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Math.Helv. 73 (1998), no. 1, p. 137-174.[7] M. Brion & I. Kausz, “Vanishing of top equivariant Chern classes of regular embeddings”, preprint arxiv.org/math.AG/0503196.[8] C. de Concini, “Equivariant embeddings of homogeneous spaces”, in Proceedingsof the International Congress of Mathematicians (Berkeley, California, USA) (Providence, RI), vol. 1,2, Amer. Math. Soc., 1986, p. 369-377.TOME 56 (2006), FASCICULE 41256Valentina KIRITCHENKO[9] C. de Concini & C.

Procesi, “Complete symmetric varieties I”, in Invariant theory(Montecatini, 1982) (Berlin), Lect. Notes in Math., vol. 996, Springer, 1983, p. 1-44.[10] ——— , “Complete symmetric varieties II Intersection theory”, in Algebraic groupsand related topics (Kyoto/Nagoya, 1983) (Amsterdam), Adv. Stud. Pure Math.,vol. 6, North-Holland, 1985, p. 481-513.[11] F. Ehlers, “Eine Klasse komplexer Mannigfaltigkeiten und die Auflösung einigerisolierter Singularitäten”, Math. Ann. 218 (1975), no. 2, p. 127-157.[12] W. Fulton, Intersection theory, Springer, Berlin, 1984.[13] I.

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Anal. Appl. 21 (1987), no. 4,p. 319-321.[18] A. G. Khovanskii, “Newton polyhedra, and the genus of complete intersections”,Funct. Anal. Appl. 12 (1978), no. 1, p. 38-46.[19] V. Kiritchenko, “A Gauss-Bonnet theorem, Chern classes and an adjunction formula for reductive groups”, PhD Thesis, University of Toronto, Toronto, Ontario,2004.[20] S. L. Kleiman, “The transversality of a general translate”, Compositio Mathematica 28 (1974), no. 3, p. 287-297.[21] F. Knop, “The Luna-Vust theory of spherical embeddings”, in Proceedings of theHyderabad Conference on Algebraic Groups (Hyderabad, 1989) (Madras), ManojPrakashan, 1991, p.

225-249.[22] ——— , “Automorphisms, root systems, and compactifications of homogeneous varieties”, J. Amer. Math. Soc. 9 (1996), no. 1, p. 153-174.[23] D. Luna, “Sur les plongements de Demazure”, J. Algebra 258 (2002), no. 1, p. 205215.[24] R. W. Richardson, “Principal orbit types for algebraic transformation spaces incharacteristic zero”, Invent. Math. 16 (1972), p. 6-14.[25] A.

Rittatore, “Reductive embeddings are Cohen-Macaulay”, Proc. Amer. Math.Soc. 131 (2003), no. 3, p. 675-684.[26] D. Timashev, “Equivariant compactifications of reductive groups”, Sb. Math. 194(2003), no. 3–4, p. 589-616.Manuscrit reçu le 21 avril 2004,révisé le 10 octobre 2005,accepté le 11 novembre 2005.Valentina KIRITCHENKOState University of New York at Stony BrookDept. of Mathematicsvkiritch@math.sunysb.eduANNALES DE L’INSTITUT FOURIERПриложение B.Статья 2.Valentina Kiritchenko “On intersection indices of subvarieties inreductive groups”Moscow Mathematical Journal, Volume 7, Number 3, July–September2007, Pages 489–505Разрешение на копирование: Согласно https://www.ams.org/distribution/mmj/копирование не требует получения разрешения, если копии используются вобразовательных и научных целях.

При копировании требуется указатьисточник.MOSCOW MATHEMATICAL JOURNALVolume 7, Number 3, July–September 2007, Pages 489–505ON INTERSECTION INDICES OF SUBVARIETIES INREDUCTIVE GROUPSVALENTINA KIRITCHENKOTo my Teacher Askold KhovanskiiAbstract. In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined earlier by the author) of anarbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Eulercharacteristic of complete intersections in reductive groups thus extending the beautiful result by D.

