Диссертация (Геометрия сферических многообразий и многогранники Ньютона-Окунькова), страница 9

This sum contains two onedimensional components p(V1 ) and p(V2 ) (which are considered as lines inΛd (V1 ⊕ V2 )). In particular, for any vector in Λd (V1 ⊕ V2 ) it makes senseto speak of its projections to p(V1 ) and p(V2 ). On V1 and V2 there aretwo special n-forms, preserved by SL(V ). These forms give rise to two 1forms l1 and l2 on p(V1 ) and p(V2 ), respectively. Consider the hyperplaneH in Λd (V1 ⊕ V2 ) consisting of all vectors v such that the functionals l1and l2 take the same values on the projections of v to p(V1 ) and p(V2 ),respectively. Then it is easy to check that p(XE ) = p(G(d, 2d)) ∩ P(H).In the next section, I will be concerned with the case when E = T G isthe tangent bundle.

In this case, the vector bundle EX is closely relatedto the tangent bundles of regular compactifications of the group G. Let usdiscuss this case in more detail.Example 3. — This example is a slightly more general version of Example 1. Let g = g0 ⊕c be the decomposition of the Lie algebra g into the directsum of the semisimple and the central subalgebras, respectively.

Denote byc the dimension of the center c. Let E = T G be the tangent bundle on G.Then ϕE maps G to the Grassmannian G(n − c, (g ⊕ g)/c). It is easy toshow that the image of the map ϕE coincides with the adjoint group of Gand the image contains only subspaces that belong to (g0 ⊕ g0 ) ⊂ (g ⊕ g)/c.Comparing this with Example 1, one can easily see that XE is isomorphicto the wonderful compactification Xcan of the adjoint group of G.In this case, the bundle EX is the direct sum of the Demazure bundle andthe trivial vector bundle of rank c. Indeed, for any subspace Λx ∈ XE 'Xcan ⊂ G(n − c, Γ(E)) its intersection with the subspace c− = {(c, −c), c ∈c} ⊂ Γ(E) is trivial. Hence, the quotient space Γ(E)/Λx coincides with thedirect sum ((g0 ⊕ g0 )/Λx ) ⊕ c− .Proof of Lemma 3.3.

The proof of Lemma 3.3 relies on the followingfact. Let Y1 and Y2 be two subvarieties of codimension i in the group G.TOME 56 (2006), FASCICULE 41240Valentina KIRITCHENKOUsing Kleiman’s transversality theorem and continuity arguments, it is easyto show that Y1 and Y2 represent the same class in the ring of conditionsC ∗ (G) if there exists an equivariant compactification X of the group Gsuch that the closures of Y1 , Y2 in X have proper intersections with allG × G-orbits (see [10] for the proof).In particular, to prove Lemma 3.3 it is enough to produce an equivariantcompactification X such that the closure of a generic Si (E) has properintersections with all G × G-orbits in X.

I claim that the compactificationXE discussed in the previous paragraph (see Proposition 3.6) satisfies thiscondition.Indeed, the closure of any Si (E) in XE coincides with the intersectionof XE with the Schubert cycle Ci corresponding to a partial flag in Γ(E).By Kleiman’s transversality theorem applied to the Grassmannian G(d −c, Γ(E)) (see Subsection 3.1), a partial flag can be chosen in such a way thatthe corresponding Schubert cycle has proper intersections with all G × Gorbits in XE .

All partial flags with such property form an open dense subsetin the space of all partial flags. Hence, for generic flags the correspondingsubvarieties Si represent the same class in the ring of conditions.In the sequel, Si (E) will denote any subvariety of the family Si (E) whoseclass in the ring of conditions coincides with the Chern class [Si (E)].Remark. — Recall that the ring of conditions C ∗ (G) can be identifiedwith the direct limit of cohomology rings of equivariant compactificationsof G (see Theorem 2.3). It follows that under this identification the Chernclass [Si (E)] ∈ C ∗ (G) corresponds to an element in the cohomology ring ofthe compactification XE . In particular for an adjoint group G, the Chernclass [Si (T X)] of the tangent bundle corresponds to some cohomology classof the wonderful compactification of G.Properties of the Chern classes of reductive groups.

The nextlemma computes the dimensions of the Chern classes. It also shows that ifG acts on V without an open dense orbit, then the higher Chern classesautomatically vanish.For any representation π : G → GL(V ), there exists an open dense Ginvariant subset in V such that the stabilizers of any two elements from thissubset are conjugate subgroups of G (see [24]). In particular, all elementsfrom this subset have isomorphic G-orbits. Such orbits are called principal.Denote by d(π) the dimension of a principal orbit of G in V . If G has anopen dense orbit in V , then d(π) = d. In my main example, when π is theadjoint representation, d(π) = n − k.ANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1241Lemma 3.8. — If i > d(π), then Si (E) is empty, and if i 6 d(π) thenthe dimension of Si (E) is equal to n − i.Proof. — Recall that Si (E) is the inverse image of Ci under the mapϕE : G → G(d − c, Γ(E)).

Here Ci is the i-th Schubert cycle correspondingto a generic partial flag in Γ(E). The codimension of Ci in the Grassmannian G(d − c, Γ(E)) is equal to i. Hence, by Kleiman’s transversalitytheorem applied to G(d − c, Γ(E)) (see Subsection 3.1), the intersectionCi ∩ ϕE (G) is either empty or proper and has codimension i in ϕE (G).Then Si (E) = ϕ−1E (Ci ∩ ϕE (G)) is either empty or has codimension i inG, because all fibers of the map ϕE are isomorphic to each other (each ofthem is isomorphic to the kernel of π). It remains to find out all i for whichSi (E) is empty.By Remark 3.5, the Chern class Si (E) consists of all elements g ∈ G suchthat the graph Γg = {(v, π(g)v), v ∈ V } ⊂ V ⊕ V of π(g) has a nontrivialintersection with a generic subspace Λd−i+1 of dimension d−i+1 in V ⊕V .For all g ∈ Si (E) r Si+1 (E) the intersection Γg ∩ Λd−i+1 has dimension1.

