Диссертация (Геометрия сферических многообразий и многогранники Ньютона-Окунькова), страница 11

This example is crucial for the study of the other regularcompactifications.For arbitrary reductive group G, denote by Xcan the wonderful compactification of the adjoint group of G. There is the following criterion ofregularity.Proposition 4.3 ([6]). — Let X be a smooth G × G-equivariant compactification of G. Then the condition that X is regular is equivalent tothe existence of a G × G-equivariant map from X to Xcan .E.g. if G is a complex torus, then the latter condition is always satisfiedbecause Xcan is a point in this case.Thus the set of regular compactifications of G consists of all smoothG × G-equivariant compactifications lying over Xcan .

In particular, for reductive groups of adjoint type the wonderful compactification is the minimal regular compactification.4.2. Demazure bundle and the Chern classes of regularcompactificationsIn this subsection, I state a formula for the Chern classes of regularcompactifications of reductive groups.

It follows from a more general resultproved for arbitrary toroidal spherical varieties by Brion [5]. This formulagives a description of the Chern classes in terms of two different collectionsof subvarieties. The first collection is given by the Chern classes of G,which are independent of a compactification, and the second is given bythe closures of codimension one orbits, which are easy to deal with (inparticular, all their intersection indices with other divisors can be computedvia the Brion-Kazarnovskii theorem).Let X be a regular compactification of G, and let O1 , . . . , Ol be theclosures of the G × G-orbits of codimension one in X.

Then the tangentbundle T X of X can be described using the Demazure vector bundle Vcanover the wonderful compactification Xcan (see Example 1 from Subsection3.2) and the line bundles corresponding to the hypersurfaces Oi .TOME 56 (2006), FASCICULE 41248Valentina KIRITCHENKOLet L(O1 ), . . . , L(Ol ) be the line bundles over X associated with thehypersurfaces O1 , . . . , Ol , respectively.

Let p : X → Xcan be the canonicalmap from Proposition 4.3, and let p∗ (Vcan ) be the pull-back of the Demazure vector bundle to X. It turns out that p∗ (Vcan ) coincides up to atrivial summand with the logarithmic tangent bundle corresponding to theboundary divisor X r G.Theorem 4.4 ([5]). — The tangent bundle T X has the same Chernclasses as the direct sum of the pull-back p∗ (Vcan ) with the line bundlesL(O1 ), .

. . , L(Ol ).In the case when G is a complex torus, Theorem 4.4 was proved byEhlers [11]. For arbitrary reductive groups, Theorem 4.4 follows from amore general result by Brion ([5], 1.6 Corollary 1).This theorem implies the following formula for the Chern classesc1 (X), . . . , cn (X) of the tangent bundle of X. Let Si = Si (T G) ⊂ G fori = 1, .

. . , n − k be the Chern classes of the tangent bundle of G definedin the previous section (see Definition 3.4). Denote by S i the closure of Siin X. Note that S i has proper intersections with all G × G-orbits in X(since this is already true for the wonderful compactification Xcan , and Xlies over Xcan ).Corollary 4.5. — The total Chern class c(X) = 1+c1 (X)+. . .+cn (X)coincides with the following product:c(X) = (1 + S 1 + . . .

+ S n−k )lY(1 + Oi ).i=1The product in this formula is the intersection product in the (co)homologyring of X.Below I sketch the proof of Theorem 4.4 following mostly the proofs byEhlers and Brion. The goal is to explain the main idea of their proofs,which is very transparent, and motivate the definition of the Chern classesSi . In the torus case, this idea can be extended to a complete elementaryproof. For more details see [11] and [5].Take n generic vector fields v1 , .

. . , vn coming from the action of G × G.It is not hard to show that v1 , . . . , vn are generic in the space of allC ∞ -smooth vector fields on X (it is enough to prove it for each affine charton X). Hence, their degeneracy loci give Chern classes of X. Note thatthese fields are not only C ∞ -smooth but also algebraic so their degeneracyloci are algebraic subvarieties in X.ANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1249The picture is especially simple in the torus case, because in this casev1 , . . . , vn−i+1 are linearly dependent precisely on all orbits of codimensiongreater than or equal to i (since they all belong to the tangent bundle ofthe orbit) and independent on the other orbits.

Hence, the i-th Chern classof X consists of all orbits of codimension at least i.In the reductive case, the situation is more complicated because thedegeneracy loci of v1 , . . . , vn have nontrivial intersections with the openorbit G ⊂ X. These intersections are exactly the Chern classes S1 , . . . , Sn−kof G. So it seems more convenient to use the method described in Subsection4.1 (see Proposition 4.1). Namely, consider the logarithmic tangent bundleVX = VX (X r G) corresponding to the boundary divisor X r G = O1 ∪. .

. ∪ Ol . Recall that c denotes the dimension of the center of G.Proposition 4.6. — The vector bundle VX is isomorphic to the directsum of the pull-back p∗ (Vcan ) with the trivial vector bundle E c of rank c.Proof. — The vector fields coming from the action of G × G on X areglobal sections of the bundle VX , since they are tangent to allG × G-orbits in X. It follows easily from condition (3) in the definitionof regular compactifications that these global sections span the fiber of VXat any point of X.

Hence, the map ϕE : G → G(n − c, (g ⊕ g)/c) consideredin Example 3 extends to a map p : X → G(n−c, (g⊕g)/c). The rest followsfrom Example 3Remark 4.7. — There is also another construction of the map p : X →Xcan by Brion (see [4]).4.3. ApplicationsIn this subsection, I prove Theorem 1.1 using the formula for the Chernclasses of regular compactifications (Corollary 4.5).

