Characterizing smooth affine spherical varieties via the automorphism group

Andriy Regeta; Immanuel van Santen

doi:10.5802/jep.149

Characterizing smooth affine spherical varieties via the automorphism group
[Caractérisation des variétés sphériques affines lisses par le groupe des automorphismes]

Andriy Regeta ; Immanuel van Santen

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 379-414.

Résumé
Abstract

Soit $G$ un groupe réductif connexe. Nous montrons que le monoïde des poids d’une variété $G$ -sphérique quasi-affine est déterminé par les poids de ses $𝔾_{a}$ -actions non triviales homogènes sous l’action d’un sous-groupe de Borel de $G$ . Comme application, nous obtenons qu’une variété sphérique affine lisse non isomorphe à un tore est déterminée par son groupe des automorphismes (considéré comme un ind-groupe) dans la catégorie des variétés irréductibles affines lisses.

Let $G$ be a connected reductive algebraic group. We prove that for a quasi-affine $G$ -spherical variety the weight monoid is determined by the weights of its non-trivial $𝔾_{a}$ -actions that are homogeneous with respect to a Borel subgroup of $G$ . As an application we get that a smooth affine spherical variety that is non-isomorphic to a torus is determined by its automorphism group (considered as an ind-group) inside the category of smooth affine irreducible varieties.

Reçu le : 2019-12-18
Accepté le : 2021-01-26
Publié le : 2021-02-19

DOI : https://doi.org/10.5802/jep.149
Classification : 14R20, 14M27, 14J50, 22F50
Mots clés : Groupes des automorphismes des variétés quasi-affines, variétés sphériques quasi-affines, sous-groupes de racines, variétés toriques quasi-affines

@article{JEP_2021__8__379_0,
     author = {Andriy Regeta and Immanuel van Santen},
     title = {Characterizing smooth affine spherical varieties via the automorphism group},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     pages = {379--414},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.149},
     language = {en},
     url = {https://jep.centre-mersenne.org/item/JEP_2021__8__379_0/}
}

Andriy Regeta; Immanuel van Santen. Characterizing smooth affine spherical varieties via the automorphism group. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 379-414. doi : 10.5802/jep.149. https://jep.centre-mersenne.org/item/JEP_2021__8__379_0/

Bibliographie

[AG10] I. V. Arzhantsev & S. A. Gaĭfullin - “Cox rings, semigroups, and automorphisms of affine varieties”, Mat. Sb. (N.S.) 201 (2010) no. 1, p. 3-24 | Article | MR 2641086 | Zbl 1201.14040

[AT03] A. Auslender & M. Teboulle - Asymptotic cones and functions in optimization and variational inequalities, Springer Monographs in Math., Springer-Verlag, New York, 2003 | Zbl 1017.49001

[BCF10] P. Bravi & S. Cupit-Foutou - “Classification of strict wonderful varieties”, Ann. Inst. Fourier (Grenoble) 60 (2010) no. 2, p. 641-681 | Article | Numdam | MR 2667789 | Zbl 1195.14068

[Ber03] F. Berchtold - “Lifting of morphisms to quotient presentations”, Manuscripta Math. 110 (2003) no. 1, p. 33-44 | Article | MR 1951798 | Zbl 1014.14026

[BP05] P. Bravi & G. Pezzini - “Wonderful varieties of type $D$ ”, Represent. Theory 9 (2005), p. 578-637 | Article | MR 2183057 | Zbl 1222.14099

[Bra07] P. Bravi - “Wonderful varieties of type $E$ ”, Represent. Theory 11 (2007), p. 174-191 | Article | MR 2346359 | Zbl 1135.14037

[Bri10] M. Brion - “Introduction to actions of algebraic groups”, in Actions hamiltoniennes: invariants et classification, vol. 1, Centre Mersenne, Grenoble, 2010, p. 1-22, https://ccirm.centre-mersenne.org/volume/CCIRM_2010__1/ | Zbl 1217.14029

[CF14] S. Cupit-Foutou - “Wonderful varieties: a geometrical realization”, 2014 | arXiv:0907.2852

[CLS11] D. A. Cox, J. B. Little & H. K. Schenck - Toric varieties, Graduate Studies in Math., vol. 124, American Mathematical Society, Providence, RI, 2011 | Article | MR 2810322 | Zbl 1223.14001

[CRX19] S. Cantat, A. Regeta & J. Xie - “Families of commuting automorphisms, and a characterization of the affine space”, 2019 | arXiv:1912.01567

[Dem70] M. Demazure - “Sous-groupes algébriques de rang maximum du groupe de Cremona”, Ann. Sci. École Norm. Sup. (4) 3 (1970), p. 507-588 | Article | Numdam | Zbl 0223.14009

