Диссертация (1136188), страница 8

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— If E = T G is the tangent bundle, then Γ(E) is a verynatural class of global sections. It consists of all vector fields coming fromthe standard action of G×G on G. Namely, with any element (X, Y ) ∈ g⊕gone can associate a vector field v ∈ Γ(E) as follows:d v(x) =[etX xe−tY ] = Xx − xY.dt t=0TOME 56 (2006), FASCICULE 41236Valentina KIRITCHENKOThis example suggests that one represent elements of Γ(E) not as sumsbut as differences of left- and right-invariant sections.The space Γ(E) can be employed to define Chern classes of E as usual.Take d generic sections v1 , . .

. , vd ∈ Γ(E). Then the i-th Chern class isthe i-th degeneracy locus of these sections. More precisely, the i-th Chernclass Si (E) ⊂ G consists of all points g ∈ G such that the first d − i + 1sections v1 (g), . . . , vd−i+1 (g) taken at g are linearly dependent. This definition almost repeats one of the classical definitions of the Chern classes inthe compact setting (see Subsection 3.1).

The only difference is that globalsections used in this definition are not generic in the space of all sections.They are generic sections of the special subspace Γ(E). If one drops thisrestriction and applies the same definition, then the result will be trivial,since the bundle E is topologically trivial. In some sense, the Chern classeswill sit at infinity in this case (the precise meaning will become clear fromthe second part of this subsection). The purpose of my definition is to pullthem back to the finite part.Thus for each i = 1, . . . , d we get a family Si (E) of algebraic subvarietiesSi (E) parameterized by collections of d − i + 1 elements from Γ(E).

In thecompact situation, all generic members of an analogous family represent thesame class in the cohomology ring. The same is true here, if one uses thering of conditions as an analog of the cohomology ring in the noncompactsetting.Lemma 3.3. — For all collections v1 , . . . , vd−i+1 belonging to some opendense subset of (Γ(E))d−i+1 the class of the corresponding subvariety Si (E)in the ring of conditions C ∗ (G) is the same.The lemma implies that the family Si (E) parameterized by elements of(Γ(E))d−i+1 provides a well-defined class [Si (E)] in the ring of conditionsC ∗ (G).Definition 3.4. — The class [Si (E)] ∈ C ∗ (G) defined by the familySi (E) is called the i-th Chern class of a vector bundle E with value in thering of conditions.Before proving the lemma let me give another description of the Chernclasses [Si (E)].Maps to Grassmannians.

In this paragraph, I apply the classical construction discussed in Subsection 3.1 to define a map from the group Gto the Grassmannian G(d − c, Γ(E)) of subspaces of dimension (d − c) inANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1237the space Γ(E). Recall that c is the dimension of the maximal trivial subrepresentation of V , and the dimension of Γ(E) is 2d − c (see the end ofSubsection 3.1).Note that the global sections from the subspace Γ(E) span the fiber ofE at each point of G. Hence, one can define a map ϕE from G to theGrassmannian G(d − c, Γ(E)) as follows.

A point g ∈ G gets mapped tothe subspace Λg ⊂ Γ(E) spanned by all global sections that vanish at g.Clearly, the dimension of Λg equals to (dim Γ(E) − d) = (d − c) for allg ∈ G. We get the mapϕE : G → G(d − c, Γ(E));ϕE : g 7→ Λg .The subspace Λg can be alternatively described using the graph of the operator π(g) in V ⊕V . Namely, it is easy to check that Λg = {(X, π(g)X), X ∈V }/C.

Then ϕE comes from the natural map assigning to the operator π(g)on V its graph in V ⊕ V .Clearly, the pull-back of the tautological quotient vector bundle overG(d, Γ(E)) is isomorphic to E. Hence, the Chern class Si (E) constructedvia elements v1 , . . . , vd is the inverse image of the Schubert cycle Ci corresponding to the partial flag hv1 i ⊂ hv1 , v2 i ⊂ . . . ⊂ hv1 , . . . , vd i ⊂ Γ(E) (seeSubsection 3.1). Here hv1 , .

. . , vi i denotes the subspace of Γ(E) spanned bythe vectors v1 , . . . , vi .Remark 3.5. — This gives the following equivalent definition of Si (E).The Chern class Si (E) consists of all elements g ∈ G such that the graphof the operator π(g) in V ⊕ V has a nontrivial intersection with a genericsubspace of dimension d − i + 1 in V ⊕ V .In particular, if the representation π : G → GL(V ) corresponding to avector bundle E has a nontrivial kernel, then the Si (E) are invariant underleft and right multiplications by the elements of the kernel (since this isalready true for the preimage ϕ−1E (Λ) of any point Λ ∈ ϕE (G)).

