Диссертация (1136188), страница 28
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Thesetwo identities imply the statement of the lemma. rThis lemma allows us to describe explicitly the action of si and hence of Ai on anypolynomial in the first Chern classes. Indeed,since for any weight lk we have si l ¼ l þ kaifor someintegerk,wecancomputeclÞ¼ c1 LðlÞ n Lðai Þ as a power series inLðs1ic1 LðlÞ and c1 Lðai Þ using the formal group law. This will be used in the proof of Proposition 4.3 below and in Subsection 5.1.4.
Chevalley–Pieri formulasA key ingredient for the classical Schubert calculus is the Chevalley–Pieri formula forthe product of the Schubert cycle with the first Chern class of the line bundle on X , see e.g.[1], Proposition 4.1, and [10], Proposition4.2. We now establish analogous formulas for theproducts of ZI and RI with c1 LðlÞ (without using that ZI ¼ RI ). At the end of this section, we explain why in the case of algebraic cobordism this alone is not enough to showthat ZI ¼ RI , hence justifying our di¤erent approach of the previous two sections.By LðDÞ denote the line bundle corresponding to the divisor D. For each l-tupleI as above, denote by I j the ðl 1Þ-tuple ða1 ; .
. . ; a^j ; . . . ; al Þ. For each root a, define thelinear function ð; aÞ (that is, the coroot) on the weight lattice of G by the propertysa l ¼ l ðl; aÞa for all weights l. (The pairing ða; bÞ is often denoted by ha; b4i or byha; bi.) Note that by definition ðl; aÞ ¼ ðwl; waÞ for all elements w of the Weyl group.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism75Proposition 4.1.
Geometric Chevalley–Pieri formula(1) for Bott–Samelson resolutions: In the Picard group of RI we haverI LðlÞ ¼lNLðRI j Þ ðl; bj Þj¼1where b j ¼ sl sjþ1 aj .(2) for Schubert cycles ([1], Proposition 4.1, [10], Proposition 4.4, [8]): In the Chowring of X we havePc1 LðlÞ XI ¼ ðl; bj ÞXI jjwhere the sum is taken over j A f1; . . . ; lg for which the decomposition defined by I j isreduced.The first part of this proposition was proved in [4], Proposition 4, in the topologicalsetting (for flag varieties of compact Lie groups). It is not hard to check that the proof carries over to algebro-geometric setting. We instead provide a shorter proof along the samelines. Our proof is based on the following lemma:Lemma 4.2 ([10], Proposition 2.1).
Let p : RI ! RI l be the natural projection (coming from the fact that we defined RI as a projective bundle over RI l ). Then we have an isomorphismrI LðlÞ G p rIl Lðsl lÞ n LðRI l Þ ðl; al Þof line bundles on RI .Proposition 4.1(1) now follows from Lemma 4.2 by induction on l. The base l ¼ 1,that is r1 LðlÞ ¼ OP 1 ð1Þ ðl; a1 Þ , follows from the fact that r1 : R1 ! X maps R1 isomorphicallyto P1 =B G P 1 , which can be regarded as the flag variety for SL2 .
Then the weight l restrictedto SL2 is equal to ðl; a1 Þ times the highest weight of the tautological representation ofSL2 , which corresponds to the line bundle OP 1 ð1Þ on P 1 . To prove the induction step, plugl 1Nthe induction hypothesis for rIl Lðsl lÞ ¼LðRI j; l Þ ðsl l; sl1 sjþ1 aj Þ into the lemma and usej¼1ðsl l; sl1 sjþ1 aj Þ ¼ ðl; bj Þ (since sl2 ¼ e) and p RI j; l ¼ RI j .Proposition 4.1(1) was used in [4] to establish an algorithm for computing c1 LðlÞ ZIin W ðX Þ [4].
