Диссертация (Геометрия сферических многообразий и многогранники Ньютона-Окунькова), страница 7

— Let H be aconnected algebraic group, and let M be a homogeneous space under H.Take two algebraic subvarieties X, Y ⊂ M . Denote by gX the left translateof X by an element g ∈ H. There exists an open dense subset of H suchthat for all elements g from this subset the intersection gX ∩ Y is proper.If X and Y are smooth, then gX ∩ Y is transverse for general g ∈ H.In particular, if X and Y have complementary dimensions (but are notnecessarily smooth), then for almost all g the translate gX intersects Ytransversally at a finite number of points, and this number does not dependon g.If X and Y have complementary dimensions, define the intersection index(X, Y ) as the number #(gX ∩Y ) of the intersection points for a generic g ∈H. If one is interested in solving enumerative problems, then it is naturalto consider algebraic subvarieties of M up to the following equivalence.Two subvarieties X1 , X2 of the same dimension are equivalent if and only iffor any subvariety Y of complementary dimension the intersection indices(X1 , Y ) and (X2 , Y ) coincide.

This relation is similar to the numericalequivalence in algebraic geometry (see [12], Chapter 19). Consider all formalTOME 56 (2006), FASCICULE 41232Valentina KIRITCHENKOlinear combinations of algebraic subvarieties of M modulo this equivalencerelation. Then the resulting group C ∗ (M ) is called the group of conditionsof M .One can define an intersection product of two subvarieties X, Y ⊂ M bysetting X · Y = gX ∩ Y , where g ∈ G is generic. However, the intersectionproduct sometimes is not well-defined on the group of conditions (see [10]for a counterexample). A remarkable fact is that for spherical homogeneousspaces the intersection product is well-defined, i.e., if one takes differentrepresentatives of the same classes, then the class of their product willbe the same [10, 8].

The corresponding ring C ∗ (M ) is called the ring ofconditions.In particular, the group of conditions C ∗ (G) of a reductive group is a ring.De Concini and Procesi related the ring of conditions to the cohomologyrings of equivariant compactifications as follows. Consider the set S of allsmooth equivariant compactifications of the group G. This set has a naturalpartial order. Namely, a compactification Xσ is greater than Xπ if Xσ liesover Xπ , i.e., if there exists a map Xσ → Xπ commuting with the action ofG × G. Clearly, such a map is unique, and it induces a map of cohomologyrings H ∗ (Xπ ) → H ∗ (Xσ ).Theorem 2.3 ([10, 8]).

— The ring of conditions C ∗ (G) is isomorphicto the direct limit over the set S of the cohomology rings H ∗ (Xπ ).De Concini and Procesi proved this theorem in [10] for symmetric spaces.In [8] De Concini noted that their arguments go verbatim for arbitraryspherical homogeneous spaces.3. Chern classes of reductive groups3.1. PreliminariesReminder about the classical Chern classes. In this paragraph, Iwill recall one of the classical definitions of the Chern classes, which I willuse in the sequel. For more details see [14].Let M be a compact complex manifold, and let E be a vector bundleof rank d over M . Consider d global sections s1 , .

. . , sd of E that are C ∞ smooth. Define their i-th degeneracy locus as the set of all points x ∈M such that the vectors s1 (x), . . . , sd−i+1 (x) are linearly dependent. Thehomology class of the i-th degeneracy locus is the same for all genericANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1233choices of the sections s1 (x), . . .

, sd (x) [14]. It is called the i-th Chern classof E.In what follows, I will only consider complex vector bundles that haveplenty of algebraic global sections (so that in the definition of the Chernclasses, it will be possible to take only algebraic global sections instead ofC ∞ -smooth ones).In particular, there is the following way to choose generic global sections.Let Γ(E) be a finite-dimensional subspace in the space of all global C ∞ smooth sections of the bundle E.

Suppose that at each point x ∈ M thesections of Γ(E) span the fiber of E at the point x. Then there is an opendense subset U in Γ(E)d such that for any collection of global sections(s1 , . . . , sd ) ⊂ U their i-th degeneracy locus is a representative of the i-thChern class of E.I will also use the following classical construction that associates with thesubspace Γ(E) a map from the variety M to a Grassmannian. Denote by Nthe dimension of Γ(E).

Let G(N − d, N ) be the Grassmannian of subspacesof dimension (N −d) in Γ(E). One can map M to G(N −d, N ) by assigningto each point x ∈ M the subspace of all sections from Γ(E) that vanishat x. By construction of the map the vector bundle E coincides with thepull-back of the tautological quotient vector bundle over the GrassmannianG(N −d, N ). Recall that the tautological quotient vector bundle over G(N −d, N ) is the quotient of two bundles.

The first one is the trivial vector bundlewhose fibers are isomorphic to Γ(E), and the second is the tautologicalvector bundle whose fiber at a point Λ ∈ G(N − d, N ) is isomorphic to thecorresponding subspace Λ of dimension N − d in Γ(E).Using the definition of the Chern classes given above, it is easy to checkthat the i-th Chern class of the tautological quotient vector bundle is thehomology class of the following Schubert cycle. Let Λ1 ⊂ . .

