On polynomial automorphisms of affine spaces

V. L. Popov

Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 569-587.

Résumé

In the first part of this paper we prove some general results on the linearizability of algebraic group actions on $\mathbb A^n$. As an application, we get a method of construction and concrete examples of non-linearizable algebraic actions of infinite non-reductive insoluble algebraic groups on $\mathbb A^n$ with a fixed point. In the second part we use these general results to prove that every effective algebraic action of a connected reductive algebraic group $G$ on the $n$-dimensional affine space $\mathbb A^n$ over an algebraically closed field $k$ of characteristic zero is linearizable in each of the following cases: 1) $n=3$; 2) $n=4$ and $G$ is not a one- or two-dimensional torus. In particular, this means that $\operatorname{GL}_3(k)$ is the unique (up to conjugacy) maximal connected reductive subgroup of the automorphism group of the algebra of polynomials in three variables over $k$.

Bibliographie

[1] Bass H., Pismo k avtoru, aprel 11, 1997

[2] Bass H., Haboush W., “Linearizing certain reductive group actions”, Trans. Amer. Math. Soc., 292:2 (1985), 463–482 | DOI | MR | Zbl

[3] Białynicki-Birula A., “Remarks on the action of an algebraic torus on $k^{n}$, I”, Bull. Acad. Polon. Sci. Ser. Sci. Math., Astr., Phys., XIV (1967), 177–188 | MR

[4] Białynicki-Birula A., “Remarks on the action of an algebraic torus on $k^{n}$, II”, Bull. Acad. Polon. Sci. Ser. Sci. Math., Astr., Phys., XV (1967), 123–125 | MR

[5] Borel A., Lineinye algebraicheskie gruppy, Mir, M., 1972 | MR | Zbl

[6] Elashvili A. G., “Kanonicheskii vid i statsionarnye podalgebry tochek obschego polozheniya dlya prostykh lineinykh grupp Li”, Funkts. analiz i ego prilozh., 6 (1972), 51–62 | MR | Zbl

[7] Elashvili A. G., “Statsionarnye podalgebry tochek obschego polozheniya dlya neprivodimykh lineinykh grupp Li”, Funkts. analiz i ego prilozh., 6 (1972), 65–78 | MR | Zbl

[8] Gorbatsevich V. V., Onischik A. L., “Gruppy Li preobrazovanii”, Sovremennye problemy matematiki. Fundamentalnye napravleniya., 20, VINITI, M., 1988, 103–240 | MR

[9] Kaliman S., Makar-Limanov L., “On the Russell–Koras contactible $3$-folds”, J. Alg. Geom., 6 (1997), 247–268 | MR | Zbl

[10] Kaliman S., Makar-Limanov L., Koras M., Russell P., “$\mathbb {C}^*$-actions on $\mathbb {C}^3$ are linearizable”, E. Res. Announc., 3 (1997), 63–71 | DOI | MR | Zbl

[11] Kambayashi T., “Automorphism group of a polynomial ring and algebraic group action on an affine space”, J. of Algebra, 60 (1979), 439–451 | DOI | MR | Zbl

[12] Kambayashi T., Russell P., “On linearizing algebraic torus actions”, J. Pure Appl. Algebra, 23 (1982), 243–250 | DOI | MR | Zbl

[13] Kimura T., “A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications”, J. Algebra, 83:1 (1983), 72–100 | DOI | MR | Zbl

[14] Koras M., “A characterization of $\mathbb {C}^2/\mathbb{ Z}_a$”, Comp. Math., 87 (1993), 241–267 | MR | Zbl

[15] Koras M., Russell P., “$\mathbb {G}_m$-actions on $\mathbb A^{3}$”, Can. Math. Soc. Conf. Proc., 10 (1989), 269–276 | MR

[16] Koras M., Russell P., $\mathbb {C}^*$-actions on $\mathbb {C}^3$: the smooth locus is not of hyperbolic type, CICMA reports No 06, 1996

[17] Koras M., Russell P., “Contractible $3$-folds and $\mathbb {C}^*$-actions on $\mathbb {C}^3$”, J. Alg. Geom., 6 (1997), 671–695 | MR | Zbl

[18] Kraft H., “Challenging problems on affine $n$-space”, Séminaire Bourbaki, 47ème année, no. 802, 1994–95, 1–19

[19] Kraft H., Popov V. L., “Semisimple group actions on the three-dimensional affine space are linear”, Comment. Math. Helvetici, 60 (1985), 466–479 | DOI | MR | Zbl

[20] Kurth A., “Nonlinear equivariant automorphisms”, Manuscr. Math., 94 (1997), 327–335 | DOI | MR | Zbl

[21] Luna D., “Slices étales”, Bull. Soc. Math. France, 33 (1973), 81–105 | MR | Zbl

[22] Masuda M., Petrie T., “Algebraic families of $O(2)$-actions on affine space $\mathbb C^4$”, Proc. Symp. Pure Math., 54:I (1994), 347–354 | MR

[23] Vinberg E. B., Gorbatsevich V. V., Onischik A. L., Stroenie grupp i algebr Li, Sovremennye problemy matematiki. Fundamentalnye napravleniya, 41, VINITI, M., 1990

[24] Panyushev D. I., “Poluprostye gruppy avtomorfizmov chetyrekhmernogo affinnogo prostranstva”, Izv. AN SSSR. Ser. matem., 47:4 (1983), 881–894 | MR

[25] Popov V. L., “Kriterii stabilnosti deistviya poluprostoi gruppy na faktorialnom mnogoobrazii”, Izv. AN SSSR. Ser. matem., 34:3 (1970), 523–531 | MR

[26] Popov V. L., “Klassifikatsiya trekhmernykh affinnykh algebraicheskikh mnogoobrazii, kvaziodnorodnykh otnositelno algebraicheskoi gruppy”, Izv. AN SSSR. Ser. matem., 39:3 (1975), 566–609 | MR | Zbl

[27] Popov V. L., Algebraic actions of connected reductive algebraic groups on $\mathbb A^{3}$ are linearizable, Preprint. June 20, 1996

[28] Popov V. L., Polynomial automorphisms of affine spaces: connected reductive subgroups of $\operatorname{Aut}\mathbb A^{3}$ and $\operatorname{Aut}\mathbb A^{4}$, Preprint. December 25, 1997 | MR

[29] Vinberg E. B., Popov V. L., “Teoriya invariantov”, Sovremennye problemy matematiki. Fundamentalnye napravleniya, 55, VINITI, M., 1989, 137–314 | MR

[30] Rosenlicht M., “Toroidal algebraic groups”, Proc. Amer. Math. Soc., 12 (1961), 984–988 | DOI | MR | Zbl

[31] Sato M., Kimura T., “A classification of irreducible prehomogeneous vector spaces and their relative invariants”, Nagoya Math. J., 65 (1977), 1–155 | MR | Zbl

[32] Shafarevich I. R., “On some infinite dimensional algebraic groups”, Rend. Math. e Appl., 25:2 (1966), 208–212 | MR | Zbl