Диссертация (1136188), страница 46

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 46)

. . , αl } of simple roots of G, there corresponds a smooth Bott–Samelson variety RI endowed with an action of B suchthat there is a B-equivariant map RI → G/B. In particular, each RI gives rise tothe cobordism class ZI = [RI → G/B] as well as to the T -equivariant cobordismclass [ZI ]T . The latter can be expressed as follows.Theorem 5.4.[ZI ]T = ∂αTl . . . ∂αT1 [pt]T .The key ingredient is the following geometric interpretation of ∂αT . Denote byPα the minimal parabolic subgroup corresponding to the root α.EQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES407Lemma 5.5. The operator ∂αT is the composition of the change of group homomorphism rTPα : Ω∗Pα (G/B) → Ω∗T (G/B) and the push-forward map rPTα : Ω∗T (G/B) →Ω∗Pα (G/B):∂α = rTPα rPTα .Note here that rPTα is defined by taking the limit over the push-forward maps onthe non-equivariant cobordism groups corresponding to the projective morphismBPαG/B × Uj → G/B × Uj (for a sequence of good pairs {(Vj , Uj )} for the actionof Pα ).

Similarly to [18, Cor. 2.3], this lemma follows from the Vishik–QuillenBPαformula [18, Prop. 2.1] applied to P1 -bundles G/B × Uj → G/B × Uj . Theorem5.4 then can be deduced from Lemma 5.5 by the same arguments as in [18, Thm.3.2].6. Cobordism ring of wonderful symmetric varietiesThe wonderful (or more generally, regular) compactifications of symmetric varieties form a large class of spherical varieties. In fact, much of the study of a verylarge class of spherical varieties can be reduced to the case of symmetric varieties(cf.

[35]). In this section, we compute the rational equivariant cobordism ring ofwonderful symmetric varieties of minimal rank (see Theorem 6.4). A presentationfor the equivariant cohomology of the wonderful group compactification analogousto Theorem 6.4 below was obtained by Littelmann and Procesi in [30], and thecorresponding result for the equivariant Chow ring was obtained by Brion in [6,Thm. 3.1]. This result of Brion was later generalized to the case of wonderfulsymmetric varieties of minimal rank by Brion and Joshua in [7, Thm. 2.2.1].Our proof of Theorem 6.4 follows the strategy of [7]. The two new ingredientsin our case are the localization theorem for torus action in cobordism (cf.

[23,Thm. 7.8]), and a divisibility result (Lemma 6.3) in the ring S = Ω∗T (k).6.1. Symmetric varietiesWe now define symmetric varieties and describe their basic structural propertiesfollowing [7]. Let k be a field of characteristic zero and let G be a connected andsplit reductive group over k. We assume throughout this section that G is ofadjoint type. Let θ be an involutive automorphism of G and let K ⊂ G be thesubgroup of fixed points Gθ . The homogeneous space G/K is called a symmetricspace.Let P be a minimal θ-split parabolic subgroup of G (a parabolic subgroup Pis θ-split if θ(P ) is opposite to P ), and L = P ∩ θ(P ) a θ-stable Levi subgroup ofP .

Then every maximal torus of L is also θ-stable. We assume that T is such atorus, so that T = T θ T −θ and the identity component A = T −θ,0 is a maximalθ-split subtorus of G (a torus is θ-split if θ acts on it via the inverse map g 7→ g −1 ).The rank of such a torus A is called the rank of the symmetric space G/K. SinceT θ ∩ T −θ is finite, we getrk(G) ≤ rk(K) + rk(G/K).(6.1)One says that the symmetric space G/K is of minimal rank if equality occurs in(6.1).

This is equivalent to saying that T θ,0 is a maximal torus of K 0 and T −θ,0408VALENTINA KIRITCHENKO, AMALENDU KRISHNAis a maximal θ-split torus (here K 0 denotes the identity component of K). Notethat K 0 is reductive. Set TK = (T ∩ K)0 . For symmetric spaces of minimal rank,the roots of (K 0 , TK ) are exactly the restrictions to TK of the roots of (G, T ) ([7,Lemma 1.4.1]). Moreover, the Weyl group of (K 0 , TK ) is identified with W θ .Let Σ+ denote the set of positive roots of G with respect to a Borel subgroupB containing T . Let ∆G = {α1 , .

