spherical variety
In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric spaces and (affine or projective) toric varieties.
There is also a notion of real spherical varieties.
A projective spherical variety is a Mori dream space.{{cite journal
| last = Brion
| first = Michel
| title = The total coordinate ring of a wonderful variety
| journal = Journal of Algebra
| volume = 313
| issue = 1
| pages = 61–99
| year = 2007
| doi=10.1016/j.jalgebra.2006.12.022
| arxiv = math/0603157
| s2cid = 15154549
}}
Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory.
In his seminal paper, {{harvtxt|Luna|2001}} developed a framework to classify complex spherical subgroups of reductive groups; he reduced the classification of spherical subgroups to wonderful subgroups. He further worked out the case of groups of type A and conjectured that combinatorial objects consisting of "homogeneous spherical data" classify spherical subgroups. This is known as the Luna Conjecture.
This classification is now complete according to Luna's program; see contributions of Bravi, Cupit-Foutou, Losev and Pezzini.
As conjectured by Knop, every "smooth" affine spherical variety is uniquely determined by its weight monoid.
This uniqueness result was proven by Losev.
{{harvtxt|Knop|2013}} has been developing a program to classify spherical varieties in arbitrary characteristic.
References
{{reflist}}
- Paolo Bravi, Wonderful varieties of type E, Representation theory 11 (2007), 174–191.
- Paolo Bravi and Stéphanie Cupit-Foutou, Classification of strict wonderful varieties, Annales de l'Institut Fourier (2010), Volume 60, Issue 2, 641–681.
- Paolo Bravi and Guido Pezzini, Wonderful varieties of type D, Representation theory 9 (2005), pp. 578–637.
- Paolo Bravi and Guido Pezzini, Wonderful subgroups of reductive groups and spherical systems, J. Algebra 409 (2014), 101–147.
- Paolo Bravi and Guido Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105–123.
- Paolo Bravi and Guido Pezzini, Primitive wonderful varieties, Arxiv 1106.3187.
- Stéphanie Cupit-Foutou, Wonderful Varieties. a geometrical realization, Arxiv 0907.2852.
- Michel Brion, "Introduction to actions of algebraic groups" [http://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf]
- {{citation |last1=Knop |first1=Friedrich |journal=Algebra & Number Theory |title=Localization of spherical varieties |volume=8|issue=3 | pages=703–728 |year=2013 |doi=10.2140/ant.2014.8.703|arxiv=1303.2561|s2cid=119293458 }}
- {{cite arXiv|last1=Losev |first1=Ivan |eprint=math/0612561 |title=Proof of the Knop conjecture |year=2006}}
- {{cite arXiv|last1=Losev |first1=Ivan |eprint=0904.2937 |title=Uniqueness properties for spherical varieties |year=2009|class=math.AG }}
- {{citation |last1=Luna|first1=Dominique| title=Variétés sphériques de type A| journal= Publications Mathématiques de l'Institut des Hautes Études Scientifiques |year=2001|volume=94| pages=161–226| doi=10.1007/s10240-001-8194-0|s2cid=123850545|url=http://www.numdam.org/item/PMIHES_2001__94__161_0/}}
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