Let G be a connected reductive algebraic group over the field of complex
numbers. A normal, irreducible algebraic variety Y endowed with an action of
G is said to be spherical if it contains an open dense orbit for some (and
hence any) Borel subgroup B of G.
In the case when G=T is a torus we have B=T, hence any toric variety is a
spherical variety.
In the talk, which will contain almost no new results, we shall consider
spherical homogeneous spaces Y=G/H, where H is an algebraic subgroup of G.
We shall introduce combinatorial invariants of spherical homogeneous spaces,
and we shall state Losev's uniqueness theorem claiming that these
combinatorial invariants uniquely determine H up to conjugacy, that is,
they uniquely determine the Gvariety Y=G/H up to an isomorphism of
Gvarieties.
In the next talk Giuliano Gagliardi will describe spherical embeddings
(nonhomogeneous sherical varieties) using the combinatorial invariants of
spherical homogeneous spaces. The spherical embeddings will be classified in
terms of colored cones and colored fans, which generalizes the description
of toric varieties in terms of cones and fans.
