Tori in the Cremona groups

V. L. Popov

Izvestiya. Mathematics , Tome 77 (2013) no. 4, pp. 742-771.

Résumé

We classify up to conjugacy all subgroups of certain types in the full, affine and special affine Cremona groups and prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results on the linearization problem by generalizing Białynicki-Birula's results of 1966–67 to disconnected groups. We prove fusion theorems for $n$-dimensional tori in the affine and special affine Cremona groups of rank $n$, and introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.

Détail

Keywords: Cremona group, algebraic torus, diagonalizable algebraic group, conjugate subgroups, fusion theorems
Mots-clés : affine Cremona group, torsion primes.

Bibliographie

[1] V. L. Popov, Problems for problem session, 2005, arXiv: math/0504517

[2] V. L. Popov, “Roots of the affine Cremona group”, Affine algebraic geometry (Seville, Spain, 2003), Contemp. Math., 369, Amer. Math. Soc., Providence, RI, 2005, 12–13 | MR | Zbl

[3] A. Liendo, “Roots of the affine Cremona group”, Transform. Groups, 16:4 (2011), 1137–1142 | DOI | MR | Zbl

[4] A. Białynicki-Birula, “Remarks on the action of an algebraic torus on $k^n$”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 14 (1966), 177–181 | MR | Zbl

[5] A. Białynicki-Birula, “Remarks on the action of an algebraic torus on $k^n$. II”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 15 (1967), 123–125 | MR | Zbl

[6] J. Blanc, “Conjugacy classes of affine automorphisms of $\mathbf K^n$ and linear automorphisms of $\mathbf P^n$ in the Cremona groups”, Manuscripta Math., 119:2 (2006), 225–241 | DOI | MR | Zbl

[7] J.-P. Serre, “Le groupe de Cremona et ses sous-groupes finis.”, Séminaire Bourbaki. Volume 2008/2009. Exposés 997-1011, Astérisque, 332, Soc. Math. de France, Paris, 2010, 75–100 | MR | Zbl

[8] J.-P. Serre, “Faisceaux algébriques cohérents”, Ann. of Math. (2), 61:2 (1955), 197–278 | DOI | MR | Zbl

[9] A. Borel, Linear algebraic groups, Grad. Texts in Math., 126, Springer-Verlag, New York, 1991 | MR | Zbl

[10] V. L. Popov, E. B. Vinberg, “Invariant theory”, Algebraic Geometry IV, Encyclopaedia Math. Sci., 55, Springer-Verlag, Berlin, 1994, 123–284 | DOI | MR | MR | Zbl | Zbl

[11] G. Laumon, “Comparaison de caracteristiques d'Euler-Poincare en cohomologie $l$-adique”, C. R. Acad. Sci. Paris Ser. I Math., 292 (1981), 209–212 | MR | Zbl

[12] H. Kraft, V. L. Popov, “Semisimple group actions on the three-dimensional affine space are linear”, Comment. Math. Helv., 60:3 (1985), 466–479 | DOI | MR | Zbl

[13] C. P. Ramanujam, “A note on automorphism groups of algebraic varieties”, Math. Ann., 156:1 (1964), 25–33 | DOI | MR | Zbl

[14] J. Blanc, “Groupes de Cremona, connexité et simplicité”, Ann. Sci. Éc. Norm. Supér. (4), 43:2 (2010), 357–364 | MR | Zbl

[15] V. L. Popov, “Some subgroups of the Cremona groups”, Proceedings of Conference on Affine Algebraic Geometry (Osaka, 2011), World Scientific

[16] E. B. Vinberg, A course in algebra, Grad. Stud. Math., 56, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[17] M. Marcus, H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, MA, 1964 | MR | MR | Zbl

[18] J.-P. Serre, “Sous-groupes finis des groupes de Lie”, Séminaire Bourbaki. Volume 1998/99. Exposés 850–864, Astérisque, 266, Paris, 2000, 415–430 | MR | Zbl

[19] I. R. Shafarevich, A. O. Remizov, Linear algebra and geometry, Springer-Verlag, Heidelberg, 2013 | MR | Zbl

[20] A. Borel, J.-P. Serre, “Théorèmes de finitude en cohomoplogie galoisienne”, Comment. Math. Helv., 39 (1964), 111–164 | DOI | MR | Zbl

[21] M. Demazure, “Sous-groupes algebriques de rang maximum du groupe de Cremona”, Ann. Sci. École Norm. Sup. (4), 3 (1970), 507–588 | MR | Zbl

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

[23] N. Gupta, A counter-example to the Cancellation Problem for the affine space ${\mathbf A}^3$ in characteristic $p$, arXiv: 1208.0483

[24] V. L. Popov, “Stability criteria for the action of a semisimple group on a factorial manifold”, Math. USSR-Izv., 4:3 (1970), 527–535 | DOI | MR | Zbl | Zbl

[25] M. F. Atiyah, I. G. McDonald, Introduction to commutative algebra, Addison-Wesley, Reading, MA, 1969 | MR | MR | Zbl | Zbl

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

[27] J. Harris, Algebraic geometry, Grad. Texts in Math., 133, Springer-Verlag, New York, 1992 | DOI | MR | Zbl

[28] H. Matsumura, “On algebraic groups of birational transformations”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 34 (1963), 151–155 | MR | Zbl

[29] R. Steinberg, “Regular elements of semisimple algebraic groups”, Inst. Hautes Études Sci. Publ. Math., 25:1 (1965), 49–80 | DOI | MR | Zbl

[30] I. R. Shafarevich, “On some infinite dimensional groups”, Rend. Mat. e Appl. (5), 25:1–2 (1966), 208–212 | MR | Zbl

[31] I. R. Šafarevič, “On some infinite-dimensional groups. II”, Math. USSR-Izv., 18:1 (1982), 185–194 | DOI | MR | Zbl | Zbl

[32] J. Blanc, “The number of conjugacy classes of elements of the Cremona group of some given finite order”, Bull. Soc. Math. France, 135:3 (2007), 419–434 | MR | Zbl

[33] T. Igarashi, Finite subgroups of the automorphism group of the affine plane, Thesis, Osaka University, 1977

[34] V. L. Popov, Tori in Cremona groups, Essential Dimension and Cremona Groups, talk delivered at the International workshop, Chern Institute of Mathematics, Nankai University, Tianjin, China, June 11–15, 2012 http://www.nim.nankai.edu.cn/activites/conferences/hy20120611/index.htm

[35] T. de Fernex, “On planar Cremona maps of prime order”, Nagoya Math. J., 174 (2004), 1–28 | MR | Zbl

[36] J. Blanc, Finite abelian subgroups of the Cremona group of the plane, Thèse No 3777, Université de Genève, Faculté des Sciences, 2006

[37] A. Beauville, “$p$-elementary subgroups of the Cremona group”, J. Algebra, 314:2 (2007), 553–564 | DOI | MR | Zbl