Classification of spherical varieties

Paolo Bravi[1]

  • [1] Dipartimento di Matematica G. Castelnuovo, Università La Sapienza, P.le Aldo Moro 5, 00185 Roma, Italy

Les cours du CIRM (2010)

  • Volume: 1, Issue: 1, page 99-111
  • ISSN: 2108-7164

Abstract

top
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.

How to cite

top

Bravi, Paolo. "Classification of spherical varieties." Les cours du CIRM 1.1 (2010): 99-111. <http://eudml.org/doc/116366>.

@article{Bravi2010,
abstract = {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.},
affiliation = {Dipartimento di Matematica G. Castelnuovo, Università La Sapienza, P.le Aldo Moro 5, 00185 Roma, Italy},
author = {Bravi, Paolo},
journal = {Les cours du CIRM},
keywords = {spherical varieties; wonderful varieties; symmetric varieties; spherical nilpotent orbits; model spaces},
language = {eng},
number = {1},
pages = {99-111},
publisher = {CIRM},
title = {Classification of spherical varieties},
url = {http://eudml.org/doc/116366},
volume = {1},
year = {2010},
}

TY - JOUR
AU - Bravi, Paolo
TI - Classification of spherical varieties
JO - Les cours du CIRM
PY - 2010
PB - CIRM
VL - 1
IS - 1
SP - 99
EP - 111
AB - 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.
LA - eng
KW - spherical varieties; wonderful varieties; symmetric varieties; spherical nilpotent orbits; model spaces
UR - http://eudml.org/doc/116366
ER -

References

top
  1. D.N. Ahiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), 49–78. Zbl0537.14033MR739893
  2. V. Alexeev, M. Brion, Moduli of affine schemes with reductive group action, J. Algebraic Geom. 14 (2005), 83–117. Zbl1081.14005MR2092127
  3. P. Bravi, Wonderful varieties of type E , Represent. Theory, 11 (2007), 174–191. Zbl1135.14037MR2346359
  4. P. Bravi, S. Cupit-Foutou, Equivariant deformations of the affine multicone over a flag variety, Adv. Math. 217 (2008), 2800–2821. Zbl1171.14029MR2397467
  5. P. Bravi, S. Cupit-Foutou, Classification of strict wonderful varieties, Ann. Inst. Fourier (Grenoble) 60 (2010), to appear. Zbl1195.14068
  6. P. Bravi, D. Luna, An introduction to wonderful varieties with many examples of type F4, J. Algebra, to appear. Zbl1231.14040
  7. P. Bravi, G. Pezzini, Wonderful varieties of type D , Represent. Theory, 9 (2005), 578–637. Zbl1222.14099MR2183057
  8. P. Bravi, G. Pezzini, Wonderful varieties of type B and C, arXiv:0909.3771v1 . 
  9. M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), 189–208. Zbl0642.14011MR906369
  10. M. Brion, Vers une généralisation des espaces symétriques, J. Algebra 134 (1990), 115–143. Zbl0729.14038MR1068418
  11. M. Brion, in this volume. 
  12. S. Cupit-Foutou, Invariant Hilbert schemes and wonderful varieties, arXiv:0811.1567v2 . Zbl1086.14039
  13. S. Cupit-Foutou, Wonderful varieties: a geometrical realization, arXiv:0907.2852v1 . Zbl1195.14068
  14. C. De Concini, M. Goresky, R. MacPharson, C. Procesi, On the geometry of quadrics and their degenerations, Comment. Math. Helv. 63 (1988), 337–413. Zbl0693.14023MR960767
  15. C. De Concini, C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982), Lecture Notes in Math. 996, 1–44, Springer, Berlin, 1983. Zbl0581.14041MR718125
  16. M. Haiman, B. Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), 725–769. Zbl1072.14007MR2073194
  17. F. Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), 153–174. Zbl0862.14034MR1311823
  18. M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), 129–153. Zbl0402.22006MR528837
  19. I.V. Losev, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147 No. 2 (2009), 315–343. Zbl1175.14035MR2495078
  20. I.V. Losev, Proof of the Knop conjecture, Ann. Inst. Fourier (Grenoble), 59 (2009), no. 3, 1105–1134. Zbl1191.14075MR2543664
  21. I.V. Losev, in this volume. 
  22. D. Luna, Toute variété magnifique est sphérique, Transform. Groups 1 (1996), 249–258. Zbl0912.14017MR1417712
  23. D. Luna, Variétés sphériques de type A , Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226. Zbl1085.14039MR1896179
  24. D. Luna, T. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245. Zbl0545.14010MR705534
  25. I.V. Mikityuk, On the integrability of invariant hamiltonian systems with homogeneous configurations spaces (in Russian), Math. Sbornik 129, 171 (1986), 514–534. Zbl0621.70005MR842398
  26. G. Pezzini, Simple immersions of wonderful varieties, Math. Z. 255 (2007), 793–812. Zbl1122.14036MR2274535
  27. G. Pezzini, in this volume. 
  28. B. Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), 375–403. Zbl0921.14031MR1424449

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.