Диссертация (1136188), страница 27
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In this case, the variety RI is a resolution of singularities for the Schubert cycle XI .Bott–Samelson resolutions were introduced by Bott and Samelson in the case ofcompact Lie groups, and by Demazure in the case of algebraic semisimple groups [10].There are several equivalent definitions, see e.g. [6], [10], [16].
We will use the definitionbelow (which follows easily from [6], 2.2), since it is most suited to our needs. Namely, RIis defined by the following inductive procedure starting from R j ¼ pt ¼ SpecðkÞ (in whatfollows we will rather denote R j by Re ). For each j-tuple J ¼ ða1 ; . . .
; aj Þ with j < l, denoteby J W f j þ 1g the ð j þ 1Þ-tuple ða1 ; . . . ; aj ; ajþ1 Þ. Define RJWf jþ1g as the fiber productRJ G=Pjþ1 G=B, where Pjþ1 is the minimal parabolic subgroup corresponding to the rootajþ1 . Then the map rJWf jþ1g : RJWf jþ1g ! X is defined as the projection to the secondfactor. In what follows,we will use that RJ can be embedded into RJWf jþ1g by sendingx A RJ to x; rJ ðxÞ A RJ G=Pjþ1 G=B.In particular, one-dimensional Bott–Samelson resolutions are isomorphic to the corresponding Schubert cycles. It is easy to show that any two-dimensional Bott–Samelsonresolution RI for a reduced I is also isomorphic to the corresponding Schubert cycle.More generally, RI is isomorphic to XI if and only if all simple roots in I are pairwise distinct (in particular, the length of I should not exceed the rank of G).
The simplest exampleBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism71where RI and XI are not isomorphic for a reduced I is G ¼ GL3 and I ¼ ðg1 ; g2 ; g1 Þ (whereg1 , g2 are two simple roots of GL3 ).E,It is easy to show that RJWf jþ1g is the projectivization of the bundle rJ pjþ1where E is the rank two vector bundle on G=Pjþ1 defined in the next subsection andpjþ1 : G=B ! G=Pjþ1 is the natural projection.
This is the definition used in [4]. In thetopological setting, the vector bundle rJ pjþ1E splits into the sum of two line bundles [4]but in the algebro-geometric setting this is no longer true (though rJ pjþ1E still contains aline subbundle, as follows from the proof of Lemma 3.4).This definition of RI allows to describe easily (by repeated use of the projectivebundle formula) the ring structure of the cobordism ring W ðRI Þ. It also implies that RI iscellular with 2 l cells labeled by all subindices J H I .The cobordism classes ZI of Bott–Samelson resolutions generate W ðX Þ but do notform a basis.
The following proposition is an immediate corollary of Theorem 2.4. Ananalogous statement for complex cobordism is proved in [4], Proposition 1, by using theAtiyah–Hirzebruch spectral sequence (as mentioned in Section 2).Proposition 3.1. As an L-module, the algebraic cobordism ring W ðX Þ of the flagvariety is freely generated by the Bott–Samelson classes ZI ðwÞ , where w A W and I ðwÞ definesa reduced decomposition for w (we choose exactly one I ðwÞ for each w).There is no canonical choice for a decomposition I ðwÞ of a given element w inthe Weyl group.
From the geometric viewpoint it is more natural to consider all Bott–Samelson classes at once (including those for non-reduced I ) even though they are notlinearly independent over L. So throughout the rest of the paper we will not put any restrictions on the multiindex I .3.2. Schubert calculus. We will now describe the cobordism classes ZI as polynomials in the first Chern classes of line bundles on X . This allows us to compute productsof Bott–Samelson resolutions and hence achieves the goal of a Schubert calculus for algebraic cobordism.We first define operators Ai on W ðX Þ following the approach of the previous section(see Definition 2.2).
