Chern classes of reductive groups and an adjunction formula

Kiritchenko, Valentina

doi:10.5802/aif.2211

Cited by 15 publications

(17 citation statements)

References 22 publications

(54 reference statements)

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“…. , S n−k (see Corollary 4.4 in [10]). Theorem 1.2 allows to compute explicitly the intersection index of c i (X) with a complete intersection of complementary dimension in X.…”

Section: Introductionmentioning

confidence: 92%

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On Intersection Indices of Subvarieties in Reductive Groups

Kiritchenko¹

2007

MMJ

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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].

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“…. , S n−k (see Corollary 4.4 in [10]). Theorem 1.2 allows to compute explicitly the intersection index of c i (X) with a complete intersection of complementary dimension in X.…”

Section: Introductionmentioning

confidence: 92%

“…

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.

…”

mentioning

confidence: 99%

On Intersection Indices of Subvarieties in Reductive Groups

Kiritchenko¹

2007

MMJ

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“…I cannot generalize Theorem 2.3 to an arbitrary spherical space X, but this formal problem is not the main difficulty with affine characteristic classes for arbitrary X. For instance, the class X , which is especially important due to property (2), has already been cosidered in [Kir06] under the name of the non-compact characteristic class of X (see also [BK05] and [BJ08] for its equivariant version), but it is not yet computed even for X = SL n with large n.…”

Section: Affine Intersection Theory and Characteristic Classesmentioning

confidence: 99%

Characteristic classes of affine varieties and Plücker formulas for affine morphisms

Esterov

2017

J. Eur. Math. Soc.

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An enumerative problem on a variety V is usually solved by reduction to intersection theory in the cohomology of a compactification of V . However, if the problem is invariant under a "nice" group action on V (so that V is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of V . We call this limit the affine cohomology of V and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus.This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities, and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety.The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials -the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties, whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials -a key ingredient is the n-ary fan, generalizing the secondary polytope.

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“…As shown by De Concini and Procesi, these questions find their proper setting in the ring of conditions C * (G/K), isomorphic to the direct limit of cohomology rings of G-equivariant compactifications X of G/K (see [DP83,DP85]). Recently, the Euler characteristic of any complete intersection of hypersurfaces in G/K has been expressed by Kiritchenko (see [Ki06]), in terms of the Chern classes of the logarithmic tangent bundle S X of any "regular" compactification X. As shown in [Ki06], these Chern classes are independent of the choice of X, and hence yield elements of C * (G/K); moreover, their determination may be reduced to the case where X is a "wonderful variety".…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the Euler characteristic of any complete intersection of hypersurfaces in G/K has been expressed by Kiritchenko (see [Ki06]), in terms of the Chern classes of the logarithmic tangent bundle S X of any "regular" compactification X. As shown in [Ki06], these Chern classes are independent of the choice of X, and hence yield elements of C * (G/K); moreover, their determination may be reduced to the case where X is a "wonderful variety".…”

Section: Introductionmentioning

confidence: 99%

Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

Brion

Joshua

2008

Transformation Groups

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Abstract. We describe the equivariant Chow ring of the wonderful compactification X of a symmetric space of minimal rank, via restriction to the associated toric variety Y . Also, we show that the restrictions to Y of the tangent bundle T X and its logarithmic analogue S X decompose into a direct sum of line bundles. This yields closed formulae for the equivariant Chern classes of T X and S X , and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.

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Chern classes of reductive groups and an adjunction formula

Cited by 15 publications

References 22 publications

On Intersection Indices of Subvarieties in Reductive Groups

On Intersection Indices of Subvarieties in Reductive Groups

Characteristic classes of affine varieties and Plücker formulas for affine morphisms

Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

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