Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations

Kazarnovskii, Boris

doi:10.1007/bf01077809

Cited by 23 publications

(21 citation statements)

References 2 publications

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“…It turned out that the formulas for the intersection indices of several hyperplane sections can be generalized to reductive groups and, more generally, to spherical homogeneous spaces. For reductive groups, this was done by B. Kazarnovskii [17]. Later, M. Brion established an analogous result for all spherical homogeneous spaces [4].…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

Chern classes of reductive groups and an adjunction formula

Kiritchenko¹

2006

Ann. inst. Fourier

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In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. In the case where the complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve.

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Section: Introduction and Main Resultsmentioning

confidence: 97%

Chern classes of reductive groups and an adjunction formula

Kiritchenko¹

2006

Ann. inst. Fourier

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“…As remarked in [5] a good upper bound can be found using a formula of Kazarnovskii (see [16]). We will give an explicit upper bound for σ(V ) expressed in the degrees of polynomials defining the group G and the representation V .…”

Section: β(V ) := Min{ D | O(v )mentioning

confidence: 99%

“…Hiss found a sharper bound for σ(V ), which doesn't depend on the dimension n of V , following ideas of Knop (see [12] and [5]). In [5] it was shown that one can find an even better bound using the formula of Kazarnovskii (see [16] and [1] for a generalization).…”

Section: Upper Bounds For σ(V )mentioning

confidence: 99%

Polynomial bounds for rings of invariants

Derksen

2000

Proc. Amer. Math. Soc.

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Abstract. Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depends only polynomially on the input data.

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“…We will denote the weight polytope by P π and its intersection with the positive Weyl chamber by P + π . As in [Kazarnovskii87], one can show that:…”

Section: 8mentioning

confidence: 82%

Complete intersections in spherical varieties

Kaveh

Khovanskiĭ

2016

Sel. Math. New Ser.

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Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G/H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete intersections are smooth varieties. We compute their arithmetic genus as well as some of their h p,0 numbers. The answers are given in terms of the moment polytopes and Newton-Okounkov polytopes associated to G-invariant linear systems. We also give a necessary and sufficient condition on a collection of linear systems so that the corresponding generic complete intersection is nonempty. This criterion applies to arbitrary quasi-projective varieties (i.e. not necessarily spherical homogeneous spaces). When the spherical homogeneous space under consideration is a complex torus (C * ) n , our results specialize to well-known results from the Newton polyhedra theory and toric varieties.

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Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations

Cited by 23 publications

References 2 publications

Chern classes of reductive groups and an adjunction formula

Chern classes of reductive groups and an adjunction formula

Polynomial bounds for rings of invariants

Complete intersections in spherical varieties

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