On symplectic and non-symplectic automorphisms of \(K3\) surfaces
type: Item
instance of: scholarly article
title: On symplectic and non-symplectic automorphisms of \(K3\) surfaces
zbMATH Open document ID: 1266.14031
DOI: 10.4171/RMI/716
author: Alice Garbagnati
author: Alessandra Sarti
published in: Revista Matemática Iberoamericana
publication date: 2013-03-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.1604
review text: Automorphisms of \(K3\) surfaces can be characterized by their action on the one dimensional space of holomorphic 2-forms. If the action is trivial, the automorphism is called \textit{symplectic}, otherwise it is referred to as \textit{non-symplectic}. Since the work of V. Nikulin in the eighties - who classified non-symplectic involutions, a lot of work has been done and there is now a complete classification of non-symplectic automorphism of prime order (see \textit{M. Artebani} et al., [Math. Z. 268, No. 1-2, 507--533 (2011; Zbl 1218.14024)]). Also, there is a complete classification, due to \textit{V. V. Nikulin}, of finite groups acting symplectically on \(K3\) surfaces [Usp. Mat. Nauk 31, No. 2(188), 223--224 (1976; Zbl 0331.14019)]. In the paper under review, it is the \textit{coexistence} of symplectic and non-symplectic automorphisms that is studied. In particular, the authors raise and address the issue on when a generic member of a family of \(K3\) surfaces having a given group of automorphism admits even more symmetry than that of the original group. The main result is a set of criteria which, given a non-symplectic automorphism of order \(p\), determine the existence of a symplectic automorphism of same order. In particular, the authors provide a no-go theorem for \(p=5\) and \(7\). A non-symplectic automorphism of this particular order prohibits the existence of symplectic automorphism of the same order. The paper is very well structured and gives a nice overview of the issue. Also, it contains numerous examples and references to related works.
reviewed by: Jimmy Dillies
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J50
zbMATH DE Number: 6143290
zbMATH Keywords: \(K3\)
zbMATH Keywords: symplectic
zbMATH Keywords: non-symplectic
zbMATH Keywords: automorphism
MaRDI profile type: Publication
arXiv ID: 1006.1604
cites work: Non-symplectic automorphisms of order 3 on \(K3\) surfaces
cites work: \(K3\) surfaces with non-symplectic automorphisms of prime order. With an appendix by Shigeyuki Kondō
cites work: The index of elliptic operators. III
cites work: Q5749404
cites work: On the Torelli problem for kählerian $K-3$ surfaces
cites work: Q3253968
cites work: Modular invariants for lattice polarized \(K3\) surfaces
cites work: Order 6 non-symplectic automorphisms of K3 surfaces
cites work: Q3595649
cites work: Classification of \(K3\)-surfaces with involution and maximal symplectic symmetry
cites work: The dihedral group $\mathcal D_{5}$ as a group of symplectic automorphisms on K3 surfaces
cites work: Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms
cites work: New families of Calabi-Yau threefolds without maximal unipotent monodromy
cites work: Symplectic automorphisms of prime order on \(K3\) surfaces
cites work: Elliptic Fibrations and Symplectic Automorphisms on K3 Surfaces
cites work: The Picard-Fuchs Equation of a Family of Calabi-Yau Threefolds without Maximal Unipotent Monodromy
cites work: Nikulin involutions on \(K3\) surfaces
cites work: Enriques surfaces and Jacobian elliptic \(K3\) surfaces
cites work: Q4407874
Recommended article: Finite groups of symplectic automorphisms of \(K3\) surfaces in positive characteristic
Recommended article: Q3596145
Recommended article: \(K3\)-surfaces with special symmetry
Recommended article: Classification of non-symplectic automorphisms of order 3 onK3 surfaces
Recommended article: Classification of \(K3\)-surfaces with involution and maximal symplectic symmetry
Recommended article: Symplectic automorphisms on Kummer surfaces
Recommended article: Finite symplectic actions on the \(K3\) lattice
Recommended article: Order 3 symplectic automorphisms on \(K3\) surfaces
Recommended article: Symplectic automorphisms of prime order on \(K3\) surfaces
Recommended article: Classification of degenerations of codimension \({\le}\,5\) and their Picard lattices for Kählerian \(K3\) surfaces with the symplectic automorphism group \((C_2)^2\)
description: scientific article