These operators generalize the divided di¤erence operators on the Chowring CH ðX Þ defined in [1], [10], [8] to algebraic cobordism.We first define operators Ai for GLn since in this case the Borel presentation allowsus to make them more explicit. We start with the subgroup B of upper triangular matricesand the diagonal torus, which yields an isomorphism W G Sn . Under this isomorphism, thereflection sa with respect to a root a ¼ ei ej goes to the transposition ði jÞ (see the end ofSection 2). For each positive root a of G, we define the operators sa and A^a on the ring offormal power series L½½x1 ; .
. . ; xn as follows:ðsa f Þðx1 ; . . . ; xn Þ ¼ f ðxsa ð1Þ ; . . . ; xsa ðnÞ Þ;1:A^a ¼ ð1 þ sa Þ F xiþ1 ; wðxi ÞBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4672Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismIt is easy to check that A^a is well-defined on the whole ring L½½x1 ; .
. . ; xn (see Section 5).Note also that under the homomorphismL½½x1 ; . . . ; xn ! L½x1 ; . . . ; xn =S G W ðX Þ thepower series F xiþ1 ; wðxi Þ maps to c1 Lðgi Þ (see the end of Section 2), so our definitionfor additive formal group law reduces to the definition of divided di¤erence operatoron the polynomial ring Z½x1 ; . . . ; xn (see [16], 2.3.1). Finally, we define the operatorAa : W ðX Þ ! W ðX Þ using the Borel presentation by the formulaAa f ðx1 ; . . . ; xn Þ ¼ A^a ð f Þðx1 ; . . . ; xn Þfor each polynomial f A L½x1 ; .
. . ; xn . Again, by degree reasons the right-hand side is apolynomial. The operator Aa is well defined (that is, does not depend on a choice of a polynomial f representing a given class in L½x1 ; . . . ; xn =S) since for any polynomial h and anysymmetric polynomial g we have A^a ðghÞ ¼ gA^a ðhÞ.We now define Ai ¼ Aai for an arbitrary reductive group G and a simple root ai .Denote by Pi H G the minimal parabolic subgroup corresponding to the root ai . ThenX ¼ G=B is a projective line fibration over G=Pi . Indeed, consider the projectionpi : G=B ! G=Pi . Take the line bundle LðrÞ on G=B corresponding to the weight r, wherer is the half-sum of all positive roots or equivalently the sum of all fundamental weightsof G (the weight r is uniquely characterized by the property that ðr; aÞ ¼ 1 for all simpleroots a).
Then it is easy to check that the vector bundle E :¼ pi LðrÞ on G=Pi has ranktwo and G=B ¼ PðEÞ. Note that tensoring E with any line bundle L on G=Pi does notchange PðEÞ ¼ PðE n LÞ so the property PðEÞ ¼ X does not uniquely define the bundleE. However, the choice E ¼ pi LðrÞ (suggested to us by Michel Brion) is the only uniformchoice for all i, since LðrÞ is the only line bundle on X with the property P pi LðrÞ ¼ Xfor all i. We now use Definition 2.2 to define an W ðG=Pi Þ-linear operator Ai :¼ Api onW ðX Þ. For G ¼ GLn , this definition coincides with the one given above.
This is easyto show using that G=Pi for ai ¼ gi is the partial flag variety whose points are flagsF ¼ ff0g ¼ F 0 H H F i1 H F iþ1 H H F n ¼ k n g.Let I ¼ ða1 ; . . . ; al Þ be an l-tuple of simple roots of G. Define the element RI inW ðX Þ by the formulaRI :¼ Al . . . A1 Ze :In the case G ¼ GLn , we can also regard RI as a polynomial in L½x1 ; . . . ; xn =S.Similar to [1], Theorem 3.15, or [4], page 807, one may describe Ze for general G usingthe formulaZe ¼ Re :¼1 Q c1 LðaÞ ;jW j a A R þwhere R þ denotes the set of positive roots of G (recall that jR þ j ¼ dim X ¼: d).
