Skip to main content
Log in

ON THE GEOMETRY OF SPHERICAL VARIETIES

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

This is a survey article on the geometry of spherical varieties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. P. Achinger, N. Perrin, Spherical multiple flag varieties, preprint arXiv:1307.7236.

  2. D. Akhiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors. Ann. Global Anal. Geom. 1 (1983), no. 1, 49–78.

    MATH  MathSciNet  Google Scholar 

  3. Д. Н. Ахиезер, О действиях с конечным числом орбит, Фуккц. анализ и его прил. 19 (1985), no. 1, 1–5. Engl. transl.: D. N. Akhiezer, Actions with a finite number of orbits, Functional Analysis and Its Applications 19 (1985), no. 1, 1–4.

    Google Scholar 

  4. V. Alexeev, M. Brion, Toric degenerations of spherical varieties. Selecta Math. (N.S.) 10 (2004), no. 4, 453–478.

  5. V. Alexeev, M. Brion, Stable spherical varieties and their moduli, IMRP Int. Math. Res. Pap. 2006, Art. ID 46293, 1–57.

    MathSciNet  Google Scholar 

  6. D. Anderson, Okounkov bodies and toric degenerations, Math. Ann. 356 (2013), no. 3, 1183–1202.

    MATH  MathSciNet  Google Scholar 

  7. E. Arrondo, J. Caravantes, On the Picard group of low-codimension subvarieties, Indiana Univ. Math. J. 58 (2009), no. 3, 1023–1050.

    MATH  MathSciNet  Google Scholar 

  8. M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15.

    MATH  MathSciNet  Google Scholar 

  9. Р. С. Авдеев, О разрешимых сферических подгруппах полупростых алгебраических групп, Труды ММО 71 (2010), 235–269. Engl. transl.: R. Avdeev, On solvable spherical subgroups of semisimple algebraic groups, Transactions of the Moscow Mathematical Society 2011, 1–44.

    Google Scholar 

  10. W Barth, Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projektiven Raum, Math. Ann. 187 (1970), 150–162.

    MATH  MathSciNet  Google Scholar 

  11. W. Barth, M. E. Larsen, On the homotopy groups of complex projective algebraic manifolds, Math. Scand. 30 (1972), 88–94.

    MATH  MathSciNet  Google Scholar 

  12. C. Benson, G. Ratcliff, A classification of multiplicity free actions, J. Algebra 181 (1996), no. 1, 152–186.

    MATH  MathSciNet  Google Scholar 

  13. A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.

    Google Scholar 

  14. F. Bien, Orbits, multiplicities and differential operators, in: Representation Theory of Groups and Algebras, Contemp. Math., Vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 199–227.

  15. F. Bien, M. Brion, Automorphisms and local rigidity of regular varieties, Compositio Math. 104 (1996), no. 1, 1–26.

    MATH  MathSciNet  Google Scholar 

  16. A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57, (1953), 115–207.

    Google Scholar 

  17. A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.

  18. A. Borel, A. Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961), 461–513.

    MATH  MathSciNet  Google Scholar 

  19. P. Bravi, Wonderful varieties of type E, Represent. Theory 11 (2007), 174–191.

    MATH  MathSciNet  Google Scholar 

  20. P. Bravi, G. Pezzini, Wonderful varieties of type D, Represent. Theory 9 (2005), 578–637.

    MATH  MathSciNet  Google Scholar 

  21. P. Bravi, G. Pezzini, A constructive approach to the classification of wonderful varieties, preprint arXiv:1103.0380.

  22. P. Bravi, G. Pezzini, Primitive wonderful varieties, preprint arXiv:1106.3187.

  23. P. Bravi, G. Pezzini, The spherical systems of the wonderful reductive subgroups, preprint arXiv:1109.6777.

  24. M. Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math. 55 (1986), no. 2, 191–198.

    MATH  MathSciNet  Google Scholar 

  25. M. Brion, Sur l’image de l’application moment, in: Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986), Lecture Notes in Math., Vol. 1296, Springer, Berlin, 1987, pp. 177–192.

