Диссертация (1136188), страница 43
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, xn ; t1 , . . . , tn ]]/(si (x1 , . . . , xn ) − si (t1 , . . . , tn ), i = 1, . . . , n),where si (x1 , . . . , xn ) denotes the i-th elementary symmetric function of the variables x1 ,. . . , xn . The isomorphism sends xi and ti , respectively, to the first T equivariant Chern classes cT1 (Li ) and cT1 (Li ).Proof. First, note that Ω∗T (X) = Ω∗B (X) by [22, Prop. 8.1], where B is a Borelsubgroup in G (we choose B to be the subgroup of the upper-triangular matrices).For N > n, we can approximate the classifying space BB by partial flag varietiesFN,n := F(N − n, N − n + 1, . . .
, N − 1, N ) consisting of all flagsF = {V N −n ( V N −n+1 ( . . . ( V N −1 ( k N }.396VALENTINA KIRITCHENKO, AMALENDU KRISHNAWe choose exactly this approximation because its cobordism ring is easier to compute via projective bundle formula than the cobordism ring of the dual flag varietyF(1, 2, . . . , n; N ) (for cohomology rings, this difference does not show up, since forthe additive formal group law, the Chern classes of dual vector bundles are thesame up to a sign).
Approximate EB by the variety EN := Hom◦ (k N , k n ) of allprojections of k N onto k n . Note that {(Hom(k N , k n ), EN )}N ≥n is a sequence ofgood pairs (as in Theorem 2.3) for the action of GLn .Denote by E the tautological quotient bundle of rank n on FN,n (i.e., the fiberof E at the point F is equal to k N /V N −n ). For the complete flag variety X, wehave that X ×B EN is the flag variety F(E) relative to the bundle E. Points of F(E)can be identified with complete flags in the fibers of E.
Hence, we can computethe cobordism ring of X ×B EN by the formula for the cobordism rings of relativeflag varieties [18, Thm. 2.6]. We getΩ∗ (X ×B EN ) = Ω∗ (F(E)) ' Ω∗ (FN,n )[x1 , ..., xn ]/I,where I is the ideal generated by the relations sk (x1 , .., xn ) = ck (E) for 1 ≤ k ≤ n.The isomorphism sends xi to the first Chern class of the line bundle Li ×B EN onX × B EN .By the repeated use of the projective bundle formula (as in the proof of [18,Thm. 2.6]) we get thatΩ∗ (FN,n ) ' Lgr [t1 , .
. . , tn ]/(hN (tn ), hN −1 (tn−1 , tn ), . . . , hN −n+1 (t1 , . . . , tn )),where ti is the first Chern class of the i-th tautological line bundle on FN,n (whosefiber at the point F is equal to V N −n+i /V N −n+i−1 ), and hk (ti , . .
. , tn ) denotesthe sum of all monomials of degree k in ti , . . . , tn .It is easy to deduce from the Whitney sum formula that ck (E) = sk (t1 , . . . , tn ).Passing to the limit, we get that ΩiB (X) := lim Ωi (X ×B EN ) consists of all←−Nhomogeneous power series of degree i in t1 , . .
. , tn and x1 , . . . , xn modulo the relations sk (x1 , . . . , xn ) = sk (t1 , . . . , tn ) for 1 ≤ k ≤ n. Indeed, all relations betweent1 , . . . , tn in Ω∗ (FN,n ) are in degree greater than i if N > i + n − 1. 3. Algebraic and complex cobordismIn this section and in Section 4, we shall assume our ground field to be the fieldof complex numbers C. A scheme will mean a quasi-projective scheme over C.
Todescribe the equivariant algebraic cobordism ring of flag varieties, we first describethe equivariant complex cobordism and then use comparison results between thealgebraic and complex cobordism. Our main goal in this section is to establishsuch comparison results.For a scheme X, let H ∗ (X, A) denote the singular cohomology of the analytic space X(C) with coefficients in an abelian group A. Let M U ∗ (X, A) denoteM U ∗ (X) ⊗Z A, where M U ∗ (−) is the complex cobordism, a generalized cohomology theory on the category of CW-complexes.Recall from [33, §2] that X 7→ M U ∗ (X(C)) is an example of an oriented cohomology theory on VCS .
In fact, it is the universal oriented cohomology theory inEQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES397the category of CW-complexes. This cohomology theory is multiplicative in thesense that it has external and internal products. One knows that X 7→ H ∗ (X, Z)is also an example of a multiplicative oriented cohomology theory on VCS .3.1. Equivariant complex cobordismRecall ([22, Sect. 7]) that if G is a complex linear algebraic group and X is a finiteCW-complex with a G-action, then its Borel style equivariant complex cobordismis defined as G∗M UG(X) := M U ∗ X × EG ,(3.1)∗where EG → BG is a universal principal G-bundle.
It is known that M UG(X) isindependent of the choice of this universal bundle.Definition 3.1. Let U = {(Vj , Uj )}j≥0 be a sequence of good pairs for G-action.For a linear algebraic group G acting on a scheme X and for any i ∈ Z, we define G iM UG(X, U) := lim M U i X × Uj(3.2)←−j≥0∗and set M UG(X, U) =Li∈ZiM UG(X, U). We also set G MΩiG (X, U) := lim Ωi X × Uj and Ω∗ (X, U) =ΩiG (X, U) .←−j≥0(3.3)i∈Z∗It is easy to check as in [22, Thm. 5.2] that M UG(−, U) and Ω∗G (−, U) haveall the functorial properties of the equivariant cobordism.
