Диссертация (1136188), страница 5

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Soc. Transl. Ser. 2, 181 (1998), 231–244Annales Scientifiques de l’ENSПриложение A.Статья 1.Valentina Kiritchenko “Chern classes of reductive groups and anadjunction formula”Annales de L’Institut Fourier, Grenoble 56, 4 (2006) 1225-1256Разрешение на копирование: Согласно http://aif.cedram.org/spip.php?rubrique12&lang=en на статью распространяется лицензия CC-BY-ND.Копирование разрешено при условии обязательного упоминания копирайта.FOURUTLES DENAAIER NANNALESSL’IN TITDEL’INSTITUT FOURIERValentina KIRITCHENKOChern classes of reductive groups and an adjunction formulaTome 56, no 4 (2006), p. 1225-1256.<http://aif.cedram.org/item?id=AIF_2006__56_4_1225_0>© Association des Annales de l’institut Fourier, 2006, tous droitsréservés.L’accès aux articles de la revue « Annales de l’institut Fourier »(http://aif.cedram.org/), implique l’accord avec les conditionsgénérales d’utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que cesoit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale.

Toutecopie ou impression de ce fichier doit contenir la présente mentionde copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Ann. Inst. Fourier, Grenoble56, 4 (2006) 1225-1256CHERN CLASSES OF REDUCTIVE GROUPSAND AN ADJUNCTION FORMULAby Valentina KIRITCHENKOAbstract. — In this paper, I construct noncompact analogs of the Chernclasses for equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the (topological)Euler characteristic of complete intersections in reductive groups.

In the case wherea complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve. I also prove that the higher Chernclasses vanish. The first and the last nontrivial Chern classes are described explicitly. An extension of these results to the setting of spherical homogeneous spacesis outlined.Résumé. — Dans cet article, je construis l’analogue non compact des classes deChern pour des fibrés vectoriel équivariants au-dessus de groupes réductifs complexes. Pour le fibré tangent, ces classes de Chern produisent une formule d’adjonction pour la caractéristique d’Euler (topologique) d’intersections complètes dansdes groupes réductifs. Dans le cas d’une intersection complète qui est une courbe,cette formule donne une réponse explicite pour la caractéristique d’Euler et legenre de la courbe.

Je démontre également que les classes de Chern supérieuressont nulles. La première et la dernière classe de Chern non nulle sont décrites explicitement. J’esquisse également une extension de ces résultats dans le cadre desespaces homogènes sphériques.1. Introduction and main resultsLet G be a connected complex reductive group. Consider a faithful finitedimensional representation π : G → GL(V ) on a complex vector spaceV .

Let H ⊂ End(V ) be a generic affine hyperplane. The hypersurfaceπ −1 (π(G) ∩ H) ⊂ G is called a hyperplane section corresponding to therepresentation π. The problem underlying this paper is how to find theKeywords: Reductive groups, hyperplane section, Chern classes.Math. classification: 14L30, 20G05.1226Valentina KIRITCHENKOEuler characteristic of a hyperplane section or, more generally, of the complete intersection of several hyperplane sections corresponding to differentrepresentations.The motivation to study such question comes from the case where thegroup G = (C∗ )n is a complex torus.

In this case, D. Bernstein, A. Khovanskii and A. Kouchnirenko found an explicit and very beautiful answer interms of the weight polytopes of representations (see [18]). E.g. the Eulercharacteristic χ(π) of a hyperplane section corresponding to the representation π is equal to (−1)n times the normalized volume of the weight polytopeof π. The proof uses an explicit relation between the Euler characteristicχ(π) and the degree of the affine subvariety π(G) in End(V ):(1.1)χ(π) = (−1)n−1 deg π(G).The degree is defined as usual. Namely, the degree of an affine subvariety X ⊂ CN equals to the number of the intersection points of X witha generic affine subspace in CN of complementary dimension. For the degree deg π(G) (that can also be interpreted as the self-intersection indexof a hyperplane section corresponding to the representation π) there is anexplicit formula proved by Kouchnirenko.

