Диссертация (1136188), страница 26

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E.g., for G ¼ GLn ðCÞ(resp. SLn ðCÞ), the maximal compact subgroup is UðnÞ (resp. SUðnÞ). This is the languagein which many of the definitions and results in [1], [2] and [4] are stated. We sometimesallow ourselves to use those definitions and results which do carry over to the ‘‘algebraic’’case (reductive groups over k) without mentioning explicitly the obvious changes that haveto be carried out.There are n natural line bundles L1 ; . . .

; Ln on X , namely, the fiber of Li at the pointF is equal to F i =F i1 . Put xi ¼ c1 ðLi Þ, where the first Chern class c1 with respect to algebraic cobordism is defined in [14]. Note that our definition of xi di¤ers by sign from the onein [16]. The following result on the algebraic cobordism ring is an analog of the Borelpresentation for the singular cohomology ring of a flag variety. In fact, it holds for anyorientable cohomology theory since its proof only uses the projective bundle formula.Theorem 2.6.

Let A ðÞ be any orientable cohomology theory (e.g. CH ðÞ orW ðÞ). Then the ring A ðX Þ is isomorphic as a graded ring to the ring of polynomials inx1 ; . . . ; xn with coe‰cients in the coe‰cient ring A ðptÞ and degðxi Þ ¼ 1, quotient by theideal S generated by the symmetric polynomials of strictly positive polynomial degree:A ðX Þ F A ðptÞ½x1 ; . . . ; xn =S:More generally, let E be a vector bundle of rank n over a smooth variety Y and FðEÞ be theflag variety relative to this bundle. Then we have an isomorphism of graded ringsA FðEÞ F A ðptÞ½x1 ; . .

. ; xn =Iwhere I is the ideal generated by the relations ek ðx1 ; . . . ; xn Þ ¼ ck ðEÞ for 1 e k e n with ekdenoting the k-th elementary symmetric polynomial.Proof. The proof of [16], Theorem 3.6.15, for the Chow ring case can be slightlymodified so that it becomes applicable to any other orientable theory A . Namely, for anarbitrary oriented cohomology theory A , it is more convenient to dualize the geometricargument in [16], Theorem 3.6.15, because we can no longer use that ci ðEÞ ¼ ð1Þ i ci ðE Þfor a vector bundle E (which is used implicitly several times in the proof of [16], Theorem3.6.15). That is, we start with the variety of partial flagsPi ¼ fF ni H F niþ1 H H F n ¼ k n g(e.g.

P1 is the variety of hyperplanes in k n and Pn1 ¼ X ). The proof of the more generalcase is completely analogous to the proof of [16], Proposition 3.8.1. Note that the proof of[16], Proposition 3.8.1 does not really use [16], Theorem 3.6.15 as an induction base (despite the claim in the proof) and in fact gives another proof for [16], Theorem 3.6.15, whichis also applicable to an arbitrary oriented cohomology theory.

rBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism69Remark 2.7. The proof immediately implies that the class of a point in A ðX Þ1Qn2n2is equal to xnn1 xn1 x2 . Since xnn1 xn1 x2 ¼ðxi xj Þ mod S (which is easy ton! i>jshow by induction on n using that ðxn xn1 Þ ðxn x1 Þ ¼ nxnn1 mod S) we also have1Qðxi xj Þ.that the class of a point can be represented by the polynomial Dn ¼n! i>jNote that the proof also gives an explicit formula for the classes of one-dimensionalSchubert cycles X1 ¼ Xsg1 ; . .

. ; Xn1 ¼ Xsgn1 in X corresponding to the simple rootsg1 ; . . . ; gn1 of GLn (see the beginning of Section 3 for the definition of the Schubert cyclesXw for w in the Weyl group of G). The cycle Xk consists of flagsF ¼ ff0g ¼ F 0 H F 1 H F 2 H H F n ¼ k n gsuch that all F i except for F k are fixed. Then the class of Xk is equal to the class of a pointdivided by xkþ1 .