  26. M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), no. 2, 189–208.

    MATH  MathSciNet  Google Scholar 

  27. M. Brion, Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J. 58 (1989), no. 2, 397–424.

    MATH  MathSciNet  Google Scholar 

  28. M. Brion, Une extension du théorème de Borel-Weil, Math. Ann. 286 (1990), no. 4, 655–660.

    MATH  MathSciNet  Google Scholar 

  29. M. Brion, Vers une généralisation des espaces symétriques, J. Algebra 134 (1990), no. 1, 115–143.

    MATH  MathSciNet  Google Scholar 

  30. M. Brion, Sur la géométrie des variétés sphériques, Comment. Math. Helv. 66 (1991), no. 2, 237–262.

    MATH  MathSciNet  Google Scholar 

  31. M. Brion, Variétés sphériques et théorie de Mori, Duke Math. J. 72 (1993), no. 2, 369–404.

    MATH  MathSciNet  Google Scholar 

  32. M. Brion, Représentations des groupes réductifs dans des espaces de cohomologie, Math. Ann. 300 (1994), no. 4, 589–604.

    MATH  MathSciNet  Google Scholar 

  33. M. Brion, Curves and divisors in spherical varieties, in: Algebraic Groups and Lie Groups, Austral. Math. Soc. Lect. Ser., Vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 21–34.

  34. M. Brion, On orbit closures of spherical subgroups in flag varieties, Comment. Math. Helv. 76 (2001), no. 2, 263–299.

    MATH  MathSciNet  Google Scholar 

  35. M. Brion, Multiplicity-free subvarieties of flag varieties, in: Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math., Vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 13–23.

  36. M. Brion, The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), no. 1, 61–99.

    MATH  MathSciNet  Google Scholar 

  37. M. Brion, Log homogeneous varieties, in: Proceedings of the XVIth Latin American Algebra Colloquium, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 1–39.

  38. M. Brion, Variétés sphériques, available at http://www-fourier.ujf-grenoble.fr/∼mbrion/spheriques.pdf.

  39. Brion, M., Spherical varieties, available at http://www-fourier.ujf-grenoble.fr/∼mbrion/notes_bremen.pdf.

  40. M. Brion, Invariant Hilbert schemes, in: Handbook of Moduli, Vol. I, Advanced Lectures in Mathematics, Vol. 24, International Press, 2013, 63–118.

  41. M. Brion, F. Knop, Contractions and flips for varieties with group action of small complexity, J. Math. Sci. Univ. Tokyo 1 (1994), no. 3, 641–655.

    MATH  MathSciNet  Google Scholar 

  42. M. Brion, S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, Vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005.

    Google Scholar 

  43. M. Brion, D. Luna, T. Vust, Espaces homogènes sphériques, Invent. Math. 84 (1986), no. 3, 617–632.

    MATH  MathSciNet  Google Scholar 

  44. M. Brion, F. Pauer, Valuations des espaces homogènes sphériques, Comment. Math. Helv. 62 (1987), no. 2, 265–285.

    MATH  MathSciNet  Google Scholar 

  45. R. Bryant, Rigidity and quasi-rigidity of extremal cycles in compact Hermitian symmetric spaces, to appear in Annals of Mathematics Studies, Vol. 153, Princeton University Press.

  46. P. Caldero, Toric degenerations of Schubert varieties, Transform. Groups 7 (2002), no. 1, 51–60.

    MATH  MathSciNet  Google Scholar 

  47. E. Cartan, Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France 54 (1926), 214–264.

    MathSciNet  Google Scholar 

  48. E. Cartan, Sur une classe remarquable d’espaces de Riemann. II, Bull. Soc. Math. France 55 (1927), 114–134.

    MathSciNet  Google Scholar 

  49. C. De Concini, C., Procesi, Complete symmetric varieties, in: Invariant Theory (Montecatini, 1982), Lecture Notes in Math., Vol. 996, Springer, Berlin, 1983, pp. 1–44.

  50. C. De Concini, T. A. Springer, Compactification of symmetric varieties, Transform. Groups 4 (1999), nos. 2–3, 273–300.

