On Automorphisms of the Affine Cremona Group

Kraft, Hanspeter; Stampfli, Immanuel

doi:10.5802/aif.2785

On Automorphisms of the Affine Cremona Group
[Sur les automorphismes du groupe de Cremona affine]

Kraft, Hanspeter ¹ ; Stampfli, Immanuel ¹

¹ Universität Basel Mathematisches Institut Rheinsprung 21, CH-4051 Basel (Suisse)

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1137-1148.

Résumé
Abstract

Nous montrons que tous les automorphismes du groupe des automorphismes polynomiaux de l’espace affine $ℂ^{n}$ sont des automorphismes intérieurs modulo des automorphismes du corps $ℂ$ , 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.

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

EuDML MR Zbl

DOI : 10.5802/aif.2785

Classification : 14R10, 14R20, 14L30
Keywords: Polynomial automorphisms, algebraic group actions, ind-varieties, affine n-space
Mot clés : Automorphismes polynomiaux, actions de groupes algébriques, variétés algébriques de dimension infinie, éspace affine

Affiliations des auteurs :

Kraft, Hanspeter ¹ ; Stampfli, Immanuel ¹

¹ Universität Basel Mathematisches Institut Rheinsprung 21, CH-4051 Basel (Suisse)

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     author = {Kraft, Hanspeter and Stampfli, Immanuel},
     title = {On {Automorphisms} of the {Affine} {Cremona} {Group}},
     journal = {Annales de l'Institut Fourier},
     pages = {1137--1148},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {3},
     year = {2013},
     doi = {10.5802/aif.2785},
     zbl = {1297.14059},
     mrnumber = {3137481},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2785/}
}

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Kraft, Hanspeter; Stampfli, Immanuel. On Automorphisms of the Affine Cremona Group. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1137-1148. doi : 10.5802/aif.2785. http://www.numdam.org/articles/10.5802/aif.2785/

Bibliographie
Cité par

[1] Bass, Hyman; Connell, Edwin H.; Wright, David The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.), Volume 7 (1982) no. 2, pp. 287-330 | DOI | MR | Zbl

[2] Białynicki-Birula, A. Remarks on the action of an algebraic torus on $k^{n}$ , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 14 (1966), pp. 177-181 | MR | Zbl

[3] Déserti, Julie Sur le groupe des automorphismes polynomiaux du plan affine, J. Algebra, Volume 297 (2006), pp. 584-599 | DOI | MR | Zbl

[4] Fogarty, John Fixed point schemes, Amer. J. Math., Volume 95 (1973), pp. 35-51 | DOI | MR | Zbl

[5] Kraft, Hanspeter; Popov, Vladimir L. Semisimple group actions on the three-dimensional affine space are linear, Comment. Math. Helv., Volume 60 (1985) no. 3, pp. 466-479 | DOI | MR | Zbl

[6] Kraft, Hanspeter; Russell, Peter Families of group actions, generic isotriviality, and linearization, 2011 (preprint, submitted to Tranformation Groups)

[7] Kraft, Hanspeter; Schwarz, Gerald W. Reductive group actions with one-dimensional quotient, Inst. Hautes Études Sci. Publ. Math. (1992) no. 76, pp. 1-97 | DOI | Numdam | MR | Zbl

[8] Kumar, Shrawan Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser Boston Inc., Boston, MA, 2002 | DOI | MR | Zbl

[9] Liendo, Alvaro Roots of the affine Cremona group, Transform. Groups, Volume 16 (2011) no. 4, pp. 1137-1142 | DOI | MR | Zbl

[10] Serre, Jean-Pierre How to use finite fields for problems concerning infinite fields, Arithmetic, geometry, cryptography and coding theory (Contemp. Math.), Volume 487, Amer. Math. Soc., Providence, RI, 2009, pp. 183-193 | DOI | MR | Zbl

[11] Smith, P. A. A theorem on fixed points for periodic transformations, Ann. of Math. (2), Volume 35 (1934) no. 3, pp. 572-578 | DOI | MR | Zbl

Cité par Sources :