surfaces

- Mathematics
- 2016

In this paper we classify complex K3 surfaces with non-symplectic automorphism of order 8 that leaves invariant a smooth elliptic curve. We show that the rank of the Picard group is either 10, 14 or…

### ON SOME ORDER 6 NON-SYMPLECTIC AUTOMORPHISMS OF ELLIPTIC K3 SURFACES.

- Mathematics
- 2012

We classify non-symplectic automorphisms of order 6 on elliptic K3 surfaces which commute with a given elliptic bration. We show how their study can be reduced to the study of non-symplectic…

### CLASSIFICATION OF ORDER SIXTEEN NON-SYMPLECTIC AUTOMORPHISMS ON K3 SURFACES

- Mathematics
- 2014

In the paper we classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and points and we completely classify…

### Automorphisms of even unimodular lattices and equivariant Witt groups

- MathematicsJournal of the European Mathematical Society
- 2020

We characterize the irreducible polynomials that occur as a characteristic polynomial of an automorphism of an even unimodular lattice of given signature, generalizing a theorem of Gross and…

### Bi-canonical representations of finite automorphisms acting on Enriques surfaces

- Mathematics
- 2015

We classify the bi-canonical representations of finite automorphisms on Enriques surfaces. There are three types of non-trivial cases and examples are given explicitly by Horikawa models. In…