  51. I. Coskun, Rigid and non-smoothable Schubert classes, J. Differential Geom. 87 (2011), no. 3, 493–514.

    MATH  MathSciNet  Google Scholar 

  52. S. Cupit-Foutou, Classification of two-orbit varieties, Comment. Math. Helv. 78 (2003), no. 2, 245–265.

    MATH  MathSciNet  Google Scholar 

  53. S. Cupit-Foutou, Wonderful varieties: A geometrical realization, preprint arXiv: 0907.2852.

  54. O. Debarre, Théorèmes de connexité pour les produits d’espaces projectifs et les grassmanniennes, Amer. J. Math. 118 (1996), no. 6, 1347–1367.

    MATH  MathSciNet  Google Scholar 

  55. T. Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339.

    MATH  MathSciNet  Google Scholar 

  56. T. Delzant, C. Wacheux, Actions hamiltoniennes, in: Actions hamiltoniennes: invariants et classification, Les cours du CIRM 1 (2010), no. 1, pp. 23–31.

    Google Scholar 

  57. R. Elkik, Singularités rationnelles et déformations, Invent. Math. 47 (1978), no. 2, 139–147.

    MATH  MathSciNet  Google Scholar 

  58. G. Ellingsrud, S. A. Strømme, Towards the Chow ring of the Hilbert scheme of \( {{\mathbb{P}}^2} \), J. Reine Angew. Math. 441 (1993), 33–44.

    MATH  MathSciNet  Google Scholar 

  59. G. Faltings, Formale Geometrie und homogene Räume, Invent. Math. 64 (1981), no. 1, 123–165.

    MATH  MathSciNet  Google Scholar 

  60. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.

    Google Scholar 

  61. W. Fulton, Intersection Theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 2, Springer-Verlag, Berlin, 1998.

  62. W. Fulton, R. Lazarsfeld, Connectivity and its applications in algebraic geometry, in: Algebraic Geometry (Chicago, Ill., 1980), Lecture Notes in Math., Vol. 862, Springer, Berlin-New York, 1981, pp. 26–92.

    Google Scholar 

  63. W. Fulton, R. MacPherson, F. Sottile, B. Sturmfels, Intersection theory on spherical varieties. J. Algebraic Geom. 4 (1995), no. 1, 181–193.

    MATH  MathSciNet  Google Scholar 

  64. W. Fulton, B. Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353.

    MATH  MathSciNet  Google Scholar 

  65. F. D. Grosshans, Contractions of the actions of reductive algebraic groups in arbitrary characteristic, Invent. Math. 107 (1992), no. 1, 127–133.

    MATH  MathSciNet  Google Scholar 

  66. V. Guillemin, S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513.

    MATH  MathSciNet  Google Scholar 

  67. V. Guillemin, S. Sternberg, Convexity properties of the moment mapping. II, Invent. Math. 77 (1984), no. 3, 533–546.

    MATH  MathSciNet  Google Scholar 

  68. R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032.

    MATH  MathSciNet  Google Scholar 

  69. R. Hartshorne, E. Rees, E. Thomas, Nonsmoothing of algebraic cycles on Grassmann varieties, Bull. Amer. Math. Soc. 80 (1974), 847–851.

    MATH  MathSciNet  Google Scholar 

  70. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Харстхорн, Алгебраическая геометрия, Мир, М., 1981.

  71. X. He, J. F. Thomsen, Geometry of B × B-orbit closures in equivariant embeddings, Adv. Math. 216 (2007), no. 2, 626–646.

    MATH  MathSciNet  Google Scholar 

  72. X. He, J. F. Thomsen, Frobenius splitting and geometry of G-Schubert varieties, Adv. Math. 219 (2008), no. 5, 1469–1512.

    MATH  MathSciNet  Google Scholar 

  73. H. Hironaka, Smoothing of algebraic cycles of small dimensions, Amer. J. Math. 90 (1968), 1–54.

    MATH  MathSciNet  Google Scholar 

  74. A. Hirschowitz, Le groupe de Chow équivariant, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 5, 87–89.

    MATH  MathSciNet  Google Scholar 

  75. J. Hong, Rigidity of smooth Schubert varieties in Hermitian symmetric spaces, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2361–2381.

    MATH  MathSciNet  Google Scholar 

  76. Y. Hu, S. Keel, Mori dream spaces and GIT. Michigan Math. J. 48 (2000), 331–348.

    MATH  MathSciNet  Google Scholar 

  77. V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213.

    MATH  MathSciNet  Google Scholar 

  78. K. Kaveh, SAGBI bases and degeneration of spherical varieties to toric varieties, Michigan Math. J. 53 (2005), no. 1, 109–121.

