Диссертация (1136188), страница 17
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Then weightsλ1 , . . . , λn−i−s are any weights of G.(2) The face F intersects a wall of the Weyl chambers and weights λ1 , . . . , λn−i−sare orthogonal to F (with respect to the inner product (· , ·) on LT ⊗ R defined inthe Introduction).4.
Proof of Theorem 1.2We use notation of Sections 2.2 and 2.3. Let X be any regular compactifica¯¯π of Hπ in X hastion lying over the compactification Xπ . Then the closure Hproper intersections with all G × G-orbits in X, and thus Si Hπn−i coincides with¯¯πn−i in the cohomology ring of X.the intersection index S¯¯i HAssume that X corresponds to a representation of G with the weight polytopeP0 . Let us compute S¯¯i Dn−i for a divisor D under the assumption that the polytope P corresponding to D is analogous to P0 . After we establish the formula ofTheorem 1.2 for such divisors, it will automatically extend to the other divisors¯¯π ) since any virtual polytope analogous to P0 is a linear com(in particular, for Hbination of polytopes analogous to P0 . Since X is regular, P0 and hence P aresimple.All computations are carried in the cohomology ring of X.
First, break Dn−i¯ i . . . Ō¯ i D(λ1 ) . . . D(λn−i−k ), where i1 , . . . , ik areinto monomials of the form Ō1kdistinct integers from 1 to l and λ1 , . . . , λn−i−k are weights. Then every such¯¯i ∩ · · · ∩ Ō¯ i is eithermonomial can be computed explicitly, since the intersection O1kempty or isomorphic to the product of two flag varieties.Since we are going to intersect Dn−i with S¯¯i we can ignore all monomials thatare annihilated by S¯¯i . Recall that S¯¯i is the i-th Chern class of the Demazure bundleover X. In particular, Corollary 3.4 implies that S¯¯i annihilates the ideal I ⊂ H ∗ (X)¯ such that either thegenerated by the monomials of the form D(λ1 ) .
. . D(λn−i−s )Ōface of P corresponding to the codimension s < k orbit O does not intersect thewalls of the Weyl chambers or, if it does, the weights λ1 , . . . , λn−i−s are orthogonalto this face.To keep track of our calculations we use a subdivision of the polytope P ∩ D intosimplices coming from the barycentric subdivision of P described below. For eachface F ⊂ P choose a point λF ∈ F as follows.
If F does not intersect the walls of theWeyl chamber D, then λF is any point in the interior of the face. Otherwise, chooseλF so that the corresponding vector is orthogonal to the face F (in particular, λFwill belong to the intersection of the face with a wall of D.) If F = P take λF = 0.An s-flag F is the collection {F1 ⊃ · · · ⊃ Fs } of s 6 k nested faces of P such¯¯Fthat each of them intersects D, and Fi has codimension i in P . Denote by Othe closure in X of the orbit corresponding to the last face Fs , and by ∆F thes-dimensional simplex with the vertices 0, λF1 , .
. . , λFs . In particular, when s = k,ON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS501the simplex ∆F has full dimension and the orbit OF is closed. The polytope D ∩ Pis the union of simplices ∆F over all possible k-flags F.Example. Take G = PSL3 (C), and let X = Xcan be its wonderful compactification. Let divisor D be a hyperplane section corresponding to the irreduciblerepresentation with a strictly dominant highest weight λ. In this case, P is ahexagon symmetric under the action of the Weyl group, with two edges Γ1 and Γ2intersecting D.
Then λi = λΓi ∈ Γi is orthogonal to Γi for i = 1, 2 and Γ1 ∩ Γ2 = λ.The subdivision of P ∩ D into simplices consists of two triangles ∆1 and ∆2 withthe vertices 0, λ1 , λ and 0, λ2 , λ, respectively (see the figure).Γ1λ10D∆1λ∆2PΓ2λ2Lemma 4.1. Denote by fd (x1 , . . . , xk ) the sum of all monomials of degree d ink variables x1 , . . . , xk . The following identity holds in the cohomology ring of Xmodulo the ideal I:XDn−i ≡ k!Vol(∆F )fn−k−i (D, D(λF1 ), . . . , D(λFk−1 ))OF (mod I),Fwhere the sum is taken over all possible k-flags F = {F1 ⊃ · · · ⊃ Fk }. The volumeform Vol is normalized so that the covolume of LT is equal to 1.Proof.
We will prove the following more general statement for s-flags. Denote byfd,s (x1 , . . . , xs ) the sum of all monomials of degree d in s variables.Recall that Γ1 , . . . , Γl denote the facets of P that intersect the Weyl chamberD. An s-flag can be alternatively described by an ordered collection of facetsΓi1 , .
