RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Journal history

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Tr. Mosk. Mat. Obs.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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.


Full text: PDF file (299 kB)
First page: PDF file
References: PDF file   HTML file

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
Received: 28.03.2017
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}


Linking options:
  • http://mi.mathnet.ru/eng/mmo598
  • http://mi.mathnet.ru/eng/mmo/v78/i2/p209

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Number of views:
    This page:18
    References:4
    First page:4

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019