V.Zhgoon: On complexity of algebraic varieties over algebraically non-closed fields

In 1986 E.B.Vinberg introduced the notion of complexity for the action of a reductive group G on an algebraic variety over an algebraically closed field, which is the transcendence degree of the field of invariants of a Borel subroup B. This number is also equal to the codimension of generic B-orbit in algebraic variety. 
Vinberg has proved that complexity does not increase after passing to the B-invariant subset of algebraic variety. 
In particular, this shows that in a spherical variety, i.e. the variety with an open B-orbit, there is only finite number of B-orbits, which was also proved independently by M.Brion.

In my talk I shall speak about generalization of this result to algebraic non-closed fields which is a joint work with F.Knop. One should recall that for algebraically non-closed field k it may happen that there are no Borel subgroups defined over k, so for defining k-complexity we consider the codimension of a generic orbit of the minimal parabolic subgroup defined over k. In the talk I shall focus on the behavior of k-complexity, in particular I shall speak about finiteness result for k-spherical varieties.
Also I shall talk about the action of the k-Weyl group on the set of P-orbits of maximal rank and homogeneity on a spherical variety which is a generalization of the action of the Weyl group on of the set of B-orbits introduced by F.Knop.
 

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Успехов,

Никон Курносов