Proof of the Knop conjecture
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 1105-1134.

In this paper we prove the Knop conjecture asserting that two smooth affine spherical varieties with the same weight monoid are equivariantly isomorphic. We also state and prove a uniqueness property for (not necessarily smooth) affine spherical varieties.

Dans cet article nous prouvons la conjecture de Knop qui affirme que deux variétés affines sphériques lisses avec le même monoïde des poids sont isomorphes de manière équivariante. On énonce et prouve également une propriété d’unicité pour des variétés affines sphériques non nécessairement lisses.

Received:
Accepted:
DOI: 10.5802/aif.2459
Classification: 14R20,  53D20
Keywords: Spherical varieties, weight monoids, systems of spherical roots, multiplicity free Hamiltonian actions
Losev, Ivan V. 1

1 Massachusetts Institute of Technology Department of Mathematics Room 2-101 77 Massachusetts Avenue Cambridge MA 02139 (USA)
@article{AIF_2009__59_3_1105_0,
     author = {Losev, Ivan V.},
     title = {Proof of the {Knop} conjecture},
     journal = {Annales de l'Institut Fourier},
     pages = {1105--1134},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     doi = {10.5802/aif.2459},
     zbl = {1191.14075},
     mrnumber = {2543664},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2459/}
}
TY  - JOUR
TI  - Proof of the Knop conjecture
JO  - Annales de l'Institut Fourier
PY  - 2009
DA  - 2009///
SP  - 1105
EP  - 1134
VL  - 59
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2459/
UR  - https://zbmath.org/?q=an%3A1191.14075
UR  - https://www.ams.org/mathscinet-getitem?mr=2543664
UR  - https://doi.org/10.5802/aif.2459
DO  - 10.5802/aif.2459
LA  - en
ID  - AIF_2009__59_3_1105_0
ER  - 
%0 Journal Article
%T Proof of the Knop conjecture
%J Annales de l'Institut Fourier
%D 2009
%P 1105-1134
%V 59
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2459
%R 10.5802/aif.2459
%G en
%F AIF_2009__59_3_1105_0
Losev, Ivan V. Proof of the Knop conjecture. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 1105-1134. doi : 10.5802/aif.2459. https://aif.centre-mersenne.org/articles/10.5802/aif.2459/

[1] Bialynicki-Birula, A. Some properties of the decompositions of algebraic varieties determined by an action of a torus, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., Volume 24 (1976), pp. 667-674 | MR: 453766 | Zbl: 0355.14015

[2] Brion, M. Sur l’image de l’application moment, Lect. Notes Math., Volume 1296, Springer Verlag, 1987 | MR: 932055 | Zbl: 0667.58012

[3] Brion, M.; Luna, D.; Vust, Th. Espaces homogènes sphériques, Invent. Math., Volume 84 (1986), pp. 617-632 | Article | MR: 837530 | Zbl: 0604.14047

[4] Camus, R. Variétés sphériques affines lisses (2001) (Ph. D. Thesis)

[5] Delzant, T. Hamiltoniens périodiques et images convexes de l’application moment, Bul. Soc. Math. France, Volume 116 (1988), pp. 315-339 | Numdam | MR: 984900 | Zbl: 0676.58029

[6] Delzant, T. Classification des actions hamiltoniennes complètement intégrables de rang deux, Ann. Global. Anal. Geom., Volume 8 (1990), pp. 87-112 | Article | MR: 1075241 | Zbl: 0711.58017

[7] Grosshans, F. Constructing invariant polynomials via Tschirnhaus transformations, Lect. Notes Math., Volume 1278, Springer Verlag, 1987 | MR: 924168 | Zbl: 0659.14010

[8] Guillemin, V.; Jeffrey, L.; Sjamaar, R. Symplectic implosion, Transform. Groups, Volume 7 (2002), pp. 155-184 | Article | MR: 1903116 | Zbl: 1015.53054

[9] Kirwan, F. Convexity properties of the moment mapping III, Invent. Math., Volume 77 (1984), pp. 547-552 | Article | MR: 759257 | Zbl: 0561.58016

[10] Knop, F. The Luna-Vust theory of spherical embeddings, Proceedings of the Hydebarad conference on algebraic groups (1991), pp. 225-249 | MR: 1131314 | Zbl: 0812.20023

[11] Knop, F. The asymptotic behaviour of invariant collective motion, Invent. Math., Volume 114 (1994), pp. 309-328 | Article | MR: 1253195 | Zbl: 0802.58024

[12] Knop, F. Automorphisms, root systems and compactifications, J. Amer. Math. Soc., Volume 9 (1996), pp. 153-174 | Article | MR: 1311823 | Zbl: 0862.14034

[13] Knop, F.; Steirteghem, B. Van Classification of smooth affine spherical varieties, Transform. Groups, Volume 11 (2006), pp. 495-516 | Article | MR: 2264463 | Zbl: 1120.14042

[14] Kramer, M. Spharische Untergruppen in kompakten zusammenhangenden Liegruppen, Compos. Math., Volume 38 (1979), pp. 129-153 | Numdam | MR: 528837 | Zbl: 0402.22006

[15] Leahy, A.S. A classification of multiplicity free representations, J. Lie Theory, Volume 8 (1998), pp. 376-391 | MR: 1650378 | Zbl: 0910.22015

[16] Losev, I. Uniqueness properties for spherical homogeneous spaces (to appear in Duke Math. J.)

[17] Luna, D. Grosses cellules pour les variétés sphériques, Austr. Math. Soc. Lect. Ser., Volume 9, Cambridge University Press, 1997 | MR: 1635686 | Zbl: 0902.14037

[18] Luna, D. Variétés sphériques de type A, IHES Publ. Math., Volume 94 (2001), pp. 161-226 | Article | Numdam | MR: 1896179 | Zbl: 1085.14039

[19] Onishchik, A.L.; Vinberg, E.B. Lie groups and algebraic groups, Springer Verlag, 1990 | MR: 1064110 | Zbl: 0722.22004

[20] Popov, V.L. Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Math. USSR-Izv, Volume 7 (1973), pp. 301-327 | Zbl: 0301.14018

[21] Popov, V.L. Contractions of the actions of reductive algebraic groups, Math. USSR Sborhik, Volume 58 (1987), pp. 311-355 | Article | MR: 865764 | Zbl: 0627.14033

[22] Vust, Th. Plongement d’espaces symétriques algébriques: une classification, Ann. Sc. Norm. Super. Pisa, Ser. IV, Volume 17 (1990), pp. 165-194 | Numdam | MR: 1076251 | Zbl: 0728.14041

[23] Wasserman, B. Wonderful varieties of rank 2, Transform. Groups, Volume 1 (1996), pp. 375-403 | Article | MR: 1424449 | Zbl: 0921.14031

[24] Woodward, C. The classification of transversal multiplicity-free Hamiltonian actions, Ann. Global. Anal. Geom., Volume 14 (1996), pp. 3-42 | Article | MR: 1375064 | Zbl: 0877.58022

[25] Woodward, C. Spherical varieties and existence of invariant Kahler structure, Duke Math. J, Volume 93 (1998), pp. 345-377 | Article | MR: 1625995 | Zbl: 0979.53085

Cited by Sources: