Диссертация (1136188), страница 42
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The techniquesof equivariant cobordism have been recently exploited to give explicit descriptionsof the ordinary cobordism rings of smooth toric varieties in [25], and that of theflag bundles over smooth schemes in [24].In this paper, we give an explicit Borel type presentation of the equivariantcobordism ring of a complete flag variety. Such a presentation for the equivariantK-theory is due to Kostant–Kumar ([21, Thm. 4.4]). The analogous results for theequivariant Chow ring (with rational coefficients) and the singular cohomology ringare, respectively, due to Brion ([5, Prop.
2]) and Holm–Sjamaar ([16, Prop. 2.2]).The ordinary cobordism rings of such varieties have been recently described byHornbostel–Kiritchenko [18] and Calmès–Petrov–Zainoulline [8]. Let B ⊂ G be aBorel subgroup containing a split maximal torus T . In Theorem 5.1, we obtain anexplicit presentation for Ω∗T (G/B) tensored with Z[t−1G ], where tG is the torsionindex of G (see Section 4 for a definition).
As a consequence, one immediatelyobtains an expression for the ordinary cobordism rings of complete flag varieties(tensored with Z[t−1G ]) using a simple relation between the equivariant and theordinary cobordism (cf. [23, Thm. 3.4]). We also outline an equivariant Schubertcalculus in Ω∗T (G/B) (see Subsection 5.2).To compute Ω∗T (G/B), we first prove some comparison theorems which relatethe equivariant algebraic and complex cobordism rings of cellular varieties overthe field of complex numbers (see Section 3). In particular, we give a Borel typepresentation for the equivariant complex cobordism M UT∗ (G/B) (see Section 4).The highlight of our proof is that it uses only elementary techniques of equivariantgeometry and does not use any computation of the non-equivariant cobordism orcohomology.
An interesting problem is to find a purely algebraic proof of ourpresentation of Ω∗T (G/B).In Section 6, we describe the rational T -equivariant cobordism rings of wonderfulsymmetric varieties of minimal rank by purely algebraic arguments. Again, thisimplies a description for their ordinary cobordism rings. In particular, one gets apresentation for the cobordism ring of the wonderful compactification of an adjointsemisimple group. The main ingredient of the proof is the localization theoremfor the equivariant cobordism rings for torus action [23, Thm.
7.8]. Once we havethis tool, the final result is obtained by adapting the argument of Brion–Joshua[7], who described the equivariant Chow ring. As it turns out, similar steps can befollowed to compute the equivariant cobordism ring of any regular compactificationof a symmetric space of minimal rank.Acknowledgements.
We are grateful to Michel Brion for useful comments on thefirst version of this paper. We are also grateful to the referees, whose commentsand suggestions helped us to improve this paper.EQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES3932. Recollection of equivariant cobordismIn this section, we recollect the basic definitions and properties of equivariantcobordism that we shall need in the sequel.
For more details, see [22]. Let k be afield of characteristic zero and let G be a linear algebraic group over k.Let Vk denote the category of quasi-projective k-schemes and let VkS denotethe full subcategory of smooth quasi-projective k-schemes. The category of quasiprojective k-schemes with an algebraic G-action and G-equivariant maps is denotedSby VG and the corresponding subcategory of smooth schemes is denoted by VG. Inthis paper, a scheme will always mean an object of Vk and a G-scheme will meanan object of VG .
For all the definitions and properties of algebraic cobordism thatare used in this paper, we refer the reader to [29]. All representations of G willbe finite-dimensional. Let L denote the Lazard ring (which is the same as thecobordism ring Ω∗ (k)).We recall the notion of a good pair from [22, § 4, p.106]. For integer j ≥ 0, letVj be a G-representation, and Uj ⊂ Vj an open G-invariant subset such that thecodimension of the complement is at least j. The pair (Vj , Uj ) is called a goodpair corresponding to j for the G-action if G acts freely on Uj and the quotientUj /G is a quasi-projective scheme.
Quotients Uj /G approximate algebraically theclassifying space BG (which is not algebraic), while Uj approximate the universalspace EG . It is known that such good pairs always exist (see [11, Lemma 9] or [38,Remark 1.4]).Let X be a smooth G-scheme. For each j ≥ 0, choose a good pair (Vj , Uj )corresponding to j. For i ∈ Z, set G Ωi X × U jΩiG (X)j =(2.1) G .F j Ωi X × U jThen it is known ([22, Lemma 4.2, Remark 4.6]) that ΩiG (X)j is independent ofthe choice of the good pair (Vj , Uj ). Moreover, there is a natural surjective mapΩiG (X)j 0 ΩiG (X)j for j 0 ≥ j ≥ 0. Here, F • Ω∗ (X) is the coniveau filtration onΩ∗ (X), i.e., F j Ω∗ (X) is the set of all cobordism cycles x ∈ Ω∗ (X) such that x diesin Ω∗ (X \ Y ), where Y ⊂ X is closed of codimension at least j (cf.
[10, Sect. 3]).Definition 2.1. Let X be a smooth k-scheme with a G-action. For any i ∈ Z, wedefine the equivariant algebraic cobordism of X to beΩiG (X) = lim ΩiG (X)j .←−(2.2)jThe reader should note from the above definition that unlike the ordinary cobordism, the equivariant algebraic cobordism ΩiG (X) can be non-zero for any i ∈ Z.We setMΩ∗G (X) =ΩiG (X).i∈ZClearly, if G is trivial, then the G-equivariant cobordism reduces to the ordinaryone.394VALENTINA KIRITCHENKO, AMALENDU KRISHNARemark 2.2. If X is a G-scheme of dimension d, which is not necessarily smooth,one defines (cf. [22, Def.
4.4]) the equivariant cobordism of X by G Ωi+lj −g X × UjΩG G ,i (X)j = lim←−j Fd+l −g−j Ωi+l −g X × Ujjj(2.3)where g = dim(G) and lj = dim(Uj ). Here, F• Ω∗ (X) is the niveau filtration onΩ∗ (X) such that Fj Ω∗ (X) is the union of the images of the natural L-linear mapsΩ∗ (Y ) → Ω∗ (X) where Y ⊂ X is closed of dimension at most j.
It is known(cf. [22, Remark 4.7]) that if X is smooth of dimension d, then ΩiG (X) ∼= ΩGd−i (X).Since we shall be dealing mostly with the smooth schemes in this paper, we do notneed this definition of equivariant cobordism.It is known that Ω∗G (X) satisfies all the properties of a multiplicative orientedcohomology theory like the ordinary cobordism.
In particular, it has pull-backs,projective push-forward, Chern class of equivariant bundles, external and internalproducts, homotopy invariance, and projection formula. We refer to [22, Thm. 5.2]for further details.The G-equivariant cobordism group Ω∗G (k) of the ground field k is denoted by∗Ω (BG ) and is called the cobordism ring of the classifying space of G.
We shalloften write it as S(G). We also recall the following result, which gives a simplerdescription of the equivariant cobordism and which will be used throughout thispaper.Theorem 2.3 (([22, Thm. 6.1])). Let {(Vj , Uj )}j≥0 be a sequence of good pairsfor the G-action such that(i) Vj+1 = Vj ⊕ Wj as representations of G with dim(Wj ) > 0,(ii) Uj ⊕ Wj ( Uj+1 as G-invariant open subsets, and(iii) codimVj+1 (Vj+1 \ Uj+1 ) > codimVj (Vj \ Uj ).Then for any smooth scheme X with a G-action, and any i ∈ Z, G ∼=ΩiG (X) −→ lim Ωi X × Uj .←−jMoreover, such a sequence {(Vj , Uj )}j≥0 of good pairs always exists.For the rest of this paper, a sequence of good pairs {(Vj , Uj )}j≥0 will alwaysmean a sequence as in Theorem 2.3.2.1.
Torus equivariant cobordism of a pointLet G = T be a split torus. The cobordism ring S(T ) was described in [22,Example 6.6]. Throughout the paper, we will use the following more invariantdescription.Let Tb denote the character lattice of T , and Sym(Tb) the symmetric algebra(over the integers) of Tb.
Consider the graded algebra Sym(Tb) ⊗ L (with respect tothe total grading, that is, the degree of an element a ⊗ b is the sum of the degreesEQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES395of a and b in Sym(Tb) and L, respectively). Then S(T ) is canonically isomorphictoMS(T ) =S i (T ),i∈ZwhereiS i (T ) := lim (Symj (Tb) ⊗ L) .←−jThe isomorphism sends a character χ ∈ Tb to the first T -equivariant Chernclass of the T -equivariant line bundle Lχ on Spec (k).
In particular, if χ1 , . . . ,χn is a basis in Tb, then S(T ) is isomorphic to the graded power series ring S =Lgr [[cT1 (Lχ1 ), . . . , cT1 (Lχn )]].Recall that for a graded ring R, the graded power series ring R gr [[x1 , . . . , xn ]]consists of all finite linear combinations of homogeneous (with respect to the totalgrading) power series (e.g., if R has no terms of negative degree then R gr [[x1 , .
. .. . . , xn ]] is just a ring of polynomials).2.2. Equivariant cobordism of the variety of complete flags in k nAs an example illustrating the definition of equivariant cobordism, we now computeΩ∗T (G/B) for G = GLn (k) directly by definition. Note that the same result canbe obtained by less computationally involved arguments (see Section 4 where wecompute Ω∗T (G/B) for all reductive groups G and also [17] for the theorem on flagbundles).We identify the points of the complete flag variety X = G/B with completeflags in k n . A complete flag F is a strictly increasing sequence of subspacesF = {{0} = V 0 ( V 1 ( V 2 ( · · · ( V n = k n }with dim(V i ) = i.
There are n natural line bundles L1 ,. . . , Ln on X, that is,the fiber of Li at the point F is equal to V i /V i−1 . These bundles are equivariantwith respect to the left action of the diagonal torus GLn (k) on X, namely, Licorresponds to the character χi of the diagonal torus T ⊂ GLn (k) given by thei-th entry of T . For each i = 1, . . . , n, consider also the T -equivariant line bundleLi on Spec (k) corresponding to the character χi .Theorem 2.4. There is the following ring isomorphismΩ∗T (X) ' Lgr [[x1 , . . .
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