Диссертация (1136188), страница 10

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It is easy to check that the weight polytope Pσ coincides with the weight polytope of the irreducible representation θ with thehighest weight 2ρ (here ρ is the half sum of all positive roots, or equivalentlythe sum of all fundamental weights). It remains to prove that S1 is generic,which means that the closure of S1 in Xσ intersects all G × G-orbits alongsubvarieties of codimension one. The proof of Lemma 3.3 implies that thisis true for the wonderful compactification, and the normalization of Xσ isthe wonderful compactification by Theorem 2.1 (since Pθ = Pσ ).It is now easy to show that the doubled sum of the closures of all codimension one Bruhat cells in G is equivalent to S1 .

This is because theclosures of codimension one Bruhat cells are generic hyperplane sectionscorresponding to the irreducible representations with fundamental highestweights.Description of Sn−k . By Corollary 3.9 the last Chern class Sn−k is thedisjoint union of translates of a maximal torus. Their number is equal tothe degree of a generic adjoint orbit in g. The latter is equal to the orderof the Weyl group W .

Denote by [T ] the class of a maximal torus in thering of conditions C ∗ (G). Then the following identity holds in C ∗ (G):[Sn−k ] = |W |[T ].The degree of π(T ) can be computed using the formula of D.Bernstein,Khovanskii and Koushnirenko [18].TOME 56 (2006), FASCICULE 41244Valentina KIRITCHENKO3.4. ExamplesG = SL2 (C). Consider the tautological embedding of G, namely, G ={(a, b, c, d) ∈ C4 : ad−bc = 1}. Since the dimension of G is 3 and the rank is1, by Lemma 3.8 we get that there are only two nontrivial Chern classes: S1and S2 . Let us apply the results of the preceding subsection to find them.The first Chern class S1 is a generic hyperplane section corresponding tothe second symmetric power of the tautological representation, i.e., to therepresentation θ : SL2 (C) → SO3 (C).

In other words, it is the intersectionof SL2 (C) with a generic quadric in C4 . The second Chern class S2 (whichis also the last one in this case) is the union of two translates of a maximaltorus (or the intersection of S1 with a hyperplane in C4 ).Let π be a faithful representation of SL2 (C). It is a direct sum of irreducible representations. Any irreducible representation of SL2 (C) is isomorphic to the i-th symmetric power of the tautological representation forsome i. Its weight polytope is the line segment [−i, i].

Hence the weightpolytope of π is the line segment [−n, n] where n is the greatest exponentof symmetric powers occurring in π. Then the matrix coefficients of π arepolynomials in a, b, c, d of degree n. In this case, it is easy to compute thedegrees of subvarieties π(G), π(S1 ) and π(S2 ) by the Bezout theorem. Thendeg π(G) = 2n3 , deg π(S1 ) = 4n2 , deg π(S2 ) = 4n. Also, if one takes another faithful representation σ with the weight polytope [−m, m], then theintersection index of S1 with two generic hyperplane sections correspondingto π and σ, equals to 4mn.Since by Theorem 1.1 the Euler characteristic χ(π) of a generic hyperplane section is equal to deg π(G) − deg π(S1 ) + deg π(S2 ), we getχ(π) = 2n3 − 4n2 + 4n.This answer was first obtained by Kaveh who used different methods [16].If π is not faithful, i.e., π(SL2 (C)) = SO3 (C), consider π as a representation of SO3 (C).

Then χ(π) is two times smaller and equals to n3 −2n2 +2n.Apply Theorem 1.1 to a curve C that is the complete intersection of twogeneric hyperplane sections corresponding to the representations π and σ.Thenχ(C) = Hπ · Hσ · Hθ − Hπ · Hσ · (Hπ + Hσ ) = −2mn(m + n − 2).G = (C∗ )n is a complex torus. In this case, all left-invariant vectorfields are also right-invariant since the group is commutative. Hence, theyare linearly independent at any point of G = (C∗ )n as long as their valuesat the identity are linearly independent. It follows that all subvarieties SiANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1245are empty, and all the Chern classes vanish.

Then Theorem 1 coincideswith a theorem of D.Bernstein and Khovanskii [18].4. Chern classes of regular compactificationsand proof of Theorem 1.14.1. PreliminariesChern classes of the tangent bundle. In this paragraph, I explain amethod from [11], which in some cases allows to find the Chern classes ofsmooth varieties.Let X be a smooth complex variety of dimension n, and let D ⊂ X be adivisor. Suppose that D is the union of l smooth irreducible hypersurfacesD1 , .

