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.
Similar content being viewed by others
Notes
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
Ahiezer, D.N.: Equivariant completions of homogeneous algebraic varieties by homogeneous divisors. Ann. Glob. Anal. Geom. 1, 49-78 (1983)
Bravi, P.: Wonderful varieties of type E. Represent. Theory 11, 174-191 (2007)
Bravi, P.: Primitive spherical systems. Trans. Am. Math. Soc. 365, 361-407 (2013)
Bravi, P., Cupit-Foutou, S.: Classification of strict wonderful varieties. Ann. Inst. Fourier (Grenoble) 60, 641-681 (2010)
Bravi, P., Luna, D.: An introduction to wonderful varieties with many examples of type F4. J. Algebra 329, 4-51 (2011)
Bravi, P., Pezzini, G.: Wonderful varieties of type D. Represent. Theory 9, 578-637 (2005)
Bravi, P., Pezzini, G.: Wonderful subgroups of reductive groups and spherical systems. J. Algebra 409, 101-147 (2014)
Bravi, P., Pezzini, G.: The spherical systems of the wonderful reductive subgroups. J. Lie Theory 25(1), 105-123 (2015)
Brion, M.: Classification des espaces homogènes sphériques. Compos. Math. 63, 189-208 (1987)
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)
Cupit-Foutou, S.: Wonderful varieties: a geometrical realization. arXiv:0907.2852v3
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)
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)
Knop, F.: The assymptotic behaviour of invariant collective motion. Invent. Math. 114, 309-328 (1994)
Knop, F.: Automorphisms, root systems, and compactifications of homogeneous varieties. J. Am. Math. Soc. 9, 153-174 (1996)
Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compositio Math. 38, 129-153 (1979)
Losev, I.V.: Uniqueness property for spherical homogeneous spaces. Duke Math. J. 147, 315-343 (2009)
Luna, D.: Toute variété magnifique est sphérique. Transform. Groups 1(3), 249-258 (1996)
Luna, D.: Variétés sphériques de type A. Publ. Math. Inst. Hautes Études Sci. 94, 161-226 (2001)
Mikityuk, I.V.: Integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Math. USSR-Sb. 57, 527-546 (1987)
Pezzini, G.: Wonderful varieties of type C, Ph.D. Thesis, Università La Sapienza, Rome (2004)
Timashev, D.: Homogeneous spaces and equivariant embeddings. In: Encycl. Math. Sci., vol. 138. Springer, Berlin (2011)
Wasserman, B.: Wonderful varieties of rank two. Transform. Groups 1, 375-403 (1996)
Acknowledgments
The second-named author was supported by the DFG Schwerpunktprogramm 1388—Darstellungstheorie.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Bravi, P., Pezzini, G. Primitive wonderful varieties. Math. Z. 282, 1067–1096 (2016). https://doi.org/10.1007/s00209-015-1578-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-015-1578-5