logo AFST

On the classification of normal G-varieties with spherical orbits
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 271-334.

Dans cet article, nous étudions la géométrie des opérations de groupes réductifs dans les variétés algébriques. Étant donné un groupe algébrique réductif connexe G, nous élaborons une approche géométrique et combinatoire basée sur la théorie de Luna–Vust pour décrire toute G-variété normale avec orbites sphériques. Cette description comprend le cas classique des variétés sphériques et la théorie des 𝕋-variétés introduite récemment par Altmann, Hausen et Süss.

In this article, we investigate the geometry of reductive group actions on algebraic varieties. Given a connected reductive group G, we elaborate on a geometric and combinatorial approach based on Luna–Vust theory to describe every normal G-variety with spherical orbits. This description encompasses the classical case of spherical varieties and the theory of 𝕋-varieties recently introduced by Altmann, Hausen, and Süss.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1632
Classification : 14L30,  14M27,  14M25,  13A18
Mots clés : action of algebraic groups, Luna–Vust theory, homogeneous spaces, valuation theory
@article{AFST_2020_6_29_2_271_0,
     author = {Kevin Langlois},
     title = {On the classification of normal $G$-varieties with spherical orbits},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {271--334},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {2},
     year = {2020},
     doi = {10.5802/afst.1632},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1632/}
}
Kevin Langlois. On the classification of normal $G$-varieties with spherical orbits. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 271-334. doi : 10.5802/afst.1632. https://afst.centre-mersenne.org/articles/10.5802/afst.1632/

[1] Dmitry Ahiezer Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom., Volume 1 (1983) no. 1, pp. 49-78 | Article | MR 739893

[2] Valery Alexeev; Michel Brion Moduli of affine schemes with reductive group action, J. Algebr. Geom., Volume 14 (2005) no. 1, pp. 83-117 | Article | MR 2092127 | Zbl 1081.14005

[3] Valery Alexeev; Michel Brion Stable spherical varieties and their moduli, IMRP, Int. Math. Res. Pap. (2006), 46293, 57 pages | MR 2268490 | Zbl 1115.14041

[4] Klaus Altmann; Jürgen Hausen Polyhedral divisors and algebraic torus actions, Math. Ann., Volume 334 (2006) no. 3, pp. 557-607 | Article | MR 2207875 | Zbl 1193.14060

[5] Klaus Altmann; Jürgen Hausen; Hendrik Süss Gluing affine torus actions via divisorial fans, Transform. Groups, Volume 13 (2008) no. 2, pp. 215-242 | Article | MR 2426131 | Zbl 1159.14025

[6] Klaus Altmann; Nathan Owen Ilten; Lars Petersen; Hendrik Süss; Robert Vollmert The geometry of T-varieties, Contributions to algebraic geometry (EMS Series of Congress Reports), European Mathematical Society, 2012, pp. 17-69 | Zbl 1316.14001

[7] Ivan V. Arzhantsev Actions of the group SL 2 that are of complexity one, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 61 (1997) no. 4, pp. 3-18 (translation in Izv. Math. 61 (1997), no. 4, p. 685-698) | MR 1480754

[8] Ivan V. Arzhantsev On the actions of reductive groups with a one-parameter family of spherical orbits, Mat. Sb., Volume 188 (1997) no. 5, pp. 3-20 (translation in Sb. Math. 188 (1997), no. 5, p. 639-655) | MR 1478627 | Zbl 0895.14015

[9] Ivan V. Arzhantsev On the normality of closures of spherical orbits, Funkts. Anal. Prilozh., Volume 31 (1997) no. 4, pp. 66-69 (translation in Funct. Anal. Appl. 31 (1997), no. 4, p. 278-280) | MR 1608908 | Zbl 0913.14014

[10] Ivan V. Arzhantsev A classification of reductive linear groups with spherical orbits, J. Lie Theory, Volume 12 (2002) no. 1, pp. 289-299 | MR 1885047 | Zbl 0999.20034

[11] Ivan V. Arzhantsev Invariant differential operators and representations with spherical orbits, Symmetry in nonlinear mathematical physics, Part 1, 2 (Kyiv, 2001) (Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Mathematics and its Applications) Volume 43(2), Institute of Mathematics of NAS of Ukraine, 2002, pp. 419-424 | MR 1915628 | Zbl 1039.20022

[12] Roman S. Avdeev On solvable spherical subgroups of semisimple algebraic groups, Trans. Mosc. Math. Soc. (2011), pp. 1-44 | MR 3184811 | Zbl 1256.20046