In particular, both areScontravariant functors on VGand Ω∗G (−, U) is also covariant for projective maps.Moreover, the pull-back and the push-forward maps commute with each other ina fiber diagram of smooth and projective morphisms.Lemma 3.2. Let U = {(Vj , Uj )}j≥1 be a sequence of good pairs for the G-action∗and let X be a smooth G-scheme such that HG(X, Z) is torsion-free. There is aniiisomorphism M UG (X) → M UG (X, U) of abelian groups for any i ∈ Z.Proof.
Since U is a sequence of good pairs for the G-action, the codimension ofthe complement of Uj in the G-representation Vj is at least j. In particular,S thepair (Vj , Uj ) is (j − 1)-connected. Taking the limit, we see that EG =Uj isj≥0contractible and hence EG → EG /G is a universal principal G-bundle and we cantake BG = EG /G. Since X(C) has the homotopy type of a finite CW-complex, weGsee that XG = X × EG has a filtration by finite subcomplexes∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ Xi ⊂ · · · ⊂ XGGwith Xj = X × Uj and XG =SXj . This yields the Milnor exact sequencej≥0i0 → lim1 M U i−1 (Xj ) → M UG(X) → lim M U i (Xj ) → 0.←−←−j≥0(3.4)j≥0∗Since HG(X, Z) = H ∗ (XG , Z) is torsion-free, it follows from [27, Cor.
1] thatthe first term in this exact sequence is zero. This proves the lemma. 398VALENTINA KIRITCHENKO, AMALENDU KRISHNA3.2. Comparison theoremRecall from [12, Example 1.9.1] that a scheme (or an analytic space) L is calledcellular, if it has a filtration ∅ = Ln+1 ( Ln ( · · · ( L1 ( L0 = L by closed subschemes (subspaces) such that each Li \Li+1 is a disjoint union of affine spaces Ari(cells).
It follows from the Bruhat decomposition that varieties G/B are cellularwith cells labelled by elements of the Weyl group. Using the above definition andthe fact that a vector bundle over an affine space is also an affine space (QuillenSuslin theorem), one checks that a vector bundle over a cellular scheme is alsocellular. The following elementary and folklore result yields more examples ofcellular schemes.Lemma 3.3. Let X be a scheme with a filtration ∅ = Xn+1 ( Xn ( · · · ( X1 (X0 = X by closed subschemes such that each Xi \ Xi+1 is a cellular scheme. ThenX is also a cellular scheme.Proof. It follows from our assumption that Xn is cellular. It suffices to prove byinduction on the length of the filtration of X that, if Y ,→ X is a closed immersionof schemes such that Y and U = X \Y are cellular, then X is also cellular.
Considerthe cellular decompositions∅ = Yl+1 ( Yl ( · · · ( Y1 ( Y0 = Y,∅ = Um+1 ( Um ( · · · ( U1 ( U0 = Uof Y and U . SetXi =(Y ∪ UiYi−m−1if 0 ≤ i ≤ m + 1if m + 2 ≤ i ≤ m + l + 2 .It is easy to verify that {Xi }0≤i≤m+l+2 is a filtration of X by closed subschemessuch that Xi \ Xi+1 is a disjoint union of affine spaces over C. Let T be a split torus of rank n and let U = {(Vj , Uj )}j≥1 be the sequence⊕nof good pairs for T -action such that each (Vj , Uj ) = (Vj0 , Uj0 ) , where Vj0 is thej-dimensional representation of Gm with all weights −1, and Uj0 is the complementof the origin and T acts on Vj diagonally.Definition 3.4.
A scheme (or a scheme over any other field) X with an actionof T is called T -equivariantly cellular, if there is a filtration ∅ = Xn+1 ( Xn (· · · ( X1 ( X0 = X by T -invariant closed subschemes such that each Xi \ Xi+1 isisomorphic to a disjoint union of representations Vi ’s of T .It is obvious that a T -equivariantly cellular scheme is cellular in the usual sense.It follows from a theorem of Bialynicki-Birula [2] that if X is a smooth projectivescheme with a T -action such that the fixed point locus X T is isolated, then X isT -equivariantly cellular. In particular, a complete flag variety G/B or a smoothprojective toric variety is T -equivariantly cellular.Proposition 3.5.
Let U = {(Vj , Uj )}j≥1 be as above, and X a smooth schemewith a T -action such that it is T -equivariantly cellular. Then the natural mapΩ∗T (X, U) → M UT∗ (X, U)is an isomorphism.EQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES399TProof. For any scheme Y with T action, we set Y j = Y × Uj for j ≥ 1. Considerthe T -equivariant cellular decomposition of X as in Definition 3.4 and set Wi =Xi \ Xi+1 . It follows immediately that X j has a filtration∅ = (X j )n+1 ( (X j )n ( · · · ( (X j )1 ( (X j )0 = X j ,Twhere (X j )i = (Xi )j = Xi × Uj and thus (X j )i \ (X j )i+1 = (Wi )j .nTSince Uj /T ∼= Pj−1 is cellular and since (Wi )j = Wi × Uj → Uj /T is adisjoint union of vector bundles, it follows that each (X j )i = (Wi )j is cellular.
Weconclude from Lemma 3.3 that X j is cellular. In particular, the map Ω∗ (X j ) →M U ∗ (X j ) is an isomorphism (cf. [18, Thm. 6.1]). The proposition now follows bytaking the limit over j ≥ 1. Lemma 3.6. Let X be a T -equivariantly cellular scheme. Then HT∗ (X, Z) is torsion-free.Proof. Let U = {(Vj , Uj )}j≥1 be a sequence of good pairs for T -action as above.∼=Since HTi (X, Z) −→ H i (X j , Z) for j 0, it suffices to show that H ∗ (X j , Z) istorsion-free for any j ≥ 0. But we have shown in Proposition 3.5 that each X j iscellular and hence H ∗ (X j , Z) is a free abelian group. Theorem 3.7. Let X be a smooth scheme with an action of a split torus T .
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