Indeed, to get the class of Xk H X we should take the point in Pnk1 corresponding to the fixed partial flag fF kþ1 H F kþ2 H H F n ¼ k n g and then take a line ina fiber of the projective bundle Pnk ! Pnk1 over this point. Namely, the line will consistof all hyperplanes in F kþ1 that contain the fixed codimension two subspace F k1 . Again itn2is easy to show by induction on n that the polynomial xnn1 xn1 x2 =xk is equal to2Dn =ðxkþ1 xk Þ modulo the ideal S.Note that the Borel presentation for singular cohomology implies, in particular, thatthe Picard group of the flag variety is generated (as an abelian group) byPthe first Chernclasses of the line bundles L1 ; . .

. ; Ln , the only nontrivial relation beingc1 ðLi Þ ¼ 0. Inwhat follows, we will also use the following alternative description of the Picard group ofX . Recall that each strictly dominant weight l of G defines an irreducible representationpl : G ! GLðVl Þ and an embedding G=B ! PðVl Þ. Hence, to each strictly dominantweight l of G we can assign a very ample line bundle LðlÞ on X by taking the pullback of the line bundle OPðVl Þ ð1Þ on PðVl Þ. The map l 7! LðlÞ extended to non-dominantweights by linearity gives an isomorphism between the Picard group of X and the weightlattice of G [5], 1.4.3.

In particular, for the line bundles above, we have Li ¼ Lðei Þ, whereei is the weight of GLn given by the i-th entry of the diagonal torus in GLn .We now compute c1 Lðai Þ as a polynomial in x1 ; . . . ; xn . Let g1 ; . . . ; gn1 be the simple roots of G (that is, gi ¼ ei eiþ1 ). We can express the line bundles Lðgi Þ in terms of theline bundles L1 ; . . .

; Ln . Since Li ¼ Lðei Þ and gi ¼ ei eiþ1 , we have that the line bundleLðgi Þ is isomorphic to Li1 n Liþ1 . In particular, we can computec1 Lðgi Þ ¼ c1 ðLi1 n Liþ1 Þ ¼ F wðxi Þ; xiþ1 :E.g. by the formulas for F ðx; yÞ and wðxÞ from [14], 2.5, the first few terms of c1 Lðgi Þlook as follows:c1 Lðgi Þ ¼ xi þ xiþ1 þ a11 xi2 a11 xi xiþ1 þ ;where a11 ¼ ½P 1 .Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4670Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismIn what follows, we will use the isomorphism W G S n .

The simple reflection sa forany root a ¼ ei ej acts on the weight lattice (spanned by the weights e1 ; . . . ; en , whichform an orthonormal basis) by the reflection in the plane perpendicular to ei ej and hencepermutes the weights e1 ; . . . ; en by the transposition ði jÞ.3.

Schubert calculus for algebraic cobordism of flag varietiesIn this section, we assume that G is an arbitrary connected split reductive group unless we explicitly mention that G ¼ GLn ðkÞ, and X ¼ G=B is the complete flag variety forG. We now investigate the ring structure of W ðX Þ in more geometric terms.3.1.

Schubert cycles and Bott–Samelson resolutions. Recall that the flag variety X iscellular with the following cellular decomposition into Bruhat cells. Let us fix a Borel subgroup B. For each element w A W of the Weyl group of G, define the Bruhat (or Schubert)cell Cw as the B-orbit of the point wB A G=B ¼ X (we identify the Weyl group withNðTÞ=T for a maximal torus T of G inside B). The Schubert cycle Xw is defined as the closure of Cw in X . The dimension of Xw is equal to the length of w [1]. Recall that the lengthof an element w A W is defined as the minimal number of factors in a decomposition of winto the product of simple reflections. Recall also that for each l-tuple I ¼ ða1 ; .

. . ; al Þ ofsimple roots of G, one can define the Bott–Samelson resolution RI (which has dimensionl) together with the map rI : RI ! X . Bott–Samelson resolutions are smooth. Consequently, for any I the map rI : RI ! X represents an element in W ðX Þ which we denoteby ZI .Denote by sa A W the reflection corresponding to a root a, and by sI the productsa1 sal . If the decomposition sI ¼ sa1 sal defined by I is reduced (that is, sI cannot bewritten as a product of less than l simple reflections, or equivalently, the length of sI isequal to l), then the image rI ðRI Þ coincides with the Schubert cycle XsI (which we willalso denote by XI ). The dimension of XI in this case is also equal to l and the maprI : RI ! XI is a birational isomorphism.

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