. ⊂ Λd ⊂ Γ(E)be a partial flag such that dim Λj = j. In the sequel, by a partial flagI will always mean a partial flag of this type. The i-th Schubert cycle Cicorresponding to such a flag consists of all points Λ ∈ G(N − d, N ) suchthat the subspaces Λ and Λd−i+1 have nonzero intersection.The following proposition relates the Schubert cycles Ci to the Chernclasses of E.Proposition 3.1 ([14]). — Let p : M → G(N − d, N ) be the mapconstructed above, and let Ci be the i-th Schubert cycle corresponding toa generic partial flag in Γ(E). Then the i-th Chern class of E coincideswith the homology class of the inverse image of Ci under the map p:ci (E) = [p−1 (Ci )].TOME 56 (2006), FASCICULE 41234Valentina KIRITCHENKOIn particular, this proposition allows to relate the definition of the Chernclasses via degeneracy loci to other classical definitions.In the sequel, the following statement will be used.

For any algebraicsubvariety X ⊂ G(N − d, N ), a partial flag can be chosen in such a waythat the corresponding Schubert cycle Ci has proper intersection with X.This follows from Kleiman’s transversality theorem, since the Grassmannian G(N −d, N ) can be regarded as a homogeneous space under the naturalaction of the group GLN . Then any left translate of a Schubert cycle Ci isagain a Schubert cycle of the same type.Equivariant vector bundles.

In this paragraph, I will recall the definition and some well-known properties of equivariant vector bundles.Let E be a vector bundle of rank d over G. Denote by Vg ⊂ E the fiberof E lying over an element g ∈ G. Assume that the standard action ofG × G on G can be extended linearly to E. More precisely, there existsa homomorphism A : G × G → Aut(E) such that A(g1 , g2 ) restricted tothe fiber Vg is a linear operator from Vg to Vg1 gg−1 . If these conditions2are satisfied, then the vector bundle E is said to be equivariant under theaction of G × G.Two equivariant vector bundles E1 and E2 are equivalent if there existsan isomorphism between E1 and E2 that is compatible with the structureof fiber bundle and with the action of G × G. The following simple andwell-known proposition describes equivariant vector bundles on G up tothis equivalence relation.Proposition 3.2.

— The equivalence classes of equivariant vector bundles of rank d are in one-to-one correspondence with the linear representations of G of dimension d.Indeed, with each representation π : G → V one can associate a bundleE isomorphic to G × V with the following action of G × G:A(g1 , g2 ) : (g, v) → (g1 gg2−1 , π(g1 )v).Then A(g, g −1 ) stabilizes the identity element e ∈ G and acts on the fiberVe = V by means of the operator π(g).E.g.

the adjoint representation of G on the Lie algebra g = T Ge corresponds to the tangent bundle T G on G. This example will be importantin the sequel.Among all algebraic global sections of an equivariant bundle E thereare two distinguished subspaces, namely, the subspaces of left- and rightinvariant sections. They consist of sections that are invariant under theANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1235action of the subgroups G × e and e × G, respectively. Both spaces can becanonically identified with the vector space V .

Indeed, any vector X ∈ Vdefines a right-invariant section vr (g) = (g, X). Then it is easy to see thatany left-invariant section vl is given by the formula vl (g) = (g, π(g)Y ) forY ∈V.Denote by Γ(E) the space of all global sections of E that are obtainedas sums of left- and right-invariant sections. Let us find the dimension ofthe vector space Γ(E). Clearly, if the representation π does not containany trivial sub-representations, then Γ(E) is canonically isomorphic to thedirect sum of two copies of V .

Otherwise, let C ⊂ V be the maximal trivialsub-representation. Embed C to V ⊕V diagonally, i.e., v ∈ C goes to (v, v).It is easy to see that Γ(E) as a G-module is isomorphic to the quotient space(V ⊕ V )/C. Denote by c the dimension of C. Then the dimension of Γ(E)is equal to 2d − c.3.2. Chern classes with values in the ring of conditionsIn this subsection, I define Chern classes of equivariant vector bundlesover G.

These Chern classes are elements of the ring of conditions C ∗ (G).Unlike the usual Chern classes in the compact situation, they measurethe complexity of the action of G × G but not the topological complexity(topologically any G×G-equivariant vector bundle over G is trivial). Whilethe definition of these classes does not use any compactification it turns outthat they are related to the usual Chern classes of certain vector bundlesover equivariant compactifications of G.Throughout this subsection, E denotes the equivariant vector bundleover G of rank d corresponding to a representation π : G → GL(V ).

Inthe subsequent sections, I will only use the Chern classes of the tangentbundle.Definition of the Chern classes. An equivariant vector bundle Ehas a special class Γ(E) of algebraic global sections. It consists of all globalsections that can be represented as sums of left- and right-invariant sections.Example.