Seminar in Real and Complex Geometry

Thursday, November 23, 2017, 11:00-12:00, Schreiber building, room 209

Mikhail Borovoi (Tel Aviv)

Spherical varieties: combinatorial invariants of spherical homogeneous spaces


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 G-variety Y=G/H up to an isomorphism of G-varieties. 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.