We now briefly recall this algorithm. By the projection formula we have c1 LðlÞ ZI ¼ ðrI Þ c1 rI LðlÞ :Note that the usual projection formula with respect to smooth projective morphismsf : X ! Y holds for algebraic cobordism as well. This follows from the definition of products via pull-backs along the diagonal and the base change axiom (A2) of [14] applied todiagpidthe cartesian square obtained from Y ! Y Y X Y .Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4676Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismOne can now use Proposition 4.1(1) and the formal group law to compute c1 rI LðlÞin terms of the Bott–Samelson classes in W ðRI Þ by an iterative procedure (since themultiplicative structure of W ðRI Þ can be determined by the projectivebundle formulaandclasses arisingthis way again have the form c1 LðlÞ for After the Chern some l).PaJ ½RJ for some aJ A L, it is easy to find ðrI Þ c1 rI LðlÞ sincec1 rI LðlÞ is written asJHIðrI Þ ½RJ ¼ ZJ .However, this procedure is ratherlengthy,and we will not use it.
Instead, we willprove a more explicit formula for c1 LðlÞ ZI (see formula (5.1) below) using our algebraicChevalley–Pieri formula together with Theorem 3.2.Proposition 4.3. Algebraic Chevalley–Pieri formula:(1) cobordism version: Let A1 ¼ Aa1 ; . . . ; Al ¼ Aal be the operators on W ðX Þ corresponding to a1 ; . . . ; al . Then we havelPc1 Lðlj Þ c1 Lðsj lj ÞAjþ1 . .
. Al Rec1 LðlÞ A1 . . . Al Re ¼A1 . . . Aj1c1 Lðaj Þj¼1in W ðX Þ, where lj ¼ sj1 s1 l and sj ¼ saj is the reflection corresponding to the root aj .(2) Chow ring version ([1], Corollary 3.7): Let A1 ¼ Aa1 ; . . . ; Al ¼ Aal be the operators on CH ðX Þ corresponding to a1 ; . . . ; al . ThenlPc1 LðlÞ A1 . . . Al Re ¼ðl; s1 sj1 aj ÞA1 . . .
A^j . . . Al Rej¼1in CH ðX Þ.c1 Lðlj Þ c1 Lðsj lj Þis a well-defined element in W ðX ÞProof. First, note thatc1 Lðaj Þbecause sj l ¼ l ðl; aj Þaj (and hence LðlÞ ¼ Lðsj lÞ n Lðaj Þ ðl; aj Þ ) and the formal group lawexpansion for c1 ðL1 n L2k Þ c1 ðL1 Þ is divisible by c1 ðL2 Þ for any integer k [14], (2.5.1). Nextwe show that c1 LðlÞ c1 Lðs1 lÞ;c1 LðlÞ A1 A1 c1 Lðs1 lÞ ¼c1 Lða1 Þwhere both sides are regarded as operators on W ðX Þ.
Indeed, by definitionA1 ¼ ð1 þ s1 Þ1c1 Lða1 Þandc1 LðlÞ s1 ¼ s1 c1 Lðs1 lÞby Lemma 3.4.Hence, we can writec1 LðlÞ c1 Lðs1 lÞA2 . . . Al Re þ A1 c1 Lðs1 lÞ A2 . . . Al Re ;c1 LðlÞ A1 . . . Al Re ¼c1 Lða1 ÞBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism77and then continue movingc1 Lðs1 lÞ to the right until we are left with the termA1 . . . Al c1 Lðsl . .
. s1 lÞ Re . This term is equal to zero since c1 Lðsl . . . s1 lÞ Re is the product of more than dim X first Chern classes, and hence its degree is greater than dim X . TheChow ring case follows immediately from the cobordism case sincec1 LðlÞ c1 Lðsj lÞ¼ ðl; aj Þc1 Lðaj Þin the Chow ring. The last identity holds because the formal group law for the Chow ring isadditive, and hence c1 LðlÞ c1 Lðsj lÞ ¼ ðl; aj Þc1 Lðaj Þ . rThe second part of this proposition was proved in [1] by more involved calculations.A calculation similar to ours was used in [17] to deduce a combinatorial Chevalley–Pieriformula for K-theory.
It would be interesting to find an analogous combinatorial interpretation of our Chevalley–Pieri formula in the cobordism case.Note that in the case of Chow groups, the algebraic Chevalley–Pieri formula forAl . . . A1 Re is exactly the same as the geometric one for the Schubert cycle XI .
