Soit un groupe réductif connexe. Nous montrons que le monoïde des poids d’une variété -sphérique quasi-affine est déterminé par les poids de ses -actions non triviales homogènes sous l’action d’un sous-groupe de Borel de . 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 be a connected reductive algebraic group. We prove that for a quasi-affine -spherical variety the weight monoid is determined by the weights of its non-trivial -actions that are homogeneous with respect to a Borel subgroup of . 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.
@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/
[AG10] - “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] - Asymptotic cones and functions in optimization and variational inequalities, Springer Monographs in Math., Springer-Verlag, New York, 2003 | Zbl 1017.49001
[BCF10] - “Classification of strict wonderful varieties”, Ann. Inst. Fourier (Grenoble) 60 (2010) no. 2, p. 641-681 | Article | Numdam | MR 2667789 | Zbl 1195.14068
[Ber03] - “Lifting of morphisms to quotient presentations”, Manuscripta Math. 110 (2003) no. 1, p. 33-44 | Article | MR 1951798 | Zbl 1014.14026
[BP05] - “Wonderful varieties of type ”, Represent. Theory 9 (2005), p. 578-637 | Article | MR 2183057 | Zbl 1222.14099
[Bra07] - “Wonderful varieties of type ”, Represent. Theory 11 (2007), p. 174-191 | Article | MR 2346359 | Zbl 1135.14037
[Bri10] - “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] - “Wonderful varieties: a geometrical realization”, 2014 | arXiv:0907.2852
[CLS11] - Toric varieties, Graduate Studies in Math., vol. 124, American Mathematical Society, Providence, RI, 2011 | Article | MR 2810322 | Zbl 1223.14001
[CRX19] - “Families of commuting automorphisms, and a characterization of the affine space”, 2019 | arXiv:1912.01567
[Dem70] - “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] - “On the geometry of the automorphism groups of affine varieties” | arXiv:1809.04175
[Fre17] - Algebraic theory of locally nilpotent derivations, Encyclopaedia of Math. Sciences, vol. 136, Springer-Verlag, Berlin, 2017 | Article | MR 3700208 | Zbl 1391.13001
[Ful93] - Introduction to toric varieties, Annals of Math. Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993 | Article | MR 1234037 | Zbl 0813.14039
[Gro61] - “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] - Algebraic homogeneous spaces and invariant theory, Lect. Notes in Math., vol. 1673, Springer-Verlag, Berlin, 1997 | Article | MR 1489234 | Zbl 0886.14020
[Hum75] - Linear algebraic groups, Graduate Texts in Math., vol. 21, Springer-Verlag, New York-Heidelberg, 1975 | MR 396773 | Zbl 0325.20039
[Kal05] - “On a theorem of Ax”, Proc. Amer. Math. Soc. 133 (2005) no. 4, p. 975-977 | Article | MR 2117196 | Zbl 1060.14002
[Kno91] - “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] - “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] - Geometrische Methoden in der Invariantentheorie, Aspects of Math., vol. D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 | Article | Zbl 0569.14003
[Kra17] - “Automorphism groups of affine varieties and a characterization of affine -space”, Trans. Moscow Math. Soc. 78 (2017), p. 171-186 | Article | MR 3738084 | Zbl 1423.14267
[KRvS19] - “Is the affine space determined by its automorphism group?”, Internat. Math. Res. Notices (2019), article ID rny281, 21 pages | Article | Zbl 1418.17051
[Lie10] - “Affine -varieties of complexity one and locally nilpotent derivations”, Transform. Groups 15 (2010) no. 2, p. 389-425 | Article | MR 2657447 | Zbl 1209.14050
[Los09a] - “Proof of the Knop conjecture”, Ann. Inst. Fourier (Grenoble) 59 (2009) no. 3, p. 1105-1134 | Article | Numdam | MR 2543664 | Zbl 1191.14075
[Los09b] - “Uniqueness property for spherical homogeneous spaces”, Duke Math. J. 147 (2009) no. 2, p. 315-343 | Article | MR 2495078 | Zbl 1175.14035
[LRU19] - “Characterization of affine surfaces with a torus action by their automorphism groups”, 2019 | arXiv:1805.03991
[Lun01] - “Variétés sphériques de type ”, Publ. Math. Inst. Hautes Études Sci. (2001) no. 94, p. 161-226 | Article | Numdam | MR 1896179 | Zbl 1085.14039
[Lun07] - “La variété magnifique modèle”, J. Algebra 313 (2007) no. 1, p. 292-319 | Article | MR 2326148 | Zbl 1116.22006
[LV83] - “Plongements d’espaces homogènes”, Comment. Math. Helv. 58 (1983) no. 2, p. 186-245 | Article | Zbl 0545.14010
[Ram64] - “A note on automorphism groups of algebraic varieties”, Math. Ann. 156 (1964), p. 25-33 | Article | MR 166198 | Zbl 0121.16103
[Reg17] - “Characterization of -dimensional normal affine -varieties”, 2017 | arXiv:1702.01173
[Ros56] - “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] - Homogeneous spaces and equivariant embeddings, Encyclopaedia of Math. Sciences, vol. 138, Springer, Heidelberg, 2011 | Article | MR 2797018 | Zbl 1237.14057