[FK] J.-P. Furter & H. Kraft - “On the geometry of the automorphism groups of affine varieties” | arXiv:1809.04175

[Fre17] G. Freudenburg - Algebraic theory of locally nilpotent derivations, Encyclopaedia of Math. Sciences, vol. 136, Springer-Verlag, Berlin, 2017 | Article | MR 3700208 | Zbl 1391.13001

[Ful93] W. Fulton - Introduction to toric varieties, Annals of Math. Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993 | Article | MR 1234037 | Zbl 0813.14039

[Gro61] A. Grothendieck - “Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes”, Publ. Math. Inst. Hautes Études Sci. 8 (1961), p. 1-222 | Numdam

[Gro97] F. D. Grosshans - Algebraic homogeneous spaces and invariant theory, Lect. Notes in Math., vol. 1673, Springer-Verlag, Berlin, 1997 | Article | MR 1489234 | Zbl 0886.14020

[Hum75] J. E. Humphreys - Linear algebraic groups, Graduate Texts in Math., vol. 21, Springer-Verlag, New York-Heidelberg, 1975 | MR 396773 | Zbl 0325.20039

[Kal05] S. Kaliman - “On a theorem of Ax”, Proc. Amer. Math. Soc. 133 (2005) no. 4, p. 975-977 | Article | MR 2117196 | Zbl 1060.14002

[Kno91] F. Knop - “The Luna-Vust theory of spherical embeddings”, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) (1991), p. 225-249 | Zbl 0812.20023

[Kno93] F. Knop - “Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins”, Math. Z. 213 (1993) no. 1, p. 33-36 | Article | MR 1217668 | Zbl 0788.14042

[Kra84] H. Kraft - Geometrische Methoden in der Invariantentheorie, Aspects of Math., vol. D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 | Article | Zbl 0569.14003

[Kra17] H. Kraft - “Automorphism groups of affine varieties and a characterization of affine $n$ -space”, Trans. Moscow Math. Soc. 78 (2017), p. 171-186 | Article | MR 3738084 | Zbl 1423.14267

[KRvS19] H. Kraft, A. Regeta & I. van Santen - “Is the affine space determined by its automorphism group?”, Internat. Math. Res. Notices (2019), article ID rny281, 21 pages | Article | Zbl 1418.17051

[Lie10] A. Liendo - “Affine $𝕋$ -varieties of complexity one and locally nilpotent derivations”, Transform. Groups 15 (2010) no. 2, p. 389-425 | Article | MR 2657447 | Zbl 1209.14050

[Los09a] I. V. Losev - “Proof of the Knop conjecture”, Ann. Inst. Fourier (Grenoble) 59 (2009) no. 3, p. 1105-1134 | Article | Numdam | MR 2543664 | Zbl 1191.14075

[Los09b] I. V. Losev - “Uniqueness property for spherical homogeneous spaces”, Duke Math. J. 147 (2009) no. 2, p. 315-343 | Article | MR 2495078 | Zbl 1175.14035

[LRU19] A. Liendo, A. Regeta & C. Urech - “Characterization of affine surfaces with a torus action by their automorphism groups”, 2019 | arXiv:1805.03991

[Lun01] D. Luna - “Variétés sphériques de type $A$ ”, Publ. Math. Inst. Hautes Études Sci. (2001) no. 94, p. 161-226 | Article | Numdam | MR 1896179 | Zbl 1085.14039

[Lun07] D. Luna - “La variété magnifique modèle”, J. Algebra 313 (2007) no. 1, p. 292-319 | Article | MR 2326148 | Zbl 1116.22006

[LV83] D. Luna & T. Vust - “Plongements d’espaces homogènes”, Comment. Math. Helv. 58 (1983) no. 2, p. 186-245 | Article | Zbl 0545.14010

[Ram64] C. P. Ramanujam - “A note on automorphism groups of algebraic varieties”, Math. Ann. 156 (1964), p. 25-33 | Article | MR 166198 | Zbl 0121.16103

[Reg17] A. Regeta - “Characterization of $n$ -dimensional normal affine ${SL}_{n}$ -varieties”, 2017 | arXiv:1702.01173

[Ros56] M. Rosenlicht - “Some basic theorems on algebraic groups”, Amer. J. Math. 78 (1956), p. 401-443 | Article | MR 82183 | Zbl 0079.25703

[Sha94] - Algebraic geometry. IV (I. R. Shafarevich, ed.), Encyclopaedia of Math. Sciences, vol. 55, Springer-Verlag, Berlin, 1994 | Article | MR 1309681

[Tim11] D. A. Timashev - Homogeneous spaces and equivariant embeddings, Encyclopaedia of Math. Sciences, vol. 138, Springer, Heidelberg, 2011 | Article | MR 2797018 | Zbl 1237.14057