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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
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})$.

References [Enhancements On Off] (What's this?)

  • [AG17] I.V.Arzhantsev and S.A.Gaĭfullin, The automorphism group of a rigid affine variety, Math. Nachr. 290 (2017), no.5-6, 662-671. MR 3636369
  • [AFK$ ^{+}$13] I.Arzhantsev, H.Flenner, S.Kaliman, F.Kutzschebauch, and M.Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no.4, 767-823. MR 3039680
  • [BKY12] A.Kanel-Belov, J.-T.Yu, and A.Elishev, On the Zariski topology of automorphism groups of affine spaces and algebras, 2012. arxiv:1207.2045v1
  • [BW00] Y.Berest and G.Wilson, Automorphisms and ideals of the Weyl algebra, Math. Ann. 318 (2000), no. 1, 127-147. MR 1785579
  • [DK08] H.Derksen and G.Kemper, Computing invariants of algebraic groups in arbitrary characteristic, Adv. Math. 217 (2008), no.5, 2089-2129. MR 2388087
  • [FK17] J.-P.Furter and H.Kraft, On the geometry of automorphism groups of affine varieties, in preparation, 2017.
  • [Igu73] J.Igusa, Geometry of absolutely admissible representations, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp.373-452. MR 0367077
  • [Jel91] Z.Jelonek, Identity sets for polynomial automorphisms, J. Pure Appl. Algebra 76 (1991), no.3, 333-337. MR 1147306
  • [Jel15] Z.Jelonek, On the group of automorphisms of a quasi-affine variety, Math. Ann. 362 (2015), no.1-2, 569-578. MR 3343890
  • [Kra84] H.Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984. MR 768181
  • [Kra16] H.Kraft, Algebraic Transformation Groups: An Introduction, Mathematisches Institut, Universität Basel, 2016.
  • [KR17] H.Kraft and A.Regeta, Automorphisms of the Lie algebra of vector fields on affine $ n$-space, J. Eur. Math. Soc. (JEMS) 19 (2017), no.5, 1577-1588. MR 3635361
  • [KRZ17] H.Kraft, A.Regeta, and S.Zimmermann, Small affine $ SL_n$-varieties, in preparation, 2017.
  • [Kum02] S.Kumar, Kac-Moody groups, their flag varieties and representation theory, Progr. Math., vol.204, Birkhäuser Boston Inc., Boston, MA, 2002. MR 1923198
  • [Lie11] A.Liendo, Roots of the affine Cremona group, Transform. Groups 16 (2011), no. 4, 1137-1142. MR 2852493
  • [Ram64] Ch.P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25-33. MR 0166198
  • [Reg17] A.Regeta, Characterization of $ n$-dimensional normal affine $ SL_n$-varieties,
  • [Sha66] I.R.Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5) 25 (1966), no.1-2, 208-212. MR 0485898
  • [Sha81] I.R.Shafarevich, On some infinite-dimensional groups. II, Math. USSR-Izv. 18 (1982), no.1, 185-194. MR 607583

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Additional Information

Hanspeter Kraft
Affiliation: Universität Basel Basel, Switzerland

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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