Then I apply it to compute the Euler characteristic and the genus of a curve in G.Proof of Theorem 1.1. First, define the notion of generic collectionof hyperplane sections used in the formulation of Theorem 1.1. A collection of m hyperplane sections H1 , . . . , Hm corresponding to representationsπ1 , . . . , πm , respectively, is called generic, if there exists a regular compactification X of G such that the closure H i of any hyperplane section Hi issmooth, and all possible intersections of H 1 , . .

. , H m with the closures ofG×G-orbits in X are transverse. E.g. one can take the compactification Xπcorresponding to the tensor product π of the representations π0 , π1 , . . . , πm ,TOME 56 (2006), FASCICULE 41250Valentina KIRITCHENKOwhere π0 is any irreducible representation with a strictly dominant highestweight. Then it is not hard to show that the set of all generic collections(with respect to the compactification Xπ ) is an open dense subset in thespace of all collections.So the closure Y = C of C = H1 ∩ . . . ∩ Hm in X is the transverseintersection of smooth hypersurfaces.

In particular, Y is smooth, and itsnormal bundle NY in X is the direct sum of m line bundles corresponding tothe hypersurfaces H i . The analogous statement is true for any subvarietyof the form Y ∩ OI , where I = {i1 , . . . , ip } is a subset of {1, . . . , l} andOI = Oi1 ∩· · ·∩Oip .

Let us find the Euler characteristic of Y ∩OI using theclassical adjunction formula. Denote by J = {1, . . . , l} r I the complementto the subset I. We get that χ(Y ∩ OI ) is the term of degree n in thedecomposition of the following intersection product in X:(1 + S 1 + · · · + S n−k )mYHs (1 + Hs )−1s=1YOii∈IY(1 + Oj ).(∗)j∈JOn the other hand, since the Euler characteristic is additive, and C =Y r (O1 ∪ · · · ∪ Ol ), one can express the Euler characteristic χ(C) in termsof the Euler characteristics χ(Y ∩ OI ) over all subsets I ⊂ {1, .

. . , l}:Xχ(C) =(−1)|I| χ(Y ∩ OI ).(∗∗)I⊂{1,...,l}Combining formulas (*) and (**), we get the formula of Theorem 1.1.Indeed, we have that χ(C) is the term of degree n in the decomposition ofthe following intersection product in X:mXYYY−1|I|1+S 1 +· · ·+S n−kOiHs (1+Hs )(−1)(1+Oj ) .s=1i∈IItJ={1,...,l}j∈JThe sum in the parentheses is equal to 1, since for any commuting variablesx1 , x2 , . .

. , xl we have the identity:1=lY((1 + xi ) − xi ) =i=1X(−1)|I|ItJ={1,...,l}Yxii∈IY(1 + xj ).j∈JComputation for a curve. Apply Theorem 1.1 and the formula forthe first Chern class S1 to a curve in G. We get that if C = H1 ∩ · · · ∩ Hn−1is a complete intersection of n − 1 generic hyperplane sections, thenχ(C) = (S1 − H1 − · · · − Hn−1 )n−1Yi=1ANNALES DE L’INSTITUT FOURIERHi .CHERN CLASSES OF REDUCTIVE GROUPS1251Since S1 is also a generic hyperplane section, the computation of χ(C) reduces to the computation of the intersection indices of hyperplane sections.Recall the Brion-Kazarnovskii formula for such intersection indices. Denote by R+ the set of all positive roots of G.

Recall that ρ denotes thehalf of the sum of all positive roots of G and LT denotes the characterlattice of a maximal torus T ⊂ G. Since G is reductive, we can assumethat g is embedded into gl(W ) so that the trace form tr(A, B) = tr(AB)for A, B ∈ gl(W ) is nondegenerate on g. Then the inner product (·, ·) onLT ⊗ R used in Theorem 4.8 is given by the trace form on g. Choose a Weylchamber D ⊂ LT ⊗ R.Theorem 4.8 ([4, 17]). — If Hπ is a hyperplane section correspondingto a representation π with the weight polytope Pπ ⊂ LT ⊗ R , then theself-intersection index of Hπ in the ring of conditions 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 in particular implies that the self-intersection index Hπndepends not on a representation but only on its weight polytope.

Note alsothat the integrand is invariant under the action of the Weyl group.Let H1 , . . . , Hn be n generic hyperplane sections corresponding to different representations π1 , . . . , πn . To compute their intersection index oneneeds to take the polarization of Hπn . Namely, the formula of Theorem4.8 gives a polynomial function D(P ) of degree n on the space of all virtual polytopes P ⊂ LT ⊗ R (the addition in this space is the Minkowskisum). The polarization Dpol is the unique symmetric n-linear form on thisspace such that Dpol (Pπ , .

. . , Pπ ) = D(Pπ ). Then Dpol (Pπ1 , . . . , Pπn ) isthe intersection index H1 · . . . · Hn . For instance, it can be found by ap1∂nplying the differential operator n!∂t1 ...∂tn to the function F (t1 , . . . , tn ) =D(t1 Pπ1 + · · · + tn Pπn ). E.g. if Pπ2 = · · · = Pπn , then the computation∂ of Dpol (Pπ1 , . . . , Pπn ) = n1 ∂tD(tPπ1 + Pπ2 ) reduces to the integrationt=0over the facets of Pπ2 .Thus we get the following answer for χ(C). For simplicity, the answer isgiven in the case when π1 = · · · = πn−1 = π. Then its polarization providesthe answer in the general case. Denote by P2ρ the weight polytope of theirreducible representation of G with the highest weight 2ρ.Corollary 4.9.