Диссертация (1136188), страница 14
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Namely, take any regular compactification X of G that lies over the compactification Xπ corresponding to therepresentation π (see Section 2.2). Then reduce the computation of Hπn to thecomputation of the intersection indices of divisors in the closed orbits of X (seeSection 4). All closed orbits are isomorphic to the product of two flag varieties.The precise algorithm for doing this was given by De Concini and Procesi [6] in theON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS491case, where X is a wonderful compactification of a symmetric space.
Then E. Bifetextended this algorithm to all regular compactifications of symmetric spaces [2].I will show that in the case, where a symmetric space is a reductive group, thisalgorithm actually produces the Brion–Kazarnovskii formula if one uses the weightpolytope of π to keep track of all transformations.Moreover, the De Concini–Procesi algorithm works not only for divisors. It canalso be carried over to the Chern classes of G (which are, in general, not linear combinations of complete intersections). In particular, there is the following explicitformula for the intersection indices of the Chern classes of G with hyperplane sections.
Assign to each lattice point (λ1 , λ2 ) ∈ LT ⊕ LT the intersection index of thei-th Chern class of the tangent bundle over G/B × G/B with the divisor D(λ1 , λ2 )corresponding to (λ1 , λ2 ), that is, the number ci (G/B × G/B)Dn−k−i (λ1 , λ2 ). Extend this function to the polynomial function on (LT ⊕ LT ) ⊗ R. Since the Chernclasses of G/B are known the resulting function can be easily computed (see Section 4). The final formula is as follows.Let D be the differential operator (on functions on (LT ⊕ LT ) ⊗ R) given by theYformulaD=(1 + ∂α )(1 + ∂eα ),α∈R+where ∂α and ∂eα are directional derivatives along the vectors (α, 0) and (0, α),respectively. Denote by [D]i the i-th degree term in D.Theorem 1.2.
If Hπ is a generic hyperplane section corresponding to a representation π with the weight polytope Pπ ⊂ LT ⊗ R, then the intersection index Si Hπn−iof the i-th Chern class of G with Hπn−i is equal toZ(n − i)![D]i F (x, x)dx.Pπ ∩DThe measure dx on LT ⊗ R is normalized so that the covolume of LT is 1.Of course, this formula also allows one to compute the intersection index Si ×Hπ1 . . . Hπn−i for any n − i generic hyperplane sections corresponding to differentrepresentations π1 , . .
. , πn−i .Since, in general, the Chern classes of G are not complete intersections, thisformula extends computation of the intersection indices to a bigger part of the ringof conditions of G. Theorem 1.2 also completes some results of [11]. Namely, theChern classes S1 , . .
. , Sn−k were used there in the following adjunction formula forthe topological Euler characteristic of complete intersections of hyperplane sectionsin G.Theorem 1.3 [11]. Let H1 , . . . , Hm be generic hyperplane sections correspondingto m (possibly different) representations of G. The Euler characteristic of thecomplete intersection H1 ∩· · ·∩Hm is equal to the term of degree n in the expansionof the following product:mY(1 + S1 + · · · + Sn−k ) ·Hi (1 + Hi )−1 .i=1The product in this formula is the intersection product in the ring of conditions.492V. KIRITCHENKOTheorem 1.2 in the present paper allows to make this formula explicit, since itkmallows to compute all terms of the form Si H1k1 .
. . Hm, where k1 + . . . + km = n − i.For example, if a complete intersection is just one hyperplane section Hπ , thenZχ(Hπ ) = (−1)n−1(n! − (n − 1)![D]1 + (n − 2)![D]2 − . . . + k![D]n−k )F (x, x) dx.Pπ ∩DThere is also a formula for the Chern classes ci (X) of the tangent bundle overany regular compactification X of G in terms of S1 , . . . , Sn−k (see Corollary 4.4in [11]). Theorem 1.2 allows to compute explicitly the intersection index of ci (X)with a complete intersection of complementary dimension in X.I am grateful to M. Brion, K.
Kaveh, B. Kazarnovskii and A. Khovanskii foruseful discussions. I would also like to thank the referees for valuable remarks.2. PreliminariesIn this section, I recall some well-known facts which are used in the proof ofTheorem 1.2. In Section 2.2, I define the regular compactification X of G associatedwith a representation π and describe the orbit structure of X in terms of the weightpolytope of the representation. In Section 2.3, the Picard group of X is related tothe space of virtual polytopes analogous to the weight polytope of π.