. . , Dl with normal crossings. One can relate the tangent bundle T Xof X to the logarithmic tangent bundle, consisting of those vector fieldsthat preserve the divisor D.Let LX (D1 ), . . . , LX (Dl ) be the line bundles over X associated withthe hypersurfaces D1 , .

. . , Dl , respectively. i.e., the first Chern class of thebundle LX (Di ) is the homology class of Di . One can also associate with Dthe logarithmic tangent bundle VX (D). It is a holomorphic vector bundleover X of rank n that is uniquely defined by the following property. Theholomorphic sections of VX (D) over an open subset U ⊂ X consist of allholomorphic vector fields v(x) on U such that v(x) restricted to U ∩ Di istangent to the hypersurface Di for any i. The precise definition is as follows.Cover X by local charts.

If a chart intersects the divisors Di1 , . . . , Dikchoose local coordinates x1 , . . . , xn such that the equation of Dij in thesecoordinates is xj = 0. Then VX is given by the collection of trivial vector∂bundles spanned by the vector fields x1 ∂x, . . . , xk ∂x∂ k , ∂x∂k+1 , . . . , ∂x∂n over1each chart with the natural transition operators.For a vector bundle E, denote by O(E) the sheaf of its holomorphicsections.Proposition 4.1 ([11]).

— There is an exact sequence of coherentsheaves0 → O(VX (D)) → O(T X) →lMO(LX (Di )) ⊗OX ODi → 0.i=1In particular, the tangent bundle T X has the same Chern classes as thedirect sum of the bundle VX (D) with LX (D1 ),. . . , LX (Dl ).TOME 56 (2006), FASCICULE 41246Valentina KIRITCHENKOProposition 4.1 gives the answer for the Chern classes of X, when theChern classes of VX (D) are known. In particular, this is the case when Xis a smooth toric variety, and D = X r (C∗ )n is the divisor at infinity.In this case, the vector bundle VX (D) is trivial, and the Chern classesof T X can be found explicitly. This was done by Ehlers [11]. A moregeneral class of examples is given by regular compactifications of reductivegroups (see the next paragraph for the definition) and, more generally, ofarbitrary spherical homogeneous spaces (see Section 5).

In this case, thevector bundle VX (D) is no longer trivial but still has a nice description,which is due to Brion [5]. I recall his result in Subsection 4.2 and use it toprove Theorem 1.1.Regular compactifications. In this paragraph, I will define the notionof regular compactifications of reductive groups following [6]. Let X be asmooth G × G-equivariant compactification of a connected reductive groupG of dimension n.

Denote by O1 , . . . , Ol the orbits of codimension one in X.Then the complement X r G to the open orbit is the union of the closuresO1 , . . . , Ol of codimension one orbits.Definition 4.2. — A smooth G × G-equivariant compactification X iscalled regular if the following three conditions are satisfied.(1)The hypersurfaces O1 , . . .

, Ol are smooth and intersect each othertransversally.(2)The closure of any G×G-orbit in X rG coincides with the intersectionof those hypersurfaces O1 , . . . , Ol that contain it.(3) For any point x ∈ X and its G × G-orbit Ox ⊂ X, the stabilizer(G × G)x ⊂ G × G acts with a dense orbit on the normal space Tx X/Tx Oxto the orbit.This definition was introduced by E.

Bifet, De Concini and Procesi in amore general setting ([2], see also Section 5).If G is a complex torus, then the regularity of X is just equivalent tothe smoothness. However, for other reductive groups, there exist compactifications that are smooth but not regular. In particular, it follows fromProposition 4.3 below that the compactification Xπ associated with a representation π : G → GL(V ) (see Section 2) is regular if and only it issmooth and none of the vertices of the weight polytope of π lies on thewalls of the Weyl chambers.Regular compactifications of reductive groups generalize smooth toricvarieties and retain many nice properties of the latter. E.g. any regularcompactification X can be covered by affine charts Xα ' Cn in such aANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1247way that only k hypersurfaces Oi1 ,.

. . , Oik intersect Xα , and intersectionsOi1 ∩ Xα , . . . , Oik ∩ Xα are k coordinate hyperplanes in Xα [9, 6]. Here kdenotes the rank of G. In particular, all G×G-orbits in X have codimensionat most k, and all closed orbits have codimension k.If G is of adjoint type, then it has the wonderful compactification Xcan ,which is regular.

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