Indeed, if dim(Γg ∩ Λd−i+1 ) > 2, then dim(Γg ∩ Λd−i ) > 1 (since thesubspace Λd−i ⊂ Λd−i+1 has codimension one in Λd−i+1 ), and g belongs toSi+1 (E). Hence, there is a well-defined mapp : Si (E) r Si+1 (E) → P(D ∩ Λd−i+1 );p : g 7→ P(Γg ∩ Λd−i+1 ).Here D ⊂ V ⊕ V is the union of all graphs Γg for g ∈ G. In particular, theChern class Si (E) is nonempty if and only if P(D ∩ Λd−i+1 ) is nonempty.We now estimate the dimension of D ∩ Λd−i+1 .

Since D is not a variety,it is more convenient to take its Zariski closure D. The subvariety D is theclosure of the image of the following morphism:F :G×V →V ×V;F : (g, v) 7→ (v, π(g)v).The source space G×V is an irreducible variety of dimension n+d, and thegeneral fibers of F are isomorphic to the principal stabilizers, of dimensionn − d(π). Hence dim D = d + d(π), that is, D has codimension d − d(π).Next, observe that D is a constructible set, invariant under scalar multiplication.

Hence it contains a dense open subset (also invariant underscalar multiplication) of the irreducible variety D. Thus a general vectorspace Λd−i+1 satisfies dim(D ∩ Λd−i+1 ) = d(π) − i + 1, if i 6 d(π), andD ∩ Λd−i+1 is dense in this intersection. In particular, if i = d(π), thenD ∩ Λd−i+1 consists of several lines whose number is equal to the degree ofD. If i > d(π), then D ∩ Λd−i+1 contains only the origin.

It follows that ifi > d(π), then Si (E) is empty.TOME 56 (2006), FASCICULE 41242Valentina KIRITCHENKOThis proof also implies the following corollary. Denote by H ⊂ G thestabilizer of an element in a principal orbit of G in V . The subgroup His defined up to conjugation so its class in the ring of conditions is welldefined.Corollary 3.9. — An open dense subset of the subvariety Si (E) admits almost a fibration whose fibers are translates of H.

Here almost meansthat the intersection of different fibers always lies in Si+1 (E) ⊂ Si (E). Inparticular, the last Chern class Sd(π) (E) admits a true fibration and coincides with the disjoint union of several translates of H. Their numberequals to the degree of a generic principal orbit of G in V .The last statement follows from the fact that the degree of D in V ⊕ V(see the proof of Lemma 3.8) is equal to the degree of a generic principalorbit of G in V .In particular, let E be the tangent bundle. Then the stabilizer of a genericelement in g is a maximal torus in G. Hence, the last Chern class Sn−k (T G)is the union of several translates of a maximal torus.

The number of translates is the cardinality of the Weyl group (the degree of a general orbit inthe adjoint representation).3.3. The first and the last Chern classesThroughout the rest of the paper, I will only consider the Chern classesSi = Si (T G) of the tangent bundle unless otherwise stated. Theorem 1.1expresses the Euler characteristic of a complete intersection via the intersection indices of the Chern classes Si with generic hyperplane sections.The question is how to compute these indices.

If [Si ] is a linear combination of complete intersections of generic hyperplane sections correspondingto some representations of G, then the answer to this question is given bythe Brion-Kazarnovskii formula. A hyperplane section corresponding to therepresentation π is called generic if its closure in the compactification Xπhas proper intersections with all G × G-orbits in Xπ .In this subsection, I describe S1 as a generic hyperplane section. Thedescription follows from a result of Rittatore [25]. One can also computethe intersection indices with the last Chern class Sn−k , because Sn−k isthe union of translates of a maximal torus (see Corollary 3.9).

However,it seems that in general the Chern class Si , for i 6= 1, is not a sum ofcomplete intersections. E.g. I can show that for G = SL3 (C) the Chernclass [S3 ] does not lie in the subring of C ∗ (G) generated by the classes ofhypersurfaces.ANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1243Description of S1 . The result of Rittatore for the first Chern class ofregular compactifications (see [25], Proposition 4) implies that the class[S1 ] in the ring of conditions can be represented by the doubled sum of theclosures of all codimension one Bruhat cells in G. Below I will deduce thisdescription directly from the definition of S1 .It is easy to show that S1 ⊂ G is given by the equation det(Ad(g) − A) =0 for a generic A ∈ End(g).

Indeed, the first Chern class S1 (E) of anyequivariant vector bundle E over G consists of elements g ∈ G such thatthe graph of the operator π(g) in V ⊕ V has a nontrivial intersection witha generic subspace of dimension n in V ⊕ V (see Remark 3.5). As a genericsubspace, one can take the graph of a generic operator A on V . Then thegraphs of operators π(g) and A have a nonzero intersection if and only ifthe kernel of the operator π(g) − A is nonzero.The function det(Ad(g)−A) is a linear combination of matrix coefficientscorresponding to all exterior powers of the adjoint representation. Hence,the equation of S1 is the equation of a hyperplane section correspondingto the sum of all exterior powers of the adjoint representation. Denote thisrepresentation by σ.