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 Tr. Mosk. Mat. Obs., 2017, Volume 78, Issue 2, Pages 209–226 (Mi mmo598)

Automorphism groups of affine varieties and a characterization of affine $n$-space

H. Kraft

Universität Basel, Basel, Switzerland

Abstract: 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 $\mathrm{Aut}(\mathbb{A}^n)$.

Key words and phrases: automorphism groups of affine varieties, ind-groups, Lie algebras of ind-groups, vector fields, affine $n$-spaces

 Funding Agency Grant Number Swiss National Science Foundation The author was partially supported by Swiss National Science Foundation.

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English version:
Transactions of the Moscow Mathematical Society, 2017, 78, 171–186

Document Type: Article
UDC: 512.745, 512.745.4, 512.714
MSC: 20G05, 20G99, 14L24, 14L30, 14L40, 14R10, 14R20, 17B40, 17B65, 17B66
Revised: 08.05.2017
Language: English

Citation: H. Kraft, “Automorphism groups of affine varieties and a characterization of affine $n$-space”, Tr. Mosk. Mat. Obs., 78, no. 2, MCCME, M., 2017, 209–226; Trans. Moscow Math. Soc., 78 (2017), 171–186

Citation in format AMSBIB
\Bibitem{Kra17} \by H.~Kraft \paper Automorphism groups of affine varieties and a~characterization of affine~$n$-space \serial Tr. Mosk. Mat. Obs. \yr 2017 \vol 78 \issue 2 \pages 209--226 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo598} \elib{http://elibrary.ru/item.asp?id=37045063} \transl \jour Trans. Moscow Math. Soc. \yr 2017 \vol 78 \pages 171--186 \crossref{https://doi.org/10.1090/mosc/262} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85037617939}