In
algebraic geometry, given a
reductive algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
''G'' and a
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgrou ...
''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 space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s and (affine or projective)
toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...
.
There is also a notion of real spherical varieties.
A projective spherical variety is a
Mori dream space
In algebraic geometry, a Mori dream space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers". showed that Mori dream spaces are quotients of affine varieties by ...
.
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, 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.
has been developing a program to classify spherical varieties in arbitrary characteristic.
References
*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
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Algebraic geometry
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