Диссертация (Геометрия сферических многообразий и многогранники Ньютона-Окунькова), страница 12

— Let C be a curve obtained as the transverse intersection of a generic collection of n − 1 hyperplane sections correspondingTOME 56 (2006), FASCICULE 41252Valentina KIRITCHENKOto the representation π. Thenχ(C) = Dpol (P2ρ , Pπ , . . . , Pπ ) − (n − 1)D(Pπ )A similar answer can be obtained for the genus of C since it is equal to thegenus of the compactified curve C ⊂ Xπ .

Hence, g(C) = g(C) = 1−χ(C)/2.To compute the Euler characteristic of C we need to sum up χ(C) and thenumber of points in C r C. The latter is the intersection index of Hπn−1with the codimension one orbits in Xπ and can be again computed by theBrion-Kazarnovskii formula. Choose l facets F1 , . . . , Fl of Pπ so that theyparameterize the codimension one orbits in Xπ . This means that each orbitof the Weyl group acting on the facets of Pπ contains exactly one Fi (seeTheorem 2.1).Corollary 4.10. — The genus g(C) of C is given by the followingformula:g(C) = 1 −ZlX1χ(C) + (n − 1)!2i=1Y (x, α)2 dx(ρ, α)2+Fi ∩D α∈RThe measure dx on a facet Fi is normalized as follows.

Let H ⊂ L ⊗ R bethe hyperplane containing Fi . Then the covolume of the sublattice L ∩ Hin H is equal to 1.In the above answer, one can rewrite the polarization Dpol (P2ρ , Pπ , . . . ,Pπ ) in terms of the integrals over the facets of Pπ . E.g. in the case when π isthe irreducible representation with a strictly dominant highest weight λ, thePkanswer takes the following form. Let 2ρ = i=1 ai αi be the decompositionof 2ρ in the basis of simple roots α1 , . .

. , αk . XZk1[aiχ(C) = n!n i=1ZY (x, α)2dx]−(n−1)(ρ, α)2+Fi ∩D α∈RY (x, α)2 dx .(ρ, α)2+Pπ ∩D α∈RZkn! 1 Xg(C) = 1 −[(ai + 1)2 n i=1Y (x, α)2dx](ρ, α)2+Fi ∩D α∈RZ− (n − 1)Y (x, α)2 dx(ρ, α)2+Pπ ∩D α∈RANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS12535. The case of regular spherical varietiesThe results of this paper concerning the Chern classes of the tangentbundle can be generalized straightforwardly to the case of arbitrary spherical homogeneous space. In this section, I briefly outline how this can bedone.Let G be a connected complex reductive group of dimension r, and let Hbe a closed algebraic subgroup of G .

Suppose that the homogeneous spaceG/H is spherical, i.e., the action of G on the homogeneous space G/Hby left multiplication is spherical. In the preceding sections, we considereda particular case of such homogeneous spaces, namely, the space (G ×G)/G ' G.The definition of the Chern classes Si of the tangent bundle T(G/H) canbe repeated verbatim for G/H.

Denote the dimension of G/H by n. Thereis a space of vector fields on G/H coming from the action of G. Take n arbitrary vector fields v1 , . . . , vn of this type. Define the subvariety Si ⊂ G/Has the set of all points x ∈ G/H such that the vectors v1 (x), . . . , vn−i+1 (x)are linearly dependent.Denote by h ⊂ g the Lie algebra of the subgroup H. Again, there is theDemazure map p : G/H → G(r − n, g), which takes g ∈ G/H to the Liesubalgebra ghg −1 .

Denote by Xcan the closure of p(X) in the GrassmannianG(r − n, g). This is a compactification of a spherical homogeneous spaceG/N (h), where N (h) ⊂ G is the normalizer of h. Brion conjectured thatif H coincides with N (H), then the compactification Xcan is smooth, andhence, regular [5]. F. Knop proved that under the same assumption thenormalization of Xcan is smooth [22]. The conjecture has been proved forsemisimple Lie algebras of type A by D.

Luna [23], and in type D byP. Bravi and G. Pezzini [3]. In the general case, one can still define theDemazure bundle over Xcan as the restriction of the tautological quotientvector bundle over G(r − n, g).Since we have not used the regularity of Xcan in the proof of Lemma 3.3the same arguments imply two facts. First, for a generic choice of vectorfields v1 , . . . , vn , the resulting subvariety Si belongs to a fixed class [Si ] inthe ring of conditions.

Second, for any compactification X of G/H lyingover Xcan the closure of a generic Si in X intersects properly any orbit ofX. Repeating the proof of Lemma 3.8 one can also show that Si is emptyunless i 6 n − k. Here k is the difference between the ranks of G andof H. Therefore, we have n − k well-defined classes [S1 ], . . . , [Sn−k ] in thering of conditions C ∗ (G/H). Recently, M.

