On Automorphisms of the Affine Cremona Group

[Sur les automorphismes du groupe de Cremona affine]

Kraft, Hanspeter ; Stampfli, Immanuel

Annales de l'Institut Fourier, Tome 63 (2013), p. 1137-1148 / Harvested from Numdam

Résumé

Nous montrons que tous les automorphismes du groupe des automorphismes polynomiaux de l’espace affine ${\mathbb{C}}^{n}$ sont des automorphismes intérieurs modulo des automorphismes du corps $\mathbb{C}$, si nous nous restreignons au sous-groupe des automorphismes modérés. Ceci généralise un résultat de Julie Déserti traitant le cas de la dimension $n = 2$. Dans ce cas, tous les automorphismes polynomiaux sont modérés. Nos méthodes sont différentes de celles de Julie Déserti et sont basées sur des arguments d’actions de groupes algébriques.

Abstract

We show that every automorphism of the group ${\mathcal{G}}_{n}:=Aut ({\mathbb{A}}^{n})$ of polynomial automorphisms of complex affine $n$-space ${\mathbb{A}}^{n}={\mathbb{C}}^{n}$ is inner up to field automorphisms when restricted to the subgroup $T{\mathcal{G}}_{n}$ of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension $n=2$ where all automorphisms are tame: $T{\mathcal{G}}_{2} = {\mathcal{G}}_{2}$. The methods are different, based on arguments from algebraic group actions.

Détail

Publié le : 2013-01-01

MR 3137481 | Zbl 1297.14059

DOI : https://doi.org/10.5802/aif.2785
Classification:  14R10,  14R20,  14L30
Mots clés: Automorphismes polynomiaux, actions de groupes algébriques, variétés algébriques de dimension infinie, éspace affine

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