. . , Γis such that their intersection Γi1 ∩ · · · ∩ Γis has codimension s. ThenFj = Γi1 ∩ · · · ∩ Γij . This is a one-to-one correspondence, since the polytope P issimple. Assign to each s-flag F the following numbercF = hi1 (P )[hi2 (P ) − hi2 (λF1 )] . . . [his (P ) − his (λFs−1 )].In particular, when s = k, i. e.
Fs is just a vertex, the number cF coincideswith the volume of ∆F times k!. Indeed, by a unimodular linear transformationof LT ⊗ R we can map the hyperplanes containing the facets Γi1 , . . . , Γis to thecoordinate hyperplanes. Then [hij (P ) − hij (λFj−1 )] is just the Euclidean distancefrom the vertex λFj−1 of ∆F to the hyperplane containing Γij .
Note that to define502V. KIRITCHENKOvolumes we do not use the inner product (· , ·) on the lattice LT . We only use thelattice itself.Then for any integer s such that 1 6 s 6 k the following is true:X¯ F (mod I),Dn−i ≡cF fn−s−i,s (D, D(λF1 ), . . . , D(λFs−1 ))Ō(1)Fwhere the sum is taken over all s-flags.Example. If G is a complex torus, formula (1) is still meaningful but looks muchsimpler and reduces toX¯¯F .Ds =cF OFProve formula (1) by induction on s. We use the notations of Section 2.3.
For¯ 1 +· · ·+hl (P )Ō¯ls = 1, the statement coincides with the decomposition D = h1 (P )Ōfrom Lemma 2.3.Assume that the formula is proved for some s < k. Prove it for s + 1. Wenow deal separately with each term on the right hand side of formula (1). First¯ F ∈ I from everysubtract the element fn−s−i,s (D(λFs ), D(λF1 ), . . . , D(λFs−1 ))Ō¯ F . This operation does not change theterm fn−s−i,s (D, D(λF1 ), .
. . , D(λFs−1 ))Ōidentity (1). A simple calculation shows thatfn−s−i,s (x, x1 , . . . , xs−1 ) − fn−s−i,s (xs , x1 , . . . , xs−1 )= (x − xs )fn−s−i−1,s+1 (x, x1 , . . . , xs−1 , xs ).Hence, after subtraction we can rewrite the difference as¯ F.(D − D(λFs ))fn−s−i−1,s+1 (D, D(λF1 ), . . . , D(λFs ))ŌSince λs lies in the intersection of s facets Γi1 , . .
. , Γis , Corollary 2.5 implies thatX¯¯j O¯¯F .¯F =[hj (P ) − hj (λFs )]O(D − D(λFs ))Ōj6=i1 ,...,is¯¯j O¯¯F is empty if and only if the intersection of Γj with Γi ∩ · · · ∩ ΓiNote that O1sis empty. Hence,X¯F =¯¯F ′ ,(D − D(λFs ))Ō[hj (P ) − hj (λFs )]OF′where the sum is taken over all (s + 1)-flags F ′ that extend F, i.
e. F ′ = {F1 ⊃· · · ⊃ Fs ⊃ Fs ∩ Γj }.It remains to compute the termS¯¯i · fn−k−i (D, D(λF1 ), . . . , D(λFk−1 ))OF(2)for each k-flag F. Suppose that the closed orbit OF is the intersection of k hy¯¯i , . . . , O¯¯i . Then for any other codimension 1 orbit Oj (such thatpersurfaces O1k¯ j is empty. Hence, D in (2) can be replacedj 6= i1 , .
. . , ik ), the intersection OF ∩ Ōby D(λFk ) sinceX¯ j.D = D(λFk ) +(hj (P ) − hj (λFk ))Ōj6=i1 ,...,ikON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS503Note also that the evaluation of (2) reduces to the computation of intersectionindices in OF , which is the product of two flag varieties. We have that S¯¯i · OF =ci (OF ) and D(λ) · OF = D(λ, λ). Here ci (OF ) is the i-th Chern class of thetangent bundle of OF , which coincides with the Demazure bundle over OF sinceOF is closed. Hence,S¯¯i fn−k−i (D(λFk ), D(λF1 ), . . .