Later D. Bernstein, and Khovanskii found an analogous formula for the intersection index of hyperplanesections corresponding to different representations.How to extend these results to the case of arbitrary reductive groups?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 doneby B. Kazarnovskii [17]. Later, M. Brion established an analogous resultfor all spherical homogeneous spaces [4]. For reductive groups, the BrionKazarnovskii theorem allows to compute explicitly the intersection indexof n generic hyperplane sections corresponding to different representations.The precise definition of the intersection index is given in Section 2.However, when G is an arbitrary reductive group, it is no longer truethat χ(π) = (−1)n−1 deg π(G).

K. Kaveh computed explicitly χ(π) anddeg π(G) for all representations π of SL2 (C). His computation shows that,in general, there is a discrepancy between these two numbers. Kaveh alsolisted some special representations of reductive groups, for which thesenumbers still coincide [16].In this paper, I will present a formula that, in particular, generalizesformula (1.1) to the case of arbitrary reductive groups. To do this I willconstruct algebraic subvarieties Si ⊂ G, whose degrees fill the gap betweenthe Euler characteristic and the degree. My construction is similar to oneANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1227of the classical constructions of the Chern classes of a vector bundle in thecompact setting (Subsection 3.1).

The subvarieties Si can be thought ofas Chern classes of the tangent bundle of G. I will also construct Chernclasses of more general equivariant vector bundles over G (Subsection 3.2).These Chern classes are in many aspects similar to the usual Chern classesof compact manifolds.

There is an analog of the cohomology ring for G,where the Chern classes of equivariant bundles live. This analog is thering of conditions constructed by C. De Concini and C. Procesi [10, 8](seeSection 2 for a reminder). It is useful in solving enumerative problems. Inparticular, the intersection product in this ring is well-defined.I now formulate the main results.

Denote by n and k the dimensionand the rank of G, respectively. Recall that the rank is the dimension ofa maximal torus in G. Denote by [S1 ], . . . , [Sn ] the Chern classes of thetangent bundle of G as elements of the ring of conditions, and denote byS1 ,. . . , Sn subvarieties representing these classes. In the case of the tangentbundle, it turns out (see Lemma 3.8) that the the higher Chern classes[Sn−k+1 ],. . .

, [Sn ] vanish. E.g. if G is a torus, then all Chern classes [Si ]vanish.Let H1 ,. . . , Hm be a generic collection of m hyperplane sections corresponding to faithful representations π1 ,. . . , πm of the group G (for theprecise meaning of “generic” see Subsection 4.3). Then the following theorem holds.Theorem 1.1. — The Euler characteristic of the complete intersectionH1 ∩ . . . ∩ Hm is equal to the term of degree n in the expansion of thefollowing product:(1 + S1 + . . .

+ Sn−k )mYHi (1 + Hi )−1 .i=1The product in this formula is the intersection product in the ring of conditions.This is very similar to the classical adjunction formula in the compactsetting.In particular, the Euler characteristic of just one hyperplane section corresponding to a representation π is equal to the following alternating sum.Put S0 = G.

Thenχ(π) =n−kXi=0TOME 56 (2006), FASCICULE 4(−1)n−i−1 deg π(Si ).1228Valentina KIRITCHENKOThe latter formula may have applications in the theory of generalized hypergeometric equations. In the torus case, I. Gelfand, M. Kapranov andA. Zelevinsky showed that the Euler characteristic χ(π) gives the numberof integral solutions of the generalized hypergeometric system associatedwith the representation π [13]. A similar system can be associated withthe representation π of any reductive group [15]. In the reductive case,the number of integral solutions of such a system is also likely to coincidewith χ(π).The proof of Theorem 1.1 is similar to the proof by Khovanskii [18] inthe torus case. Namely, Theorem 1.1 follows from the adjunction formulaapplied to the closure of a complete intersection in a suitable regular compactification of G (see Subsection 4.3).

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