Диссертация (1136188), страница 16

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. . , xk ). There is a useful formula expressing the integral of f over a simplexin Rk in terms of the polarization of f . Recall that the polarization of f is theunique symmetric d-linear form fpol on Rk such that the restriction of fpol to thediagonal coincides with f . One can define fpol explicitly as follows:fpol (v1 , . .

. , vd ) =1∂df,d! ∂v1 . . . ∂vdwhere ∂vi is the directional derivative along the vector vi .Let ∆ ⊂ Rk be a k-dimensional simplex with vertices a0 , . . . , ak and let dx =dx1 ∧ dx2 ∧ · · · ∧ dxk be the standard measure on Rk .ON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS497Proposition 2.6 [3]. Let fpol be the polarization of f . It can be regarded as alinear function on the d-th symmetric power of V . Then the average value of f onthe simplex ∆ coincides with the average value of fpol on all symmetric products ofd vectors from the set {a0 , .

. . , ak }:ZX11fpol (a0 , . . . , a0 , . . . , ak , . . . , ak ).f (x)dx = d+k| {z }| {z }Vol(∆)∆ki0 +···+ik =di0ik3. Chern ClassesIn this section, I recall the definition of the Chern classes of spherical homogeneous spaces (see [11] for more details). In the sequel, only Chern classes of G × Gorbits in regular compactifications of G will be used. For these Chern classes, Iprove a vanishing result for their intersection indices with certain weight divisors inregular compactifications. This result will be important in Section 4 when applyingthe De Concini–Procesi algorithm to the Chern classes of G.Let G/H be a spherical homogeneous space under G of dimension d. The i-thChern class Si (G/H) of G/H is the i-th degeneracy locus of n generic vector fieldsv1 , . . .

, vn coming from the action of G, that is, Si (G/H) = {x ∈ G/H : v1 (x), . . . ,vd−i+1 (x) are linearly dependent}. In what follows we will use the following reformulation of this definition. Denote by g and h the Lie algebras of G and H,respectively, and denote by m the dimension of h. Define the Demazure map ϕfrom G/H to the Grassmannian G(m, g) of m-dimensional subspaces in g as follows:ϕ : G/H → G(m, g); ϕ : gH → ghg −1 .Let Ci ⊂ G(m, g) be the Schubert cycle corresponding to a generic subspace Λi ⊂ gof codimension m + i − 1, i. e. Ci = {Λ ∈ G(m, g) : dim(Λ ∩ Λi ) > 1}. Then it iseasy to see that the i-th Chern class Si (G/H) of G/H is the preimage of Ci underthe map ϕ:Si (G/H) = ϕ−1 (Ci ).The class of Si (G/H) in the ring of conditions of G/H is the same for all genericCi [11].

It is related to the Chern classes of the tangent bundles over regularcompactifications of G [4], [11]. Namely, if X is a regular compactification of G/H,then the closure of Si (G/H) in X is the i-th Chern class of the logarithmic tangentbundle over X that corresponds to the divisor X \ (G/H). This vector bundle isgenerated by all vector fields on X that are tangent to G-orbits in X. In whatfollows, this bundle will be called the Demazure bundle of X.Let X = G/H and Y = G/P be two spherical homogeneous spaces under G.Suppose that H is a subgroup of P . Consider the G-equivariant mapf: X →Y;f : gH 7→ gP.In general, it is not true that Si (X) is the inverse image under the map f of asubset in Y . However, under the assumption that H contains a regular element ofG, the intersection of Si (X) (when it is nonempty) with a fiber of f has dimensionat least rk(P ) − rk(H).498V.

KIRITCHENKOExample. In what follows, we will mostly deal with the case, where X and Yare spherical homogeneous spaces under the doubled group G × G. Namely, X isa G × G-orbit O of a regular compactification of the group G and Y is a partialflag variety constructed as follows. Let H ⊂ G × G be the stabilizer of a pointin O. Take the minimal parabolic subgroup P ⊂ G × G that contains H and setY = (G × G)/P . It easily follows from an explicit description of the stabilizer H(see [14], Theorem 8) that H does contain a regular element of G × G.Lemma 3.1. For a generic Si (X), there exists an open dense subset of Si (X) suchthat for any element x of this subset the intersection of the fiber xP with Si (X) hasdimension greater than or equal to the rk(P ) − rk(H).

In particular, the dimensionof f (Si (X)) satisfies the inequalitydim f (Si (X)) 6 dim Si (X) − (rk(P ) − rk(H)).Proof. Choose a generic vector space Λ ⊂ g of codimension dim H + i − 1. Denoteby h and p the Lie algebras of H and P respectively. Then by definition Si (X)consists of all cosets gH such that ghg −1 has a nontrivial intersection with Λ, orequivalently h ∩ g −1 Λg is nontrivial.Let gH be any element of Si (X). Estimate the dimension of the intersection ofSi (X) with the fiber gP of the map f .

