On Intersection Indices of Subvarieties in Reductive Groups

Abstract

I will present an explicit formula for the intersection indices of the Chern classes (defined in [10]) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion-Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of De Concini and Procesi for computing such intersection indices. In particular, it is shown that this algorithm produces the Brion-Kazarnovskii formula.a nondegenerate symmetric bilinear form on the Lie algebra of G that is invariant under the adjoint action of G (such a form exists since G is reductive).Theorem 1.1. [3,9] If H π is a hyperplane section corresponding to a representation π with the weight polytope P π ⊂ L T ⊗ R , then the self-intersection index of H π in the ring of conditions is equal toThe measure dx on L T ⊗ R is normalized so that the covolume of L T is 1.This theorem was first proved by B.Kazarnovskii [9]. Later, M.Brion proved an analogous formula for arbitrary spherical varieties using a different method [3].

Overdetermined Systems of Equations on Toric, Spherical, and Other Algebraic Varieties

“…The first result in this direction was a generalization of the BKK Theorem and was obtained by Brion and Kazarnovskii in [Bri89,Kaz87]. For more results see for example [Kir06,Kir07,KK16]. The role of the Newton polytope in these results is played by the Newton-Okounkov polytope, which is a polytope fibered over the moment polytope with string polytopes as fibers.…”

Complete intersections in spherical varieties

“…Much later the first author showed ( [Kaveh04]) that the straightforward generalization of the formula for the (topological) Euler characteristic of a complete intersection in a torus (C * ) n to a reductive group fails and such a formula should be more complicated than in the torus case. The corresponding formula was soon found by V. Kiritchenko ([Kiritchenko06,Kiritchenko07]). Her unexpected and beautiful result was at the same time slightly disappointing: the formula turns out to be unavoidably too complicated.…”

Moment polytopes, semigroup of representations and Kazarnovskii’s theorem

“…The weight polytope (or moment polytope) of a representation also plays important role in questions related to the geometry of the group G, its compactifications and its subvarieties. For some interesting results in this direction see [Kapranov97,Kiritchenko06,Kiritchenko07,Timashev03].…”