We give a short introduction to the problem of classification of spherical varieties, by presenting the Luna conjecture about the classification of wonderful varieties and illustrating some of the related currently known results.
@article{CCIRM_2010__1_1_99_0, author = {Bravi, Paolo}, title = {Classification of spherical varieties}, booktitle = {Actions hamiltoniennes~: invariants et classification. 6 {\textendash} 10 avril 2009}, series = {Les cours du CIRM}, pages = {99--111}, publisher = {CIRM}, number = {1}, year = {2010}, doi = {10.5802/ccirm.5}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ccirm.5/} }
TY - JOUR AU - Bravi, Paolo TI - Classification of spherical varieties BT - Actions hamiltoniennes : invariants et classification. 6 – 10 avril 2009 AU - Collectif T3 - Les cours du CIRM PY - 2010 SP - 99 EP - 111 IS - 1 PB - CIRM UR - http://www.numdam.org/articles/10.5802/ccirm.5/ DO - 10.5802/ccirm.5 LA - en ID - CCIRM_2010__1_1_99_0 ER -
%0 Journal Article %A Bravi, Paolo %T Classification of spherical varieties %B Actions hamiltoniennes : invariants et classification. 6 – 10 avril 2009 %A Collectif %S Les cours du CIRM %D 2010 %P 99-111 %N 1 %I CIRM %U http://www.numdam.org/articles/10.5802/ccirm.5/ %R 10.5802/ccirm.5 %G en %F CCIRM_2010__1_1_99_0
Bravi, Paolo. Classification of spherical varieties, in Actions hamiltoniennes : invariants et classification. 6 – 10 avril 2009, Les cours du CIRM, no. 1 (2010), pp. 99-111. doi : 10.5802/ccirm.5. http://www.numdam.org/articles/10.5802/ccirm.5/
[A] D.N. Ahiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), 49–78.
[AB] V. Alexeev, M. Brion, Moduli of affine schemes with reductive group action, J. Algebraic Geom. 14 (2005), 83–117.
[BC08] P. Bravi, S. Cupit-Foutou, Equivariant deformations of the affine multicone over a flag variety, Adv. Math. 217 (2008), 2800–2821.
[BC10] P. Bravi, S. Cupit-Foutou, Classification of strict wonderful varieties, Ann. Inst. Fourier (Grenoble) 60 (2010), to appear.
[BL09] P. Bravi, D. Luna, An introduction to wonderful varieties with many examples of type F4, J. Algebra, to appear.
[BP05] P. Bravi, G. Pezzini, Wonderful varieties of type , Represent. Theory, 9 (2005), 578–637.
[BP09] P. Bravi, G. Pezzini, Wonderful varieties of type B and C, arXiv:0909.3771v1 .
[Bra] P. Bravi, Wonderful varieties of type , Represent. Theory, 11 (2007), 174–191.
[Bri87] M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), 189–208.
[Bri90] M. Brion, Vers une généralisation des espaces symétriques, J. Algebra 134 (1990), 115–143.
[Bri] M. Brion, in this volume.
[C08] S. Cupit-Foutou, Invariant Hilbert schemes and wonderful varieties, arXiv:0811.1567v2 .
[C09] S. Cupit-Foutou, Wonderful varieties: a geometrical realization, arXiv:0907.2852v1 .
[DGMP] C. De Concini, M. Goresky, R. MacPharson, C. Procesi, On the geometry of quadrics and their degenerations, Comment. Math. Helv. 63 (1988), 337–413.
[DP] C. De Concini, C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982), Lecture Notes in Math. 996, 1–44, Springer, Berlin, 1983.
[HS] M. Haiman, B. Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), 725–769.
[Kn] F. Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), 153–174.
[Kr] M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), 129–153.
[Lo09] I.V. Losev, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147 No. 2 (2009), 315–343.
[Lo10] I.V. Losev, Proof of the Knop conjecture, Ann. Inst. Fourier (Grenoble), 59 (2009), no. 3, 1105–1134.
[Lo] I.V. Losev, in this volume.
[Lu96] D. Luna, Toute variété magnifique est sphérique, Transform. Groups 1 (1996), 249–258.
[Lu01] D. Luna, Variétés sphériques de type , Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226.
[LV] D. Luna, T. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245.
[Mi] I.V. Mikityuk, On the integrability of invariant hamiltonian systems with homogeneous configurations spaces (in Russian), Math. Sbornik 129, 171 (1986), 514–534.
[P07] G. Pezzini, Simple immersions of wonderful varieties, Math. Z. 255 (2007), 793–812.
[P] G. Pezzini, in this volume.
[W] B. Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), 375–403.
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