As in theChow ring case, there is also the formulaZe ¼1LðrÞ d :d!Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism73Both formulas immediately follow from the analogous formulas for the Chow ring [1], Theorem 3.15, Corollary 3.16, since W d ðX Þ F CH d ðX Þ (as follows from [14], Theorem 1.2.19,Remark 4.5.6). Note that for GLn , the formula for Re reduces to Re ¼ Dn sincec1 Lðei ej Þ ¼ xj xi þ higher-order terms, and hence the equality Ze ¼ Re followsfrom Remark 2.7. In particular, by the same remark Re modulo S has a denominator-freen2expression xnn1 xn1 x2 .We now prove an algebro-geometric version of [4], Corollary 1, Proposition 3, usingour algebraic operators Ai .Theorem 3.2.
The cobordism class ZI ¼ ½rI : RI ! X of the Bott–Samelson resolution RI is equal to RI .Proof. The essential part of the proof is the formula for the push-forward as statedin Corollary 2.3. Once this formula is established it is not hard to show that Ai ZI ¼ ZI Wfigfor all I by exactly the same methods as in the Chow ring case [16] and in the complexcobordism case [4]. Namely, we have the following cartesian square:p2G=B G=Pi G=B ! G=B??????pip1 yypiG=B ! G=Pi :E.g., if G ¼ GLn we get exactly the diagram of [16], proof of Lemma 3.6.20. Using thiscommutative diagram and the definition of Bott–Samelson resolutions, it is easy toshow that pi pi ZI ¼ ZI Wfig [4], proof of Proposition 2.1. We now apply Corollary 2.3and get that Ai ¼ pi pi . It follows by induction on the length of I that ZI ¼ Al .
. . A1 Ze .rRemark 3.3. Note that if we apply the base change formula [14], Definition1.1.2(A2), to the cartesian diagram from the proof of Theorem 3.2, we get p1 p2 ¼ pi pi ,where the right-hand side is precisely the definition of the ‘‘geometric’’ operator denoted Aiin [4], while the left-hand side is the operator denoted di in [16], proof of Theorem 3.6.18.Hence Manivel and Bressler–Evens consider the same operators.We now compute the action of the operator Ai on polynomials in the first Chernclasses (this computation will be used in Sections 4 and 5). Consider the operator si :¼ spiagain defined as in Definition 2.2.
Note that si corresponds to the simple reflection si :¼ saiin the following sense:Lemma 3.4. For any line bundle LðlÞ on X , we have si c1 LðlÞ ¼ c1 Lðsi lÞ :Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4674Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismProof. Since X ¼ PðEÞ (recall that E ¼ pi LðrÞ), the bundle pi E on X admits theusual short exact sequence0 ! tE ! pi E ! OE ð1Þ ! 0;where tE is the tautological line bundle on X (that is, the fiber of tE at the pointx A X ¼ PðEÞ is the line in E represented by x). Note that in our case PðEÞ ¼ PðE dual Þsince E is of rank two (thus hyperplanes in E are the same as lines in E).
It is easy toshow that there is an isomorphism of line bundlestE1 n OE ð1Þ ¼ Lðai Þ:(Moreover, one can show that tE ¼ Lðr ai Þ and OE ð1Þ ¼ LðrÞ.) Indeed,tE1 n OE ð1Þ F Hom tE ; OE ð1Þcan be thought of as the bundle of tangents along the fibers of pi . The latter is the line bundleassociated with the B-module pi =b, which has weight ai (see [5], Remark 1.4.2, for analternative definition of the line bundles LðlÞ in terms of the one-dimensional B-modules).Here pi and b denote the Lie algebras of Pi and B, respectively. By definition, si switches c1 ðtE Þ and c1 OE ð1Þ . Hence, si c1 Lðai Þ ¼ c1 Lðai Þ .Since the Picard group of G=Pi can be identified with the sublattice fl j ðl; ai Þ ¼ 0g of theweight lattice of G (this follows fromafterProposition1.3.6, combined with [5], [5], remarkProposition 1.4.3) we also have si c1 LðlÞ ¼ c1 LðlÞ for all l perpendicular to ai .
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