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.

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.