    MATH  MathSciNet  Google Scholar 

  79. K. Kaveh, Note on cohomology rings of spherical varieties and volume polynomial, J. Lie Theory 21 (2011), no. 2, 263–283.

    MATH  MathSciNet  Google Scholar 

  80. K. Kaveh, Crystal bases and Newton-Okounkov bodies, preprint arXiv:1101.1687.

  81. K. Kaveh, A. G. Khovanskiĭ, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2) 176 (2012), no. 2, 925–978.

  82. K. Kaveh, A. G. Khovanskiĭ, Convex bodies associated to actions of reductive groups, Mosc. Math. J. 12 (2012), no. 2, 369–396, 461.

  83. Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, in: Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math., Vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360.

  84. G. R. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), no. 3, 229–239.

    MATH  MathSciNet  Google Scholar 

  85. V. Kiritchenko, Gelfand-Zetlin polytopes and flag varieties, Int. Math. Res. Not. IMRN (2010), no. 13, 2512-2531.

  86. В. А. Кириченко, Е. Ю. Смирнов, В. А. Тиморин, Исчисление Шуберта и многогранники Гельфанда-Цетлина, УМН 67 (2012), вып. 4(406), 89–128. Engl. transl.: V. Kiritchenko, E. Smirnov, V. Timorin, Schubert calculus and Gelfand-Zetlin polytopes, Russ. Math. Surv. 67(2012), no. 67(4), 685–719.

  87. F. Kirwan, Convexity properties of the moment mapping. III, Invent. Math. 77 (1984), no. 3, 547–552.

    MATH  MathSciNet  Google Scholar 

  88. K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Classics in Mathematics, Springer-Verlag, Berlin, 2005.

    Google Scholar 

  89. F. Knop, Weylgruppe und Momentabbildung, Invent. Math. 99 (1990), no. 1, 1–23.

    MATH  MathSciNet  Google Scholar 

  90. F. Knop, Weylgruppe, Momentabbildung und äquivariante Einbettung, Habilitation, Universität Basel, 1990, available at http://www.algeo.math.uni-erlan-gen.de/fileadmin/algeo/users/knop/papers/papers/habil.pdf.

  91. F. Knop, F., The Luna-Vust theory of spherical embeddings, in: Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, 1991, pp. 225–249.

  92. F. Knop, Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins, Math. Z. 213 (1993), no. 1, 33–36.

    MATH  MathSciNet  Google Scholar 

  93. F. Knop, Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind, Math. Ann. 295 (1993), no. 2, 333–363.

    MATH  MathSciNet  Google Scholar 

  94. F. Knop, The asymptotic behavior of invariant collective motion, Invent. Math. 116 (1994), no. 1, 309–328.

    MATH  MathSciNet  Google Scholar 

  95. F. Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. (2) 140 (1994), no. 2, 253–288.

  96. F. Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995), no. 2, 285–309.

    MATH  MathSciNet  Google Scholar 

  97. F. Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), no. 1, 153–174.

    MATH  MathSciNet  Google Scholar 

  98. F. Knop, Automorphisms of multiplicity free Hamiltonian manifolds, J. Amer. Math. Soc. 24 (2011), no. 2, 567–601.

    MATH  MathSciNet  Google Scholar 

  99. F. Knop, Localization of spherical varieties, preprint arXiv:1303.2561.

  100. F. Knop, Spherical roots of spherical varieties, preprint arXiv:1303.2466.

  101. F. Knop, B. Van Steirteghem, Classification of smooth affine spherical varieties, Transform. Groups 11 (2006), no. 3, 495–516.

