The goal of these lectures is to explain speaker’s results on uniqueness properties of spherical varieties. By a uniqueness property we mean the following. Consider some special class of spherical varieties. Define some combinatorial invariants for spherical varieties from this class. The problem is to determine whether this set of invariants specifies a spherical variety in this class uniquely (up to an isomorphism). We are interested in three classes: smooth affine varieties, general affine varieties, and homogeneous spaces.
Mots clés : Reductive groups, spherical varieties, combinatorial invariants
@article{CCIRM_2010__1_1_113_0,
author = {Ivan Losev},
title = {Uniqueness properties for spherical varieties},
journal = {Les cours du CIRM},
pages = {113--120},
publisher = {CIRM},
volume = {1},
number = {1},
year = {2010},
doi = {10.5802/ccirm.6},
language = {en},
url = {https://ccirm.centre-mersenne.org/articles/10.5802/ccirm.6/}
}
Ivan Losev. Uniqueness properties for spherical varieties. Les cours du CIRM, Tome 1 (2010) no. 1, pp. 113-120. doi : 10.5802/ccirm.6. https://ccirm.centre-mersenne.org/articles/10.5802/ccirm.6/
[AB] V. Alexeev, M. Brion. Moduli of affine schemes with reductive group action. J. Algebraic Geom. 14(2005), 83-117.
[Br1] M. Brion. Classification des espaces homogènes sphériques. Compositio Math. 63(1987), 189-208.
[Br2] M. Brion. Vers une généralisation des espaces symmétriques. J. Algebra, 134(1990), 115-143.
[D] T. Delzant. Classification des actions hamiltoniennes complètement intégrables de rang deux. Ann. Global. Anal. Geom. 8(1990), 87-112.
[GS] V. Guillemin, Sh. Sternberg. Symplectic techniques in physics. Cambridge University Press, 1984.
[Kn1] F. Knop. The Luna-Vust theory of spherical embeddings. Proceedings of the Hydebarad conference on algebraic groups. Madras: Manoj Prokashan 1991. Available at: http://www.mi.uni-erlangen.de/ knop/papers/LV.html
[Kn2] F. Knop. Über Bewertungen, welche unter einer reductiven Gruppe invariant sind. Math. Ann., 295(1993), p. 333-363.
[Kn3] F. Knop. The asymptotic behaviour of invariant collective motion. Invent. Math. 1994. V.114. p. 309-328.
[Kn4] F. Knop. Automorphisms, root systems and compactifications. J. Amer. Math. Soc. 9(1996), n.1, p. 153-174.
[Kr] M. Krämer. Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math. 38 (1979), 129-153.
[Lo1] I.V. Losev. Proof of the Knop conjecture. Preprint(2006) arXiv:math.AG/0612561v5, 20 pages. To appear in Ann. Inst. Fourier, 59(2009), n.3.
[Lo2] I.V. Losev. Uniqueness property for spherical homogeneous spaces. Preprint (2007), arXiv:math/0703543. Duke Math. J, 147(2009), n.2, 315-343.
[Lo3] I.V. Losev. Demazure embeddings are smooth. Preprint (2007), arXiv:math.AG/0704.3698. International Mathematics Research Notices 2009; doi: 10.1093/imrn/rnp027, 9 pages.
[Lu1] D. Luna. Grosses cellules pour les variétés sphériques. Austr. Math. Soc. Lect. Ser., v.9, 267-280. Cambridge University Press, Cambridge, 1997.
[Lu2] D. Luna. Variétés sphériques de type A. IHES Publ. Math., 94(2001), 161-226.
[Lu3] D. Luna. Sur le plongements de Demazure. J. of Algebra, 258(2002), p. 205-215.
[LV] D. Luna, T. Vust. Plongements d’espaces homogènes. Comment. Math. Helv., 58(1983), 186-245.
[M] I.V. Mikityuk. On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Matem. Sbornik 129(1986), n. 4, p. 514-534. English translation in: Math. USSR Sbornik 57(1987), n.2, 527-546.
[T] D. Timashev. Homogeneous spaces and equivariant embeddings. Preprint, arXiv:math/0602228.
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