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Primitive wonderful varieties

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Abstract

We complete the classification of wonderful varieties initiated by D. Luna. We review the results that reduce the problem to the family of primitive varieties, and report the references where some of them have already been studied. Finally, we analyze the rest case-by-case.

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Notes

  1. Here two morphisms \(f_1:X\rightarrow Y_1\), \(f_2:X\rightarrow Y_2\) are G-isomorphic if there is a G-equivariant isomorphism \(\varphi :Y_1\rightarrow Y_2\) such that \(f_2=\varphi \circ f_1\).

References

  1. Ahiezer, D.N.: Equivariant completions of homogeneous algebraic varieties by homogeneous divisors. Ann. Glob. Anal. Geom. 1, 49-78 (1983)

    Article  MathSciNet  Google Scholar 

  2. Bravi, P.: Wonderful varieties of type E. Represent. Theory 11, 174-191 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bravi, P.: Primitive spherical systems. Trans. Am. Math. Soc. 365, 361-407 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bravi, P., Cupit-Foutou, S.: Classification of strict wonderful varieties. Ann. Inst. Fourier (Grenoble) 60, 641-681 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bravi, P., Luna, D.: An introduction to wonderful varieties with many examples of type F4. J. Algebra 329, 4-51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bravi, P., Pezzini, G.: Wonderful varieties of type D. Represent. Theory 9, 578-637 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bravi, P., Pezzini, G.: Wonderful subgroups of reductive groups and spherical systems. J. Algebra 409, 101-147 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bravi, P., Pezzini, G.: The spherical systems of the wonderful reductive subgroups. J. Lie Theory 25(1), 105-123 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Brion, M.: Classification des espaces homogènes sphériques. Compos. Math. 63, 189-208 (1987)

    MathSciNet  Google Scholar 

  10. M. Brion, On spherical varieties of rank one (after D. Ahiezer, A. Huckleberry, D. Snow), Group actions and invariant theory (Montreal, PQ, 1989), CMS Conf. Proc., 10, Amer. Math. Soc. Providence, RI, pp. 31-41 (1988)

  11. Cupit-Foutou, S.: Wonderful varieties: a geometrical realization. arXiv:0907.2852v3

  12. De Concini, C., Procesi, C.: Complete symmetric varieties. In: Gherardelli, F. (ed.) Invariant Theory, Proceedings, Montecatini. Lect. Notes Math. 996, pp. 1-44. Springer, Berlin (1983)

    Google Scholar 

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

  14. Knop, F.: The assymptotic behaviour of invariant collective motion. Invent. Math. 114, 309-328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Knop, F.: Automorphisms, root systems, and compactifications of homogeneous varieties. J. Am. Math. Soc. 9, 153-174 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compositio Math. 38, 129-153 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Losev, I.V.: Uniqueness property for spherical homogeneous spaces. Duke Math. J. 147, 315-343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luna, D.: Toute variété magnifique est sphérique. Transform. Groups 1(3), 249-258 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Mikityuk, I.V.: Integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Math. USSR-Sb. 57, 527-546 (1987)

    Article  MATH  Google Scholar 

  21. Pezzini, G.: Wonderful varieties of type C, Ph.D. Thesis, Università La Sapienza, Rome (2004)

  22. Timashev, D.: Homogeneous spaces and equivariant embeddings. In: Encycl. Math. Sci., vol. 138. Springer, Berlin (2011)

  23. Wasserman, B.: Wonderful varieties of rank two. Transform. Groups 1, 375-403 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The second-named author was supported by the DFG Schwerpunktprogramm 1388—Darstellungstheorie.

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Correspondence to P. Bravi or G. Pezzini.

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Bravi, P., Pezzini, G. Primitive wonderful varieties. Math. Z. 282, 1067–1096 (2016). https://doi.org/10.1007/s00209-015-1578-5

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