    MATH  MathSciNet  Google Scholar 

  102. J. Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171–202.

  103. H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984.

    MATH  Google Scholar 

  104. M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), no. 2, 129–153.

    MATH  MathSciNet  Google Scholar 

  105. R. Lazarsfeld, Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 48. Springer-Verlag, Berlin, 2004.

  106. R. Lazarsfeld, M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835.

  107. A. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), no. 2, 367–391.

    MATH  MathSciNet  Google Scholar 

  108. P. Littelmann, On spherical double cones, J. Algebra 166 (1994), no. 1, 142–157.

    MATH  MathSciNet  Google Scholar 

  109. P. Littelmann, Cones, crystals, and patterns, Transform. Groups 3 (1998), no. 2, 145–179.

    MATH  MathSciNet  Google Scholar 

  110. I. Losev, Demazure embeddings are smooth, Int. Math. Res. Not. IMRN (2009), no. 14, 2588–2596.

  111. I. Losev, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147 (2009), no. 2, 315–343.

    MATH  MathSciNet  Google Scholar 

  112. I. Losev, Proof of the Knop conjecture, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 1105–1134.

    MATH  MathSciNet  Google Scholar 

  113. I. Losev, Uniqueness properties for spherical varieties, in: Actions hamiltoniennes: invariants et classification, Les cours du CIRM, 1 (2010), no. 1, pp. 113–120.

    Google Scholar 

  114. D. Luna, Sous-groupes sphériques résolubles, Prépublication de l’Institut Fourier, no. 241, 1993.

  115. D. Luna, Toute variété magnifique est sphérique, Transform. Groups 1 (1996), no. 3, 249–258.

    MATH  MathSciNet  Google Scholar 

  116. D. Luna, Variétés sphériques de type A, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226.

    MATH  MathSciNet  Google Scholar 

  117. D. Luna, T. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245.

    MATH  MathSciNet  Google Scholar 

  118. P. Magyar, J. Weyman, A. Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), no. 1, 97–118.

    MATH  MathSciNet  Google Scholar 

  119. P. Magyar, J. Weyman, A. Zelevinsky, Symplectic multiple flag varieties of finite type, J. Algebra 230 (2000), no. 1, 245–265.

    MATH  MathSciNet  Google Scholar 

  120. K. Matsuki, Introduction to the Mori Program, Universitext., Springer-Verlag, New York, 2002.

    MATH  Google Scholar 

  121. M. McConnell, The rational homology of toric varieties is not a combinatorial invariant, Proc. Amer. Math. Soc. 105 (1989), no. 4, 986–991.

    MATH  MathSciNet  Google Scholar 

  122. V. B. Mehta, A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40.

  123. И. В. Микитюк, Об интегрируемости инвариантных гамильтоновых систем с однородными конфигурационными пространствами, Матем. сб. 129(171) (1986), номер 4, 514–534. Engl. transl.: I. V. Mikityuk, On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math. of the USSR-Sbornik 57 (1987), no. 2, 527–546.

  124. L. Ness, A stratification of the null cone via the moment map, with an appendix by David Mumford, Amer. J. Math. 106 (1984), no. 6, 1281–1329.

    MathSciNet  Google Scholar 

  125. T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Bd. 15, Springer-Verlag, Berlin, 1988.

  126. A. Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405–411.

    MATH  MathSciNet  Google Scholar 

  127. A. Okounkov, Why would multiplicities be log-concave?, in: The Orbit Method in Geometry and Physics (Marseille, 2000), Progress in Mathematics, Vol. 213, Birkhäuser Boston, Boston, MA, 2003, pp. 329–347.

    Google Scholar 

  128. F. Pauer, “Caractérisation valuative” d’une classe de sous-groupes d’un groupe algébrique, C. R. 109e Congrès nat. Soc. sav. 3 (1984), 159–166.

  129. B. Pasquier, Variétés horosphériques de Fano, available at http://tel.archives-ouvertes.fr/docs/00/11/69/77/PDF/Pasquier2006.pdf.

  130. B. Pasquier, Variétés horosphériques de Fano, Bull. Soc. Math. France 136 (2008), no. 2, 195–225.

    MATH  MathSciNet  Google Scholar 

  131. B. Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344 (2009), no. 4, 963–987.

    MATH  MathSciNet  Google Scholar 

  132. B. Pasquier, An approach of the Minimal Model Program for horospherical varieties via moment polytopes, to appear in J. für die reine und ang. Math., preprint arXiv:1211.6229.