[13] Roman S. Avdeev Strongly solvable spherical subgroups and their combinatorial invariants, Sel. Math., New Ser., Volume 21 (2015) no. 3, pp. 931-993 | Article | MR 3366923 | Zbl 1355.14034

[14] Victor V. Batyrev Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebr. Geom., Volume 3 (1994) no. 3, pp. 493-535 | MR 1269718 | Zbl 0829.14023

[15] Paolo Bravi; Guido Pezzini Wonderful subgroups of reductive groups and spherical systems, J. Algebra, Volume 409 (2014), pp. 101-147 | Article | MR 3198836 | Zbl 1303.14060

[16] Paolo Bravi; Guido Pezzini The spherical systems of the wonderful reductive subgroups, J. Lie Theory, Volume 25 (2015) no. 1, pp. 105-123 | MR 3345829 | Zbl 1349.14164

[17] Paolo Bravi; Guido Pezzini Primitive wonderful varieties, Math. Z., Volume 282 (2016) no. 3-4, pp. 1067-1096 | Article | MR 3473657

[18] Michel Brion Sur la géométrie des variétés sphériques, Comment. Math. Helv., Volume 66 (1991) no. 2, pp. 237-262 | Article | Zbl 0741.14027

[19] Michel Brion Invariants et covariants des groupes algébriques réductifs., Summer course note at Monastir, 1996 (https://www-fourier.ujf-grenoble.fr/~mbrion/monastirrev.pdf)

[20] Michel Brion Curves and divisors in spherical varieties, Algebraic groups and Lie groups (Australian Mathematical Society Lecture Series) Volume 9, Cambridge University Press, 1997, pp. 21-34 | MR 1635672 | Zbl 0883.14024

[21] Michel Brion; Dominique Luna; Thierry Vust Espaces homogènes sphériques, Invent. Math., Volume 84 (1986) no. 3, pp. 617-632 | Article | Zbl 0604.14047

[22] Michel Brion; Franz Pauer Valuations des espaces homogènes sphériques, Comment. Math. Helv., Volume 62 (1987) no. 2, pp. 265-285 | Article | Zbl 0627.14038

[23] Jean-Louis Colliot-Thélène; Boris Kunyavskiĭ; Vladimir L. Popov; Zinovy Reichstein Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?, Compos. Math., Volume 147 (2011) no. 2, pp. 428-466 | Article | MR 2776610 | Zbl 1218.14010

[24] Stéphanie Cupit-Foutou Wonderful varieties: A geometrical realization (2009) (https://arxiv.org/abs/0907.2852)

[25] Corrado De Concini; Claudio Procesi Complete symmetric varieties, Invariant theory (Montecatini, 1982) (Lecture Notes in Mathematics) Volume 996, Springer, 1983, pp. 1-44 | Article | MR 718125 | Zbl 0581.14041

[26] Michel Demazure Anneaux gradués normaux, Introduction à la théorie des singularités, II (Travaux en Cours) Volume 37, Hermann, 1988, pp. 35-68 | Zbl 0686.14005

[27] Igor V. Dolgačev Automorphic forms, and quasihomogeneous singularities, Funkts. Anal. Prilozh., Volume 9 (1975) no. 2, p. 67-68 | MR 568895 | Zbl 0321.14003

[28] Hubert Flenner; Mikhail Zaidenberg Normal affine surfaces with -actions, Osaka J. Math., Volume 40 (2003) no. 4, pp. 981-1009 | MR 2020670 | Zbl 1093.14084

[29] William Fulton Introduction to toric varieties, Annals of Mathematics Studies, Volume 131, Princeton University Press, 1993 (The William H. Roever Lectures in Geometry) | MR 1234037 | Zbl 0813.14039

[30] William Fulton; Robert MacPherson; Frank Sottile; Bernd Sturmfels Intersection theory on spherical varieties, J. Algebr. Geom., Volume 4 (1995) no. 1, pp. 181-193 | MR 1299008 | Zbl 0819.14019

[31] Giuliano Gagliardi A combinatorial smoothness criterion for spherical varieties, Manuscr. Math., Volume 146 (2015) no. 3-4, pp. 445-461 | Article | MR 3312454 | Zbl 1327.14229

[32] Giuliano Gagliardi; Johannes Hofscheier The generalized Mukai conjecture for symmetric varieties, Trans. Am. Math. Soc., Volume 369 (2017) no. 4, pp. 2615-2649 | Article | MR 3592522 | Zbl 1387.14134

