Abstract
These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.
Similar content being viewed by others
References
Ahiezer, D. N.: Equivariant completions of homogeneous algebraic varieties by homogeneous divisors. Ann. Global Anal. Geom., 1 (1), 49–78 (1983)
Avdeev, R.: Strongly solvable spherical subgroups and their combinatorial invariants. Selecta Math. (N.S.), 21 (3), 931–993 (2015)
Bravi, P.: Primitive spherical systems. Trans. Amer. Math. Soc., 365, 361–407 (2013)
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 subgroups of reductive groups and spherical systems. J. Algebra, 409, 101–147 (2014)
Bravi, P., Pezzini, G.: Primitive wonderful varieties. Math. Z., 282(3–4), 1067–1096 (2016)
Brion, M.: On spherical varieties of rank one (after D. Ahiezer, A. Huckleberry, D. Snow), Group actions and invariant theory (Montreal, PQ, 1988), CMS Conf. Proc., 10, Amer. Math. Soc., Providence, RI, 1989, 31–41
Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J., 58 (2), 397–424 (1989)
Brion, M.: Vers une généralisation des espaces symétriques. J. Algebra, 134 (1), 115–143 (1990)
Brion, M.: Log homogeneous varieties, Actas del XVI Coloquio Latinoamericano de Álgebra, Revista Matemática Iberoamericana, Madrid, 2007, 1–39
Brion, M., Luna, D., Vust, Th.: Espaces homogènes sphériques. Invent. Math., 84, 617–632 (1986)
Brion, M., Pauer, F.: Valuations des espaces homogènes sphériques. Comment. Math. Helvetici, 62, 265–285 (1987)
De Concini, C., Procesi, C.: Complete symmetric varieties. Invariant theory (Montecatini, 1982), Lecture Notes in Math., 996, Springer, Berlin, 1983, 1–44
Knop., F.: The Luna–Vust theory of spherical embeddings. Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, 1991, 225–249
Knop, F.: Automorphisms, root systems, and compactifications of homogeneous varieties. J. Amer. Math. Soc., 9 (1), 153–174 (1996)
Knop, F.: Spherical roots of spherical varieties. Ann. Inst. Fourier (Grenoble), 64, 2503–2526 (2014)
Knop, F., Kraft, H., Vust, Th.: The Picard group of a G-variety, Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy, T. A. Springer, eds.), DMV Seminar, vol. 13, Birkhäuser, Basel-Boston-Berlin, 1989, 77–88
Losev, I.: Uniqueness property for spherical homogeneous spaces. Duke Math. J., 147 (2), 315–343 (2009)
Losev, I.: Demazure embeddings are smooth. Internat. Math. Res. Notices, 14, 2588–2596 (2009)
Luna, D.: Slices étales. Mémoires de la S.M.F., 33, 81–105 (1973)
Luna, D.: Toute variété magnifique est sphérique. Transform. Groups, 1 (3), 249–258 (1996)
Luna, D.: Grosses cellules pour les variétés sphériques, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997, 267–280
Luna, D.: Variétés sphériques de type A. Inst. Hautes’ Etudes Sci. Publ. Math., 94, 161–226 (2001)
Moser-Jauslin, L.: Some almost homogeneous group actions on smooth complete rational surfaces. L’Enseignement Mathématique, 34, 313–332 (1988)
Pezzini, G.: Automorphisms of wonderful varieties. Transform. Groups, 14 (3), 677–694 (2009)
Semple, J. G.: The variety whose points represent complete collineations of Sr on Sr. Univ. Roma, Ist. Naz. Alta Mat. Rend. Mat. e Appl., 10 (5), 201–208 (1951)
Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ., 14, 1–28 (1974)
Wasserman, B.: Wonderful varieties of rank two. Transform. Groups, 1 (4), 375–403 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pezzini, G. Lectures on Wonderful Varieties. Acta. Math. Sin.-English Ser. 34, 417–438 (2018). https://doi.org/10.1007/s10114-017-7214-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-7214-z