On symplectic and non-symplectic automorphisms of \(K3\) surfaces (Q1941171): Difference between revisions

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.