Bernstein and Khovanskii, which holdsfor a complex torus. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classesof the compactification with hypersurfaces. The formula is similar tothe Brion–Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of De Conciniand Procesi for computing such intersection indices. In particular, it isshown that this algorithm produces the Brion–Kazarnovskii formula.2000 Math. Subj.

Class. 14L30.Key words and phrases. Reductive groups, Chern classes, Euler characteristic of hyperplane sections.1. IntroductionLet G be a connected complex reductive group of dimension n, and let π : G →GL(V ) be a faithful representation of G. A generic hyperplane section Hπ corresponding to π is the preimage π −1 (H) of the intersection of π(G) with a genericaffine hyperplane H ⊂ End(V ).

There is a nice explicit formula for the selfintersection index Hπn of Hπ in G, and more generally, for the intersection index ofn generic hyperplane sections corresponding to different representations (see Theorem 1.1 below) in terms of the weight polytopes of the representations [3], [9]. Inthis paper, I give a similar formula for the intersection indices of the Chern classesof G (defined in [11]) with generic hyperplane sections (see Theorem 1.2).The Chern classes of G can be defined using the Chern classes of the logarithmictangent bundle over a regular compactification of G (see Section 3 for a precisedefinition). They were introduced in [11] as main ingredients in a formula for theEuler characteristic of a generic hyperplane section and of complete intersectionsReceived June 11, 2006.c2007Independent University of Moscow489490V.

KIRITCHENKOof several hyperplane sections. In the case where a reductive group is a complextorus (C∗ )n , there are beautiful explicit formulas for the Euler characteristic dueto D. Bernstein and A. Khovanskii [10]. The result of the present paper combinedwith [11] provides analogous formulas in the case of an arbitrary reductive group.Denote by k the rank of G, i.

e. the dimension of a maximal torus in G. Only thefirst (n−k) Chern classes are not trivial [11]. These Chern classes are elements of thering of conditions of G, which was introduced by C. De Concini and C. Procesi [7](see also Section 2.4 for a brief reminder). They can be represented by subvarietiesS1 , . . . , Sn−k ⊂ G, where Si has codimension i. All enumerative problems for G,such as the computation of the intersection index Si Hπn−i , make sense in the ringof conditions.First, I recall the usual Brion–Kazarnovskii formula for the intersection indicesof hyperplane sections.

Choose a maximal torus T ⊂ G, and denote by LT itscharacter lattice. Choose also a Weyl chamber D ⊂ LT ⊗ R. Denote by R+ the setof all positive roots of G and denote by ρ the half of the sum of all positive rootsof G. The inner product (· , ·) on LT ⊗ R is given by a nondegenerate symmetricbilinear form on the Lie algebra of G that is invariant under the adjoint action ofG (such a form exists since G is reductive).Theorem 1.1 [3], [9].

If Hπ is a hyperplane section corresponding to a representation π with the weight polytope Pπ ⊂ LT ⊗ R, then the self-intersection index Hπnof Hπ is equal toZY (x, α)2n!dx.(ρ, α)2+Pπ ∩D α∈RThe measure dx on LT ⊗ R is normalized so that the covolume of LT is 1.This theorem was first proved by B. Kazarnovskii [9]. Later, M. Brion provedan analogous formula for arbitrary spherical varieties using a different method [3].The integrand in this formula has the following interpretation.

The direct sumLT ⊕ LT can be identified with the Picard group of the product G/B × G/B oftwo flag varieties. Here B is a Borel subgroup of G. Hence, to each lattice point(λ1 , λ2 ) ∈ LT ⊕ LT one can assign the self-intersection index of the correspondingdivisor in G/B × G/B. The resulting function extends to the polynomial function(n − k)! F on (LT ⊕ LT ) ⊗ R, whereY (x, α)(y, α)F (x, y) =.(ρ, α)2+α∈RNote that the integrand is the restriction of F onto the diagonal {(x, x) : x ∈LT ⊗ R}.This interpretation leads to another proof of the Brion–Kazarnovskii formula(different from those of Kazarnovskii and Brion).