K3 surfaces with maximal finite automorphism groups containing $M_{20}$

Bonnafé, Cédric; Sarti, Alessandra

Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 711-730.

Mukai a montré que l’ordre maximal d’un groupe fini agissant fidèlement et symplectiquement sur une surface K3 est $960$ et que, si un tel groupe a pour ordre $960$, alors il est isomorphe au groupe de Mathieu $M_{20}$. Kondo a ensuite montré que l’ordre maximal d’un groupe fini agissant fidèlement sur une K3 surface est $3840$ et qu’un tel groupe contient $M_{20}$ comme sous-groupe d’indice $4$. Kondo a aussi montré qu’il existe une unique surface K3 sur laquelle ce groupe agit fidèlement : c’est la surface de Kummer $\mathrm{Km}(E_i\times E_i)$. Dans cet article, nous décrivons deux autres surfaces K3 admettant un groupe fini d’automorphismes d’ordre $1920$, ces deux groupes et ces deux surfaces K3 étant uniques. Ce résultat a été obtenu indépendamment par S. Brandhorst and K. Hashimoto dans un article à venir, dont le but est de classifier les groupes finis agissant fidèlement sur des K3 surfaces et dont la partie symplectique est maximale.

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that if such a group has order $960$, then it is isomorphic to the Mathieu group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3840$ and this group contains $M_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\mathrm{Km}(E_i\times E_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1920$, both groups contains $M_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.

Classification : 14J28, 14J40, 14J10
Keywords: K3 surfaces, automorphisms
Mot clés : Surfaces K3, automorphismes

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