Togetherwith the Borel presentation this easily implies that the polynomial Al . . . A1 Re representsthe Schubert cycle XI whenever I defines a reduced decomposition [1]. Indeed, we canproceed by the induction on l. Algebraic and geometric Chevalley–Pieri formulas allow usto compute the intersection indices of Al . . . A1 Re and of XI , respectively, with the productof k first Chern classes, and the result is the same in both cases by the induction hypothesis(for all k > 0Þ. By the Borel presentation we know that the products of first Chern classesdLspanCH i ðX Þ. Hence, by the non-degeneracy of the intersection form on CH ðX Þ (thati¼1is, by Poincaré duality) we have that Al .
. . A1 Re XI must lie in CH d ðX Þ ¼ Z½ pt (that is,dLin the orthogonal complement toCH i ðX Þ). This is only possible if Al . . . A1 Re XI ¼ 0i¼1(unless l ¼ 0, which is the induction base). Note that the only geometric input in this proofis the geometric Chevalley–Pieri formula.In the cobordism case, it is not immediately clear why geometric and algebraicChevalley–Pieri formulas are the same (though, of course, it follows from Theorem 3.2).But even without using RI ¼ ZI , it might be possibleto showPthat both formulas havethe same structure coe‰cients, that is, if c1 LðlÞ ZI ¼aJ ZJ then necessarilyJHIPc1 LðlÞ RI ¼aJ RJ with the same coe‰cients aJ A L.
However, this does not lead toJHIthe proof of RI ¼ ZI as in the case of the Chow ring. The reason is that even though thereis an analog of Poincaré duality for the cobordism rings of cellular varieties, this only yieldsan equality RI ¼ ZI up to a multiple of ½ pt, which is not enough to carry out the desiredinduction argument. For the Chow ring, Poincaré duality also yields only an equality upto the class of a point, but unless I ¼ j, the di¤erence RI ZI (where ZI now means theSchubert cycle and not the Bott–Samelson class) cannot be a non-zero multiple of ½ ptbecause the coe‰cient ring CH ð½ ptÞ ¼ CH ðkÞ G Z is concentrated in degree zero, andhas hence no non-zero elements in the corresponding degree l d.
However, for algebraiccobordism, the coe‰cient ring W ðkÞ G L does contain plenty of elements of negative degree, so one cannot deduce RI ¼ ZI .Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4678Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism5. Computations and examplesUntil now, we used the formal group law of algebraic cobordism (i.e., the universalone) as little as possible in order to make our presentation simpler.
In this section, we makethe results of the previous section more explicit using this formal group law. In particular,we give an explicit formula for the products of a Bott–Samelson resolution with the firstChern class of a line bundle in terms of other Bott–Samelson resolutions (see formula(5.1) below). Using this formula, we give an algorithm for computing the product of twoBott–Samelson resolutions.First, we show that the operator A from Section 2 and the operator A^a from Section 3are well-defined.We use the notation of Subsection 2.1, so F ðu; vÞ is the universal formal grouplaw andwðuÞ is the inverse for the universal formal group law defined by the identity F u; wðuÞ ¼ 0.1 is well defined on W ðX Þ½½y1 ; y2 , it isTo show that the operator A ¼ ð1 þ sÞ F y1 ; wðy2 Þenough to show that AðmÞ is a formal power series for any monomial m ¼ y1k1 y2k2 .
We compute Aðy1k1 y2k2 Þ usingy1 ¼ F ðx; y2 Þ ¼ y2 þ wðxÞpðx; y2 Þandy2 ¼ F wðxÞ; y1 ¼ y1 þ wðxÞp wðxÞ; y1F ðu; vÞ uwhere x ¼ F y1 ; wðy2 Þ and pðu; vÞ ¼vis a well-defined power series (since F ðu; vÞ u contains only terms u i v j for j f 1). Wegety1k1 y2k2y k1 y k2 y k2 y k1¼ 1 2 þ 1 2xxwðxÞk ky2 þ wðxÞpðx; y2 Þ 1 y1 þ wðxÞp wðxÞ; y1 Þ 2 y2k1 y1k2þ¼xwðxÞAðy1k1 y2k2 Þ ¼ ð1 þ sÞ terms divisible by x or by wðxÞ:¼ y2k1 y1k2 q x; wðxÞ þxThe second term in the last expression is a power series since the formal group law expansion for wðxÞ is divisible by x [14], (2.5.1).A similar argument shows that the operator A^a from Section 3 is indeed well-definedon the whole ring L½½x1 ; .
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