E.g. theChern classes Si (T G) are invariant under multiplication by the elementsof the center of G.We can now relate the Chern classes Si (E) to the usual Chern classes ofa vector bundle over a compact variety.Denote by XE the closure of ϕE (G) in the Grassmannian G(d − c, Γ(E)),and denote by EX the restriction of the tautological quotient vector bundleto XE . We get a vector bundle on a compact variety. The i-th Chern classof EX is the homology class of Ci ∩ XE for a generic Schubert cycle Ci(see Proposition 3.1).

By Kleiman’s transversality theorem applied to theGrassmannian G(d−c, Γ(E)) (see Subsection 3.1), a generic Schubert cycleTOME 56 (2006), FASCICULE 41238Valentina KIRITCHENKOCi has a proper intersection with the boundary divisor XE rϕE (G). Hence,there is the following relation between the Chern classes of EX and genericmembers of the family Si (E).Proposition 3.6. — For a generic Si (E) the homology class of theclosure of ϕE (Si (E)) in XE coincides with the i-th Chern class of EX .Thus the Chern classes [Si (E)] can be described via the usual Chernclasses of the bundle EX over the compactification XE .Let us study the variety XE in more detail. It is a G × G-equivariantcompactification of the group ϕE (G). Indeed, the action of G × G onϕE (G) can be extended to the Grassmannian G(d, Γ(E)) as follows.

Identify Γ(E) with (V ⊕ V )/C (see the end of Subsection 3.1). The doubledgroup G × G acts on V ⊕ V by means of the representation π ⊕ π, i.e.,(g1 , g2 )(v1 , v2 ) = (g1 v1 , g2 v2 ) for g1 , g2 ∈ G, v1 , v2 ∈ V . The subspaceC ⊂ V ⊕ V is invariant under this action. Hence, the group G × G actson Γ(E). This action provides an action of G × G on the GrassmannianG(d − c, Γ(E)).

Clearly, the subvariety XE is invariant under this action.Example 1 (Demazure embedding). — Let G be a group of adjointtype, and let π be its adjoint representation on the Lie algebra g. Thecorresponding vector bundle E coincides with the tangent bundle of G. Thecorresponding map ϕE : G → G(n, g ⊕ g) coincides with the embeddingconstructed by Demazure [9]. The Demazure map takes an element g ∈ Gto the Lie subalgebra gg = {(gXg −1 , X), X ∈ g} ⊂ g ⊕ g. Clearly, theDemazure map provides an embedding of G into G(n, g ⊕ g).It is easy to check that the Lie subalgebra gg is the Lie algebra of thestabilizer of an element g ∈ G under the standard action of G × G.

Thus forany A ∈ gg the corresponding vector field vanishes at g, and the Demazureembedding coincides with ϕE . The compactification XE in this case isisomorphic to the wonderful compactification Xcan of the group G [9]. Inparticular, it is smooth.Definition 3.7. — Let G and E be as in Example 1.

The restrictionof the tautological quotient vector bundle to XE ' Xcan is called theDemazure bundle and is denoted by Vcan .If E is the tangent bundle, then Proposition 3.6 implies that the Chernclass Si (E) is the inverse image of the usual i-th Chern class of the Demazure bundle. The Demazure bundle is considered in [5], where it is related to the tangent bundles of regular compactifications of the group G.Example 2.a) Let G be GL(V ) and let π be its tautological representation on theANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1239space V of dimension d.

Then ϕE is an embedding of GL(V ) into theGrassmannian G(d, 2d). Notice that the dimensions of both varieties arethe same. Hence, the compactification XE coincides with G(d, 2d).b) Take SL(V ) instead of GL(V ) in the previous example. Its compactification XE is a hypersurface in the Grassmannian G(d, 2d) which can bedescribed as a hyperplane section of the Grassmannian in the Plücker embedding.

Consider the Plücker embedding p : G(d, 2d) → P(Λd (V1 ⊕ V2 )),where V1 and V2 are two copies of V . Then p(XE ) is a special hyperplane section of p(G(d, 2d)). Namely, the decomposition V1 ⊕ V2 yields adecomposition of Λd (V1 ⊕V2 ) into a direct sum.

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