Because of the assumption on H statedabove, for all g from a dense open subset of Si (X), the intersection h ∩ g −1 Λgcontains an element v that is regular in g. Denote by C the centralizer in P ofv ∈ h ⊂ p. Then dim(C ∩ H) = rk(H) while C has dimension at least rk(P ).Note that for any c ∈ C the coset gcH still belongs to Si (X) since c−1 g −1 Λgccontains c−1 vc = v. Hence, Si (X) ∩ gP contains a set gCH of dimension at leastrk(P ) − rk(H).Lemma 3.1 is crucial for proving the following two vanishing results, which extendProposition 9.1 from [6] and rely on the same ideas. Let X be a regular compactification of G, and let p : X → Xcan be its equivariant projection to the wonderfulcompactification.

Denote by c1 , . . . , cn−k the Chern classes of the Demazure vectorbundle over X.¯ ⊂ X its cloLemma 3.2. Let O be a G×G-orbit in X of codimension s < k, and Ōsure. Suppose that the image p(O) under the map p : X → Xcan coincides with theclosed orbit of Xcan . In terms of polytopes, this means that the face correspondingto O does not intersect the walls of the Weyl chambers.Let λ be any weight of G, and D(λ) the corresponding weight divisor.

Then the¯¯ i. e. the following intersection index ishomology class ci Dn−i−s (λ) vanishes on O,zero:¯¯ = 0.ci D(λ)n−i−s O¯ is the i-th Chern class of theProof. First of all, the intersection product ci · Ō¯¯Demazure bundle over O (see [1], Proposition 2.4.2). Hence, it can be realized as¯¯ of the i-th Chern class Si (O) of the spherical homogeneous spacethe closure in On−i−s¯O. The computation of the intersection index ci ŌD(λ)in X thus reducesn−i−s¯¯ The latter isto the computation of the intersection index Si (O)D(λ)in O.ON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS499equal to the intersection index Si (O)D(λ)n−i−s in the ring of conditions of O since¯ \ O.D(λ) and Si (O) have proper intersections with the boundary Ōn−i−sTo compute Si (O)D(λ)we use the restriction of the map p : X → Xcan¯ By the hypothesis the image p(Ō)¯ is the closed orbit F in Xcan , so it isto Ō.isomorphic to the product G/B × G/B of two flag varieties.

Then the divisor D(λ)¯ is the inverse image under the map p of the divisor D(λ, λ) in F .restricted to ŌeeIndeed, D(λ) = p−1 (D(λ)),where D(λ)is the weight divisor in Xcan correspondingeto λ. It is easy to check that D(λ)∩ F = D(λ, λ) (see Proposition 8.1 in [6]).Hence, all the intersection points in Si (O)D(λ)n−i−s are contained in the preimage of p(Si (O))D(λ, λ)n−i−s . But the latter is empty. Indeed, since O has positiverank and F has zero rank, Lemma 3.1 implies thatdim p(Si (O)) < dim Si (O) = n − i − s.It remains to deal with the orbits in X whose image under the map p is notthe closed orbit in Xcan .

In this case, the face corresponding to such an orbitintersects the walls of the Weyl chambers, and hence, it is orthogonal to some of thefundamental weights ω1 , . . . , ωk . Note that the codimension one orbits O1 , . . . , Okin Xcan are in one-to-one correspondence with the fundamental weights ω1 , . . . , ωk .¯ 1, . . . , O¯¯k be theNamely, the facet corresponding to Oi is orthogonal to ωi .

Let Ōclosures in Xcan of O1 , . . . , Ok , respectively.Lemma 3.3. Let O be a G × G-orbit in X of codimension s < k. Suppose that theimage p(O) under the map p : X → Xcan is not closed and lies in the intersection¯i ∩···∩O¯¯i . In terms of polytopes, this means that the face corresponding to OŌ1sis orthogonal to the weights ωi1 , . . . , ωis .Let λ be any linear combination of the weights ωi1 , .

. . , ωis . Then¯¯ = 0.ci D(λ)n−i−s O¯ i ∩ · · · ∩ Ō¯ i to a partial flagProof. We use the G × G-equivariant map r from Ō1svariety G/P × G/P constructed in [6] (see [6], Lemma 5.1 for details). Considerthe compactification Xi1 ,...,is of G corresponding to the irreducible representationπi1 ,...,is whose highest weight lies strictly inside the cone spanned by ωi1 , . .

. , ωis .This compactification has a unique closed orbit G/P × G/P , where P ⊂ G isthe stabilizer of the highest weight vector in the representation πi1 ,...,is . Clearly,the fan of the Weyl chambers and their faces subdivides the normal fan of theweight polytope of πi1 ,...,is . Hence, by Theorem 2.1 there is an equivariant map¯¯i ∩ · · · ∩ Ō¯ i to the closed orbit G/P ×r : Xcan → Xi1 ,...,is . This map takes O1sG/P .The composition rp maps the orbit O to the closed orbit G/P × G/P of Xi1 ,...,is .It is easy to show that the divisor D(λ) restricted to O is the preimage of the divisor D(λ, λ) ⊂ G/P × G/P under this map (see [6] Section 8.1).

Now repeat thearguments of the proof of Lemma 3.2.These two lemmas imply the following vanishing result.500V. KIRITCHENKOCorollary 3.4. Let O be any G × G-orbit in X of codimension s < k, and let Fbe the face of the polytope of X that corresponds to O. The intersection index¯ci D(λ1 ) . . . D(λn−i−s )Ōvanishes in the cohomology ring of X in the following two cases:(1) The face F does not intersect the walls of the Weyl chambers.

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