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Transactions of the Moscow Mathematical Society

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


Author: Hanspeter Kraft
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 2.
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
Published electronically: December 1, 2017
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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})\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 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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Additional Information

Hanspeter Kraft
Affiliation: Universität Basel Basel, Switzerland
Email: Hanspeter.Kraft@unibas.ch

DOI: https://doi.org/10.1090/mosc/262
Keywords: Automorphism groups of affine varieties, ind-groups, Lie algebras of ind-groups, vector fields, affine $n$-spaces
Published electronically: December 1, 2017
Additional Notes: The author was partially supported by the Swiss National Science Foundation
Dedicated: Dedicated to Ernest Vinberg at the occasion of his 80th birthday
Article copyright: © Copyright 2017 H.Kraft

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