. . , αn } be the set of positive simple roots whichform a basis of the root system, and {sα1 , . . . , sαn } the set of associated reflections.Since G is adjoint, ∆G is also a basis of the character group Tb. Recall that W = WGdenotes the Weyl group of G. Let ΣL ⊂ Σ be the set of roots of L, and ∆L ⊂ ∆Gthe subset of simple roots of L.b is aIf G/K is of minimal rank then the image of the restriction map p : Tb → Areduced root system (denoted by ΣG/K ), and ∆G/K := p (∆G \ ∆L ) is a basis ofΣG/K ([7, Lemma 1.4.3]). This set is also identified with {α − θ(α)|α ∈ ∆G \ ∆L }under the projection p. Moreover, in the exact sequencep1 → WL → W θ −→ WG/K → 1,(6.2)a representative of the reflection of WG/K associated to the root α − θ(α) ∈ ∆G/Kis sα sθ(α) .Definition 6.1.

A smooth projective G-variety X over k will be called a wonderfulsymmetric variety, if there is a symmetric space G/K such that the following hold.(1) There is a dense open orbit of G in X isomorphic to G/K.(2) The complement to this open orbit is the union of r = rk(G/K) smoothprime divisors {X1 , · · · , Xr } with strict normal crossings.(3) The G-orbit closures in X are precisely the various intersections of the aboveprime divisors. In particular, all G-orbit closures are smooth.A wonderful symmetric variety X as above is said to be of minimal rank, if sois the dense open orbit G/K. It is known from the work of De Concini–Procesi[9] that every symmetric space G/K has a unique G-equivariant compactification(called wonderful compactification) that is a wonderful symmetric variety.Possibly the simplest example of symmetric varieties of minimal rank is whenG = G × G where G is a semisimple group of adjoint type, and θ interchanges thefactors.

In this case, we have K = diag(G) and G/K ∼= G, where G acts by leftand right multiplications. Furthermore, T = T × T where T is a maximal torusof G. Thus, TK = diag(T), A = {(x, x−1 )|x ∈ T}, L = T , and WK = WG/K =diag(WG ) ⊂ WG × WG = W . In this case, the variety X is called the wonderfulgroup compactification. For instance, complete collineations arise this way for G =PGLn . We refer to [7, Example 1.4.4] for an exhaustive list of symmetric spacesof minimal rank. A well-known example of a wonderful symmetric variety that isnot of minimal rank is the space of complete conics.Let X be a wonderful symmetric variety of minimal rank with a dense openorbit G/K. Let Y ⊂ X denote the closure of T /TK in X. It is known that Y issmooth and is the toric variety associated to the fan of the Weyl chambers andtheir faces of the root datum (G/K, ΣG/K ).

Recall here that if M (G) is the rootlattice of (G/K, ΣG/K ) with the dual lattice N (G), then every set S of simpleEQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES409roots determines a cone σS = {v ∈ N (G)Q |hu, vi ≥ 0 ∀u ∈ S}. This cone is calledthe Weyl chamber corresponding to S.Let z denote the unique T -fixed point of the affine T -stable open subset Y0 ofY given by the positive Weyl chamber of ΣG/K . It is well known that X has anisolated set of fixed points for the T -action. Moreover, it is also known by [37, §10]that X contains only finitely many T -stable curves.

We shall need the followingdescription of the fixed points and T -stable curves.Lemma 6.2 ([7, Lemma 2.1.1]).(i) The T -stable points in X (resp. Y ) are exactly the points w · z, where w ∈W (resp. WK ) and these fixed points are parameterized by W/WL (resp.WG/K ).(ii) For any α ∈ Σ+ \ Σ+L , there exists a unique irreducible T -stable curve C z,αwhich contains z and on which T acts through the character α. The T -fixedpoints in Cz,α are z and sα · z.(iii) For any γ = α − θ(α) ∈ ∆G/K , there exists a unique irreducible T -stablecurve Cz,γ which contains z and on which T acts through its character γ.The T -fixed points in Cz,γ are exactly z and sα sθ(α) · z.(iv) The irreducible T -stable curves in X are the W -translates of the curves C z,αand Cz,γ .

They are all isomorphic to P1 .(v) The irreducible T -stable curves in Y are the WG/K -translates of the curvesCz,γ .6.2. Cobordism ring of wonderful symmetric varietiesTo prove our main result, we will also need the following result on divisibility inthe ring S = S(T ). We use notations of Subsection 5.2.Lemma 6.3. For any f ∈ S and any root α, we havef ≡ sα (f ) (mod xα ).(6.3)Proof. It is enough to check this lemma for all monomials in xχ for χ ∈ Tb.First, we check the case f = xχ . We have sα χ = χ − (χ, α)α, where (χ, α) is aninteger.

Характеристики

Список файлов диссертации