arXiv subjects
Publications and source records attributed to Askold G. Khovanskii.
We prove a version of the BKK theorem for the ring of conditions of a spherical homogeneous space $G/H$. We also introduce the notion of ring of complete intersections, firstly for a spherical homogeneous space and secondly for an arbitrary variety. Similarly to the ring of conditions of the torus, the ring of complete intersections of $G/H$ admits a description in terms of volumes of polytopes.
We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in the graded algebra. We extend the notion of Duistermaat-Heckman measure to graded G-algebras and prove a Fujita type approximation theorem and a Brunn-Minkowski inequality for this measure. This in particular applies to arbitrary G-line bundles giving an equivariant version of the theory of volumes of line bundles. We generalize the Brion-Kazarnowski formula for the degree of a spherical variety to arbitrary G-varieties. Our approach follows some of the previous works of A. Okounkov. We use the asymptotic theory of semigroups of integral points and Newton-Okounkov bodies developed in our ealier work arXiv:0904.3350
Two representations of a reductive group G are spectrally equivalent if the same irreducible representations appear in both of them. The semigroup of finite dimensional representations of G with tensor product and up to spectral equivalence is a rather complicated object. We show that the Grothendieck group of this semigroup is more tractable and give a description of it in terms of moment polytopes of representations. As a corollary, we give a proof of the Kazarnovskii theorem on the number of solutions in G of a system f_1(x) = ... = f_m(x) = 0, where m=dim(G) and each f_i is a generic function in the space of matrix elements of a representation pi_i of G.
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).