, D(λFk−1 ))OF == ci (G/B × G/B)fn−k−i (D(λF1 , λF1 ), . . . , D(λFk , λFk )). (3)The intersection product in the right hand side of this formula is taken in G/B ×G/B.The function Fi (λ) = ci (G/B × G/B)D(λ, λ)n−k−i can be expressed explicitlyin terms of the function F defined in the Introduction, since the i-th Chern classof G/B × G/B is the term of degree i in the intersection productY(1 + D(α, 0))(1 + D(0, α)).α∈R+One way to compute Fi is as follows. Let D and [D]i be the differential operatorsdefined in the Introduction. ThenFi (x) = (n − k − i)![D]i F (x, x).This easily follows from the formula for the polarization mentioned in Section 2.5and the fact that Dn−k (λ, λ) = (n − k)!F (λ, λ).We can now apply Proposition 2.6 to convert the sum (3) into the integral overthe simplex ∆F .
Indeed, by definition of the function fn−k−i we have that (3) canbe rewritten asX(Fi )pol (λF1 , . . . , λF1 , . . . , λFk , . . . , λFk ).|{z}{z}|i1 +···+ik =n−k−ii1ikThis is equal to the integralZn−iFi (x)dx/Vol(∆F )k∆Fby Proposition 2.6 applied to the simplex ∆F (with the vertices 0, λF1 , . . . , λFk )and to the function Fi (x). Combining this with Lemma 4.1 we getZZ(n − i)! XS¯¯i Dn−i =Fi (x)dx = (n − i)![D]i F (x, x)dx.(n − k − i)!F ∆FP ∩DNote that when i = 0, we get the Brion–Kazarnovskii formula.5.
ExampleIn this section, I give an example of computation of the Euler characteristic usingTheorems 1.2 and 1.3. Namely, for the group G = SL3 (C), I compute the Eulercharacteristic of a hyperplane section corresponding to an irreducible representation.504V. KIRITCHENKOLet π be the irreducible representation of SL3 (C) with the highest weight λ =mω1 + nω2 , where ω1 , ω2 are the fundamental weights of SL3 (C), and m and nare nonnegative integers. Take a generic hyperplane section Hπ corresponding tothe representation π. We first find Si Hπn−i .
The dimension of SL3 (C) is 8, and therank is 2, so there are 6 nontrivial Chern classes S1 , . . . , S6 .Let us compute all ingredients of the formula of Theorem 1.2. The weight polygon Pπ of π is depicted on the figure above. The domain Pπ ∩ D is the unionof two triangles ∆1 and ∆2 . The positive roots of SL3 (C) are α1 = 2ω1 − ω2 ,α2 = 2ω2 − ω1 and α1 + α2 . Write D and F in the coordinates (s, t; s̃, t̃ ) in(LT ⊕ LT ) ⊗ R associated with the basis {α1 , α2 } in LT ⊗ R:D = (1 + ∂s )(1 + ∂t )(1 + ∂s + ∂t )(1 + ∂s̃ )(1 + ∂t̃ )(1 + ∂s̃ + ∂t̃ ).Since ρ = α1 + α2 and (α1 , ω1 ) = (α2 , ω2 ) = 1, (α1 , ω2 ) = (α2 , ω1 ) = 0 we havethat1F = (2s − t)(2t − s)(t + s)(2s̃ − t̃ )(2t̃ − s̃)(t̃ + s̃).4If we plug D and F in the formula of Theorem 1.2 and integrate, we get thatHπ8 = 3(m8 + 16m7 n + 112m6 n2 + 448m5 n3 + 700m4 n4 + 448m3 n5 + 112m2 n6 +16mn7 + n8 );S1 Hπ7 = 18(m + n)(m6 + 13m5 n + 71m4 n2 + 139m3 n3 + 71m2 n4 + 13mn5 + n6 );S2 Hπ6 = 54(m6 + 12m5 n + 50m4 n2 + 80m3 n3 + 50m2 n4 + 12mn5 + n6 );S3 Hπ5 = 90(m + n)(m4 + 9m3 n + 19m2 n2 + 9mn3 + n4 );S4 Hπ4 = 18(5m4 + 40m3 n + 72m2 n2 + 40mn3 + 5n4 );S5 Hπ3 = 54(m + n)(m2 + 5mn + n2 );S6 Hπ2 = 18(m2 + 4mn + n2 ).We now apply Theorem 1.3 to obtainχ(Hπ ) = −3(m8 + 16m7 n + 112m6 n2 + 448m5 n3 + 700m4 n4 + 448m3 n5 +112m2 n6 +16mn7 +n8 + 18(m6 +12m5 n+50m4 n2 +80m3 n3 +50m2 n4 +12mn5 +n6 )+6(5m4 +40m3 n+72m2 n2 +40mn3 +5n4 )+ 6(m2 +4mn+n2 ) −6(m+n)(m6 +13m5 n+71m4 n2 +139m3 n3 +71m2 n4 +13mn5 +n6 + 5(m4 +9m3 n+19m2 n2 +9mn3 +n4 )+3(m2 + 5mn + n2 ))).References[1] F.
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