  133. B. Pasquier, N. Perrin, Local rigidity of quasi-regular varieties, Math. Z. 265 (2010), no. 3, 589–600.

    MATH  MathSciNet  Google Scholar 

  134. S. Payne, Frobenius splittings of toric varieties, Algebra Number Theory 3 (2009), no. 1, 107–119.

    MATH  MathSciNet  Google Scholar 

  135. N. Perrin, Courbes rationnelles sur les variétés homogènes, Ann. Inst. Fourier 52 (2002), no. 1, 105–132.

    MATH  MathSciNet  Google Scholar 

  136. N. Perrin, Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group \( \mathbb{Z} \), C. R. Math. Acad. Sci. Paris 345 (2007), no. 3, 155–160.

    MATH  MathSciNet  Google Scholar 

  137. N. Perrin, Small codimension subvarieties in homogeneous spaces, Indag. Math. (N.S.) 20 (2009), no. 4, 557–581.

  138. N. Perrin, Spherical varieties and Wahl’s conjecture, to appear in Ann. Inst. Fourier, preprint arXiv:1202.3236.

  139. E. Ponomareva, Classification of double flag varieties of complexity 0 and 1, preprint arXiv:1204.1988.

  140. В. Л. Попов, Стягивание действий редуктивных алгебраических групп, Матем. сб. 130(172) (1986), вып. 3(7), 310–334. Engl. transl.: V. L. Popov, Contraction of the actions of reductive algebraic groups, Math. of USSR-Sbornik 58 (1987), no. 2, 311–335.

  141. V. L. Popov, Generically multiple transitive algebraic group actions, in: Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces (Mumbai, 2004), Tata Inst. Fund. Res. Stud. Math., Vol. 19, Narosa Publishing House, Internat. distrib. by AMS, New Delhi, 2007, pp. 481–523.

    Google Scholar 

  142. А. В. Пухликов, А. Г. Хованский, Теорема Римана-Роха для интегралов и сумм квазиполиномов по виртуальным многогранникам, Алгебра и анализ 4 (1992), вып. 4, 189–216. Engl. transl.: A. V. Pukhlikov, A. G. Khovanskiĭ, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, St. Petersburg Math. J. 4 (1993), no. 4, 789–812.

    MathSciNet  Google Scholar 

  143. A. Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61–90.

    MathSciNet  Google Scholar 

  144. N. Ressayre, About Knop’s action of the Weyl group on the set of orbits of a spherical subgroup in the flag manifold, Transform. Groups 10 (2005), no. 2, 255–265.

    MATH  MathSciNet  Google Scholar 

  145. N. Ressayre, Spherical homogeneous spaces of minimal rank, Adv. Math. 224 (2010), no. 5, 1784–1800.

    MATH  MathSciNet  Google Scholar 

  146. R. W. Richardson, T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), 389–436.

    MATH  MathSciNet  Google Scholar 

  147. R. W. Richardson, T. A. Springer, Complements to: “The Bruhat order on symmetric varieties”, Geom. Dedicata 49 (1994), no. 2, 231–238.

    MATH  MathSciNet  Google Scholar 

  148. A. Rittatore, Reductive embeddings are Cohen-Macaulay, Proc. Amer. Math. Soc. 131 (2003), no. 3, 675–684.

    MATH  MathSciNet  Google Scholar 

  149. C. Robles, Schur flexibility of cominuscule Schubert varieties, Comm. Anal. Geom. 21, no. 5, 979–1013.

  150. C. Robles, D. The, Rigid Schubert varieties in compact Hermitian symmetric spaces, Selecta Math. (N.S.) 18 (2012), no. 3, 717–777.

  151. M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci. 35 (1963), 487–489.

    MATH  MathSciNet  Google Scholar 

  152. A. Ruzzi, Geometrical description of smooth projective symmetric varieties with Picard number one, Transform. Groups 15 (2010), no. 1, 201–226.

    MATH  MathSciNet  Google Scholar 

  153. A. Ruzzi, Smooth projective symmetric varieties with Picard number one, Internat. J. Math. 22 (2011), no. 2, 145–177.

    MATH  MathSciNet  Google Scholar 

  154. A. Ruzzi, A., Effective and big divisors on a projective symmetric variety, J. Algebra 354 (2012), 20–35.

    MATH  MathSciNet  Google Scholar 

  155. B. Schalke, Sphärische Einbettungen in positiver Charakteristik, Diplomarbeit (2011), Universität Erlangen-Nürnberg.