[33] Alexander Grothendieck Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math., Inst. Hautes Étud. Sci., Volume 8 (1961), pp. 1-222 | Numdam | Zbl 0118.36206

[34] Alexander Grothendieck Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5, Séminaire de Géométrie Algébrique, Volume 1960/61, Institut des Hautes Études Scientifiques, 1963

[35] Jürgen Hausen; Hendrik Süss The Cox ring of an algebraic variety with torus action, Adv. Math., Volume 225 (2010) no. 2, pp. 977-1012 | Article | MR 2671185 | Zbl 1248.14008

[36] George Kempf; Finn Faye Knudsen; David Mumford; Bernard Saint-Donat Toroidal embeddings. I, Lecture Notes in Mathematics, Volume 339, Springer, 1973 | MR 335518 | Zbl 0271.14017

[37] Friedrich Knop Weylgruppe und Momentabbildung, Invent. Math., Volume 99 (1990) no. 1, pp. 1-23 | Article | MR 1029388 | Zbl 0726.20031

[38] Friedrich Knop The Luna–Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) (1991), pp. 225-249 | Zbl 0812.20023

[39] Friedrich Knop Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind, Math. Ann., Volume 295 (1993) no. 2, pp. 333-363 | Article | Zbl 0789.14040

[40] Friedrich Knop Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins, Math. Z., Volume 213 (1993) no. 1, pp. 33-36 | Article | Zbl 0788.14042

[41] János Kollár; Shigefumi Mori Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Volume 134, Cambridge University Press, 1998 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | MR 1658959 | Zbl 0926.14003

[42] Kevin Langlois Clôture intégrale et opérations de tores algébriques de complexité un dans les variétés affines, Transform. Groups, Volume 18 (2013) no. 3, pp. 739-765 | Article | MR 3084333 | Zbl 1286.13003

[43] Kevin Langlois Polyhedral divisors and torus actions of complexity one over arbitrary fields, J. Pure Appl. Algebra, Volume 219 (2015) no. 6, pp. 2015-2045 | Article | MR 3299717 | Zbl 1333.14061

[44] Kevin Langlois Singularités canoniques et actions horosphériques, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 4, pp. 365-369 | Article | MR 3634672 | Zbl 1394.14028

[45] Kevin Langlois; Clélia Pech; Michel Raibaut Stringy invariants for horospherical varieties of complexity one, Algebr. Geom., Volume 6 (2019) no. 3, pp. 346-383 | MR 3938623 | Zbl 1436.14086

[46] Kevin Langlois; Ronan Terpereau On the geometry of normal horospherical G-varieties of complexity one, J. Lie Theory, Volume 26 (2016) no. 1, pp. 49-78 | MR 3384981 | Zbl 1391.14091

[47] Kevin Langlois; Ronan Terpereau The Cox ring of a complexity-one horospherical variety, Arch. Math., Volume 108 (2017) no. 1, pp. 17-27 | Article | MR 3599538 | Zbl 1430.14094

[48] Qing Liu Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, Volume 6, Oxford University Press, 2002 (Translated from the French by Reinie Erné, Oxford Science Publications) | MR 1917232 | Zbl 0996.14005

[49] Ivan V. Losev Uniqueness property for spherical homogeneous spaces, Duke Math. J., Volume 147 (2009) no. 2, pp. 315-343 | Article | MR 2495078 | Zbl 1175.14035

[50] Dominique Luna Toute variété magnifique est sphérique, Transform. Groups, Volume 1 (1996) no. 3, pp. 249-258 | Article | Zbl 0912.14017

[51] Dominique Luna Grosses cellules pour les variétés sphériques, Algebraic groups and Lie groups (Australian Mathematical Society Lecture Series) Volume 9, Cambridge University Press, 1997, pp. 267-280

[52] Dominique Luna Variétés sphériques de type A, Publ. Math., Inst. Hautes Étud. Sci., Volume 94 (2001), pp. 161-226 | Article | Numdam | Zbl 1085.14039

[53] Dominique Luna; Thierry Vust Plongements d’espaces homogènes, Comment. Math. Helv., Volume 58 (1983) no. 2, pp. 186-245 | Article | Zbl 0545.14010

[54] Hideyuki Matsumura Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8, Cambridge University Press, 1989 (Translated from the Japanese by M. Reid) | MR 1011461 | Zbl 0666.13002

[55] David Mumford Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, Volume 5, Oxford University Press, 1970 | MR 282985 | Zbl 0223.14022

