Geodesic
Automorphism groups of affine varieties and a~characterization of affine~$n$-space
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 209-226.

Voir la notice de l'article provenant de la source Math-Net.Ru

We show that the automorphism group of affine $n$-space $\mathbb{A}^n$ determines $\mathbb{A}^n$ up to isomorphism: If $X$ is a connected affine variety such that $\mathrm{Aut}(X) \simeq \mathrm{Aut}(\mathbb{A}^n)$ as ind-groups, then $X \simeq \mathbb{A}^n$ as varieties. We also show that every torus appears as $\mathrm{Aut}(X)$ for a suitable irreducible affine variety $X$, but that $\mathrm{Aut}(X)$ cannot be isomorphic to a semisimple group. In fact, if $\mathrm{Aut}(X)$ is finite dimensional and if $X \not\simeq \mathbb{A}^1$, then the connected component $\mathrm{Aut}(X)^{\circ}$ is a torus. Concerning the structure of $\mathrm{Aut}(\mathbb{A}^n)$ we prove that any homomorphism $\mathrm{Aut}(\mathbb{A}^n) \to \mathcal{G}$ of ind-groups either factors through $\mathrm{jac}\colon{\mathrm{Aut}(\mathbb{A}^n)} \to {\Bbbk^*}$ where $\mathrm{jac}$ is the Jacobian determinant, or it is a closed immersion. For $\mathrm{SAut}(\mathbb{A}^n):=\ker(\mathrm{jac})\subset \mathrm{Aut}(\mathbb{A}^n)$ we show that every nontrivial homomorphism $\mathrm{SAut}(\mathbb{A}^n) \to \mathcal{G}$ is a closed immersion. Finally, we prove that every non-trivial homomorphism $\phi\colon{\mathrm{SAut}(\mathbb{A}^n)} \to\mathrm{SAut}(\mathbb{A}^n)$ is an automorphism, and that $\phi$ is given by conjugation with an element from $\mathrm{Aut}(\mathbb{A}^n)$. 
@article{MMO_2017_78_2_a1,
     author = {H. Kraft},
     title = {Automorphism groups of affine varieties and a~characterization of affine~$n$-space},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {209--226},
     publisher = {mathdoc},
     volume = {78},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a1/}
}
TY  - JOUR
AU  - H. Kraft
TI  - Automorphism groups of affine varieties and a~characterization of affine~$n$-space
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2017
SP  - 209
EP  - 226
VL  - 78
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a1/
LA  - en
ID  - MMO_2017_78_2_a1
ER  - 
%0 Journal Article
%A H. Kraft
%T Automorphism groups of affine varieties and a~characterization of affine~$n$-space
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2017
%P 209-226
%V 78
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a1/
%G en
%F MMO_2017_78_2_a1
H. Kraft. Automorphism groups of affine varieties and a~characterization of affine~$n$-space. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 209-226. http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a1/

[1] I. V. Arzhantsev, S. A. Gaĭfullin, “The automorphism group of a rigid affine variety”, Math. Nachr., 290:5–6 (2017), 662–671 | DOI | MR | Zbl

[2] I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, M. Zaidenberg, “Flexible varieties and automorphism groups”, Duke Math. J., 162:4 (2013), 767–823 | DOI | MR | Zbl

[3] A. Kanel-Belov, J.-T. Yu, A. Elishev, On the {Z}ariski topology of automorphism groups of affine spaces and algebras, 2012, arXiv: 1207.2045v1

[4] Y. Berest, G. Wilson, “Automorphisms and ideals of the Weyl algebra”, Math. Ann., 318:1 (2000), 127–147 | DOI | MR | Zbl

[5] H. Derksen, G. Kemper, “Computing invariants of algebraic groups in arbitrary characteristic”, Adv. Math., 217:5 (2008), 2089–2129 | DOI | MR | Zbl

[6] J.-P. Furter, H. Kraft, On the geometry of automorphism groups of affine varieties, in preparation, 2017 | MR

[7] J. Igusa, “Geometry of absolutely admissible representations”, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, 373–452 | MR

[8] Z. Jelonek, “Identity sets for polynomial automorphisms”, J. Pure Appl. Algebra, 76:3 (1991), 333–337 | DOI | MR | Zbl

[9] Z. Jelonek, “On the group of automorphisms of a quasi-affine variety”, Math. Ann., 362:1–2 (2015) | MR | Zbl

[10] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, 1, Friedr. Vieweg Sohn, Braunschweig, 1984 | MR

[11] H. Kraft, Algebraic Transformation Groups: An Introduction, , Mathematisches Institut, Universität Basel, 2016 http://kraftadmin.wixsite.com/hpkraft | Zbl

[12] H. Kraft, A. Regeta, “Automorphisms of the Lie algebra of vector fields on affine $n$-space”, J. Eur. Math. Soc. (JEMS), 19:5 (2017), 1577–1588 | DOI | MR | Zbl

[13] H. Kraft, A. Regeta, S. Zimmermann, Small affine $SL_n$-varieties, in preparation, 2017

[14] S. Kumar, Kac–Moody groups, their flag varieties and representation theory, Progr. Math., 204, Birkhäuser Boston Inc., Boston, MA, 2002 | MR | Zbl

[15] A. Liendo, “Roots of the affine Cremona group”, Transform. Groups, 16:4 (2011), 1137–1142 | DOI | MR | Zbl

[16] Ch. P. Ramanujam, “A note on automorphism groups of algebraic varieties”, Math. Ann., 156 (1964), 25–33 | DOI | MR | Zbl

[17] A. Regeta, Characterization of $n$-dimensional normal affine $SL_n$-varieties, arXiv: 1702.01173 [math.AG]

[18] I. R. Shafarevich, “On some infinite-dimensional groups”, Rend. Mat. e Appl. (5), 25:1–2 (1966), 208–212 | MR | Zbl

[19] I. R. Shafarevich, “On some infinite-dimensional groups. II”, Math. USSR-Izv., 18:1 (1982), 185–194. | DOI | MR | Zbl