  156. J.-P. Serre, Espaces fibrés algébriques, in: Séminaire Claude Chevalley 3 (1958), Exposé No. 1, 37 pp.

    Google Scholar 

  157. А. В. Смирнов, Квазизамкнутые орбиты в проективных представлениях полупростых комплексных групп Ли, Труды ММО 64 (2002), 213–270. Engl. transl.: A. V. Smirnov, Quasiclosed orbits in projective representation of semi-simple complex Lie groups, Transactions of the Moscow Math. Soc. 2003, 193–247.

    Google Scholar 

  158. A. J. Sommese, A. van de Ven, Homotopy groups of pullbacks of varieties, Nagoya Math. J. 102 (1986), 79–90.

    MATH  MathSciNet  Google Scholar 

  159. J. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404–439.

    MATH  MathSciNet  Google Scholar 

  160. E. Strickland, A vanishing theorem for group compactifications, Math. Ann. 277 (1987), no. 1, 165–171.

    MATH  MathSciNet  Google Scholar 

  161. H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.

    MATH  MathSciNet  Google Scholar 

  162. H. Sumihiro, Equivariant completion. II, J. Math. Kyoto Univ. 15 (1975), no. 3, 573–605.

    MATH  MathSciNet  Google Scholar 

  163. R. Tange, On embeddings of certain spherical homogeneous spaces in prime characteristic, Transform. Groups 17 (2012), no. 3, 861–888.

    MATH  MathSciNet  Google Scholar 

  164. Д. А. Тимашёв, G-многообразия сложности 1, УМН 51 (1996), вып. 3(309), 213–214. Engl. transl.: D. A. Timashëv, G-manifolds of complexity 1, Russian Math. Surveys 51 (1996), no. 3, 567–568.

    MATH  MathSciNet  Google Scholar 

  165. Д. А. Тимашёв, Классификация G-многообразия сложности 1, Изв. РАН. Сер. мат. 61 (1997), вып. 2, 127–162. Engl. transl.: D. A. Timashëv, Classification of G-manifolds of complexity 1, Izv.: Math. 61 (1997), no. 2, 363–397.

  166. D. A. Timashëv, Equivariant Embeddings of Homogeneous Spaces, in: Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics, London Math. Soc. Lecture Note Ser., Vol. 338, Cambridge Univ. Press, Cambridge, 2007, pp. 226–278.

  167. D. A. Timashëv, Homogeneous Spaces and Equivariant Embeddings, Encyclopaedia of Mathematical Sciences, Vol. 138, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. 8, Springer, Heidelberg, 2011.

  168. E. Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8.

    MATH  MathSciNet  Google Scholar 

  169. Э. Б. Винберг, Коммутативные однородные пространства и коизотропные симплектические действия, УМН 56 (2001), вып. 1(337), 3–62. Engl. transl.: È. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Russian Math. Surveys 56 (2001), no. 1, 1–60.

    MathSciNet  Google Scholar 

  170. Э. Б. Винберг, Сложность действий редуктивных групп, Функц. анализ и его прил. 20 (1986), вып. 1, 1–13. Engl. transl.: È. Vinberg, Complexity of actions of reductive groups, Funct. Analysis Appl. 20 (1986), no. 1, 1–11.

  171. Э. Б. Винберг, Б. Н. Кимельфельд, Однородные области на флаговых многообразиях и сферические подѕруппы полупростых групп Ли, Функц. анализ и его прил. 12 (1978), вып. 3, 12–19. Engl. transl.: È. Vinberg, B. N. Kimel’fel’d, Homogeneous domains on flag manifolds and spherical subsets of semisimple Lie groups, Funct. Analysis Appl. 12 (1978), no. 3, 168–174.

  172. T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317–333.

    MATH  MathSciNet  Google Scholar 

  173. T. Vust, Plongements d’espaces symétriques algébriques: une classification, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 2, 165–195.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to NICOLAS PERRIN.

Rights and permissions

Reprints and permissions

About this article

Cite this article

PERRIN, N. ON THE GEOMETRY OF SPHERICAL VARIETIES. Transformation Groups 19, 171–223 (2014). https://doi.org/10.1007/s00031-014-9254-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-014-9254-0

Keywords

Navigation