[56] Boris Pasquier Variétés horosphériques de Fano, Bull. Soc. Math. Fr., Volume 136 (2008) no. 2, pp. 195-225 | Article | Numdam | MR 2415341 | Zbl 1162.14030

[57] Boris Pasquier The pseudo-index of horospherical Fano varieties, Int. J. Math., Volume 21 (2010) no. 9, pp. 1147-1156 | Article | MR 2725271 | Zbl 1200.14079

[58] Franz Pauer Normale Einbettungen von G/U, Math. Ann., Volume 257 (1981) no. 3, pp. 371-396 | Article | MR 637959 | Zbl 0461.14013

[59] Nicolas Perrin On the geometry of spherical varieties, Transform. Groups, Volume 19 (2014) no. 1, pp. 171-223 | Article | MR 3177371 | Zbl 1309.14001

[60] Lars Petersen; Hendrik Süss Torus invariant divisors, Isr. J. Math., Volume 182 (2011), pp. 481-504 | Article | MR 2783981 | Zbl 1213.14084

[61] Henry Pinkham Normal surface singularities with C * action, Math. Ann., Volume 227 (1977) no. 2, pp. 183-193 | Article | MR 432636 | Zbl 0338.14010

[62] Roger W. Richardson Jr. Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math., Volume 16 (1972), pp. 6-14 | Article | MR 294336 | Zbl 0242.14010

[63] Maxwell Rosenlicht A remark on quotient spaces, Anais Acad. Brasil. Ci., Volume 35 (1963), pp. 487-489 | MR 171782 | Zbl 0123.13804

[64] Ichirô Satake On representations and compactifications of symmetric Riemannian spaces, Ann. Math., Volume 71 (1960), pp. 77-110 | Article | MR 118775 | Zbl 0094.34603

[65] Jean-Pierre Serre Galois cohomology, Springer, 1997 (Translated from the French by Patrick Ion and revised by the author) | Zbl 0902.12004

[66] Tonny A. Springer Aktionen reduktiver Gruppen auf Varietäten, Algebraische Transformationsgruppen und Invariantentheorie (DMV Seminar) Volume 13, Birkhäuser, 1989, pp. 3-39 | Article | MR 1044583 | Zbl 0706.14029

[67] Hideyasu Sumihiro Equivariant completion, J. Math. Kyoto Univ., Volume 14 (1974), pp. 1-28 | Article | MR 337963

[68] Hendrik Süss Fano threefolds with 2-torus action: a picture book, Doc. Math., Volume 19 (2014), pp. 905-940 | MR 3262075 | Zbl 1341.14021

[69] Dmitry A. Timashëv Classification of G-manifolds of complexity 1, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 61 (1997) no. 2, pp. 127-162 (translation in Izv. Math., 61 (1997), no. 2, p. 363-397) | MR 1470147 | Zbl 0911.14022

[70] Dmitry A. Timashëv Cartier divisors and geometry of normal G-varieties, Transform. Groups, Volume 5 (2000) no. 2, pp. 181-204 | Article | MR 1762117 | Zbl 1034.14004

[71] Dmitry A. Timashëv Torus actions of complexity one, Toric topology (Contemporary Mathematics) Volume 460, American Mathematical Society, 2008, pp. 349-364 | Article | MR 2428367 | Zbl 1151.14037

[72] Dmitry A. Timashëv Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, Volume 138, Springer, 2011 | MR 2797018 | Zbl 1237.14057

[73] Èrnest B. Vinberg Complexity of actions of reductive groups, Funkts. Anal. Prilozh., Volume 20 (1986) no. 1, p. 1-13, 96 | Article | MR 831043

[74] Èrnest B. Vinberg; Vladimir L. Popov A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 749-764 | MR 313260

[75] Èrnest B. Vinberg; Vladimir L. Popov Invariant theory, Algebraic geometry, 4 (Russian) (Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya), Vsesoyuznyĭ Institut Nauchnoĭ i Tekhnicheskoĭ Informatsii, 1989, pp. 137-314 | Zbl 0735.14010

[76] Thierry Vust Plongements d’espaces symétriques algébriques: une classification, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (1990) no. 2, pp. 165-195 | Numdam | MR 1076251 | Zbl 0728.14041

[77] Ben Wasserman Wonderful varieties of rank two, Transform. Groups, Volume 1 (1996) no. 4, pp. 375-403 | Article | MR 1424449 | Zbl 0921.14031