Automorphism groups of affine varieties and a characterization of affine $n$-space
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- by Hanspeter Kraft
- Trans. Moscow Math. Soc. 2017, 171-186
- DOI: https://doi.org/10.1090/mosc/262
- Published electronically: December 1, 2017
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 $\mathrm {SAut}({\mathbb {A}^n}):=\mathrm {ker}(\mathrm {jac})\subseteq \mathrm {Aut}({\mathbb {A}^n})$ we show that every nontrivial homomorphism $\mathrm {SAut}({\mathbb {A}^n}) \to \mathcal {G}$ is a closed immersion.
Finally, we prove that every nontrivial homomorphism $\varphi \colon \mathrm {SAut}({\mathbb {A}^n}) \to \mathrm {SAut}({\mathbb {A}^n})$ is an automorphism, and that $\varphi$ is given by conjugation with an element from $\mathrm {Aut}({\mathbb {A}^n})$.
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Bibliographic Information
- Hanspeter Kraft
- Affiliation: Universität Basel, Basel, Switzerland
- Email: Hanspeter.Kraft@unibas.ch
- Published electronically: December 1, 2017
- Additional Notes: The author was partially supported by the Swiss National Science Foundation
- © Copyright 2017 H. Kraft
- Journal: Trans. Moscow Math. Soc. 2017, 171-186
- MSC (2010): Primary 20G05, 20G99, 14L24, 14L30, 14L40, 14R10, 14R20, 17B40, 17B65, 17B66
- DOI: https://doi.org/10.1090/mosc/262
- MathSciNet review: 3738084
Dedicated: Dedicated to Ernest Vinberg at the occasion of his 80th birthday