The notion ofanalogous polytopes is discussed in Section 2.1. In Section 2.4, I recall the definitionof the ring of conditions of G. Section 2.5 contains a formula for the integral of apolynomial function over a simplex, which is used to interpret the computation ofintersection indices in terms of integrals over the weight polytope.2.1. Polytopes.
Let P ⊂ Rk be a convex polytope. Define the normal fan P ∗ ofP . This is a fan in the dual space (Rk )∗ . To each face F i ⊂ P of dimension i therecorresponds a cone Fi∗ of dimension (n − i) in P ∗ defined as follows. The cone Fi∗consists of all linear functionals in (Rk )∗ whose maximum value on P is attainedon the interior of the face F i . In particular, to each facet of P there correspondsa one-dimensional cone, i.
e. a ray, in P ∗ . If the dual space (Rk )∗ is identified withRk by means of the Euclidean inner product, the ray corresponding to a facet isspanned by a normal vector to the facet.Two convex polytopes are called analogous if they have the same normal fan.All polytopes analogous to a given polytope P form a semigroup SP with respectto Minkowski sum. This semigroup is also endowed with the action of the multiplicative group R>0 (polytopes can be dilated). Hence, SP can be regarded as acone in the vector space VP , where VP is the minimal group containing SP (i. e.the Grothendieck group of SP ).
The elements of VP are called virtual polytopesanalogous to P .We now introduce special coordinates in the vector space VP in the case whereP is simple. A polytope in Rk is called simple if it is generic with respect toparallel translations of its facets. Namely, exactly k facets must meet at eachvertex. This implies that any other face is also the transverse intersection of thosefacets that contain it.
Let Γ1 , . . . , Γl be the facets of P , and let Γ∗1 , . . . , Γ∗l be thecorresponding rays in P ∗ . Choose a non-zero functional hi ∈ Γ∗i in each ray. Call hiON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS493a support function corresponding to the facet Γi . For any polytope Q analogous toP , denote by hi (Q) the maximal value of hi on the polytope Q. For instance, if hiis normalized so that its value on the external unit normal to the facet Γi is 1, thenhi (P ) is up to a sign the distance from the origin to the hyperplane that containsthe facet Γi (the sign is positive if the origin and the polytope P are to the sameside of this hyperplane, and negative otherwise).
The numbers h1 (Q), . . . , hl (Q)are called the support numbers of Q. Clearly, the polytope Q is uniquely definedby its support numbers. The coordinates h1 (Q), . . . , hl (Q) can be extended to thespace VP , providing the isomorphism between VP and the coordinate space Rl .In what follows, we will deal with integer polytopes, i. e. polytopes whose verticesbelong to a given lattice Zk ⊂ Rk .
For such polytopes, the natural way to normalizethe support functions is to require that hi (P ) be equal to the integral distancefrom the origin to the hyperplane that contains the facet Γi . Suppose that ahyperplane H not passing through the origin is spanned by lattice vectors. Thenthe integral distance from the origin to the hyperplane H is the index in Zk ofthe subgroup spanned by H ∩ Zk .
To compute the integral distance one can applya unimodular (with respect to the lattice Zk ) linear transformation of Rk so thatH becomes parallel to a coordinate hyperplane. Then the integral distance is theusual Euclidean distance from the origin to this coordinate hyperplane.2.2. Regular compactifications of reductive groups. With any representation π : G → GL(V ) one can associate the following compactification of π(G). Takethe projectivization P(π(G)) of π(G) (i. e.
the set of all lines in End(V ) passingthrough a point of π(G) and the origin), and then take its closure in P(End(V )).We obtain a projective variety Xπ ⊂ P(End(V )) with a natural action of G × Gcoming from the left and right action of π(G) × π(G) on End(V ). For example,when G = (C∗ )n is a complex torus, all projective toric varieties can be constructedin this way.Assume that P(π(G)) is isomorphic to G. Consider all weights of the representation π, i. e.
all characters of the maximal torus T occurring in π. Take their convexhull Pπ in LT ⊗R. Then it is easy to see that Pπ is a polytope invariant under the action of the Weyl group of G. It is called the weight polytope of the representation π.The polytope Pπ contains information about the compactification Xπ .Theorem 2.1. (1) ([14], Proposition 8) The subvariety Xπ consists of a finitenumber of G × G-orbits.
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