Brion and I. Kausz proved thatTOME 56 (2006), FASCICULE 41254Valentina KIRITCHENKOthe G-equivariant Chern classes of the Demazure bundle also vanish fori > n − k [7].To extend Theorem 1.1 to an arbitrary spherical homogeneous space onecan use the same description of the Chern classes of its regular compactifications.

The definition of regular compactifications repeats Definition 4.2.Theorem 5.1. — Let X be a regular compactification of G/H. Thenthe total Chern class of X equals to(1 + S 1 + · · · + S n−k )lY(1 + Oi ).i=1This description also follows from Subsection 4.1. The proof uses themethods mentioned in Subsection 4.2.

In fact, regular compactifications ofspherical homogenous spaces arise naturally when one try to apply thesemethods to a wider class of varieties with a group action. Namely, supposethat a connected complex affine group G of dimension r acts on a compactsmooth irreducible complex variety X with a finite number of orbits.

Thenthere is a unique open orbit in X isomorphic to G/H for some subgroupH ⊂ G, so X can be regarded as a compactification of G/H. Denote byO1 , . . . , Ol the orbits of codimension one in X. Then one can describe thetangent bundle of X exactly by the methods mentioned in Subsection 4.2if the following conditions hold. First, the hypersurfaces O1 , . . .

, Ol aresmooth and intersect each other transversally (this allows to apply Ehlers’method to the divisor X r (G/H) = O1 ∪ · · · ∪ Ol ). Second, the vectorbundle VX (defined as in Subsection 4.2) is generated by its global sectionsv1 , . . . , vr , where v1 , . . . , vr are infinitesimal generators of the action of G onX (this allows to give a uniform description of VX for all compactificationsof G/H satisfying these conditions). It is not hard to check that these twoconditions are equivalent to the definition of regular compactifications.It turns out that a homogeneous space G/H admits a regular compactification if and only if G/H is spherical [1].

Regular compactifications ofarbitrary spherical homogeneous spaces are exactly their smooth toroidalcompactifications [1]. A compactification X of the spherical homogeneousspace G/H is called toroidal if for any codimension one orbit of a Borelsubgroup of G acting on G/H, its closure in X does not contain any G-orbitin X.The proof of Theorem 1.1 goes without any change for complete intere1, . . . , Hemsections in arbitrary spherical homogeneous space G/H. Let HANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1255be smooth hypersurfaces in some regular compactification of G/H.

Supe i with the closures of G-orbits arepose that all possible intersections of Htransverse.ei ∩Theorem 5.2. — Let H1 , . . . , Hm ⊂ G/H be the hypersurfaces H(G/H), and let C = H1 ∩ · · · ∩ Hm be their intersection. Then the Eulercharacteristic of C equals to the term of degree n in the decomposition of(1 + S1 + · · · + Sn−k )mYHi (1 + Hi )−1 .i=1The products are taken in the ring of conditions C ∗ (G/H).For instance, if G/H is compact, then the Si become the usual Chernclasses and the above formula coincides with the classical adjunction formula. However, if G/H is noncompact then the Chern classes in the usualsense (as degeneracy loci of generic vector fields on G/H) do not usuallyyield the adjunction formula (although they do for G = (C∗ )n ).

Indeed,when the homogeneous space is a noncommutative reductive group, allusual Chern classes are trivial but as we have seen χ(H) 6= (−1)n H n evenfor one smooth hypersurface H. Theorem 5.2 shows that the adjunctionformula still holds for noncompact spherical homogeneous spaces, if onereplaces the usual Chern classes with the refined Chern classes Si that aredefined as the degeneracy loci of the vector fields coming from the actionof G.BIBLIOGRAPHY[1] F. Bien & M. Brion, “Automorphisms and local rigidity of regular varieties”,Compositio Math. 104 (1996), no. 1, p. 1-26.[2] E.

Bifet, C. de Concini & C. Procesi, “Cohomology of regular embeddings”,Adv. in Math. 82 (1990), no. 1, p. 1-34.[3] P. Bravi & G. Pezzini, “Wonderful varieties of type D”, arXiv.org/math.AG/0410472.[4] M. Brion, “Groupe de Picard et nombres caractéristiques des variétés sphériques”,Duke Math J. 58 (1989), no. 2, p. 397-424.[5] ——— , “Vers une généralisation des espaces symétriques”, J. Algebra 134 (1990),no. 1, p. 115-143.[6] ——— , “The behaviour at infinity of the Bruhat decomposition”, Comment.