Диссертация (1136188), страница 23

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Annales Scientifiques del’École Normale Supérieure 7, sup. 4 (1974): 53–88.[8]Fulton, W. “Introduction to Toric Varieties.” In The William H. Roever Lectures in Geometry.Annals of Mathematics Studies, 131. Princeton, NJ: Princeton University Press, 1993.[9]Gelfand, I.M., and M.L. Cetlin. “Finite dimensional representations of the group of unimodular matrices.” Doklady Akademii Nauk SSSR 71 (1950): 825–28.[10] Kiritchenko, V. “Flag varieties and Gelfand–Zetlin polytopes.” Oberwolfach Reports, Report01/2009, 16–19.[11] Kiritchenko, V., E.

Smirnov, and V. Timorin. Schubert Calculus on Gelfand–Zetlin Polytopes(in preparation).[12] Khovanskii, A. “Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups inLobachevskii space.” Functional Analysis and Its Applications 20, no. 1 (1986): 41–50.[13] Kogan, M. “Schubert geometry of flag varieties and Gelfand–Cetlin theory.” Ph.D Thesis, Massachusetts Institute of Technology, 2000.[14] Kogan, M., and E. Miller. “Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes.” Advances in Mathematics 193, no.

1 (2005): 1–17.[15] Manivel, L. Symmetric Functions, Schubert Polynomials and Degeneracy Loci. Translatedfrom the 1998 French original by John R. Swallow. Société Mathématique de France/AmericanMathematical Society Texts and Monographs 6. Cours Spécialisés 3. Providence, RI: American Mathematical Society/Paris: Société Mathématique de France, 2001.Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012[7]Приложение D.Статья 4.Jens Hornbostel and Valentina Kiritchenko “Schubert calculus for algebraiccobordism”Journal für die reine und angewandte Mathematik 656 (2011), 59—85Разрешение на копирование: Согласноhttps://www.degruyter.com/dg/page/308/copyright-agreement? автор статьи можетполучить письменное согласие издателя на копирование статьи, если послеиздания статьи прошло больше года.Journal für die reine undangewandte MathematikJ. reine angew.

Math. 656 (2011), 59—85DOI 10.1515/CRELLE.2011.043( Walter de GruyterBerlin New York 2011Schubert calculus for algebraic cobordismBy Jens Hornbostel at Wuppertal and Valentina Kiritchenko at MoscowAbstract. We establish a Schubert calculus for Bott–Samelson resolutions in thealgebraic cobordism ring of a complete flag variety G=B extending the results of Bressler–Evens [4] to the algebro-geometric setting.1. IntroductionWe fix a base field k of characteristic 0.

Algebraic cobordism W ðÞ has beeninvented some years ago by Levine and Morel [14] as the universal oriented algebraiccohomology theory on smooth varieties over k. In particular, its coe‰cient ring W ðkÞ isisomorphic to the Lazard ring L (introduced in [12]). In a recent article [15], Levine andPandharipande show that algebraic cobordism W n ðX Þ allows a presentation with generators being projective morphisms Y ! X of relative codimension nð:¼ dimðX Þ dimðY ÞÞbetween smooth varieties and relations given by a refinement of the naive algebraic cobordism relation (involving double point relations). A recent result of Levine [13] which relies onunpublished work of Hopkins and Morel asserts an isomorphism W n ðÞ G M GL 2n; n ðÞbetween Levine–Morel and Voevodsky algebraic cobordism for smooth quasiprojectivevarieties.

In particular, algebraic cobordism is representable in the motivic stable homotopycategory.In short, algebraic cobordism is to algebraic varieties what complex cobordismMU ðÞ is to topological manifolds.The above fundamental results being established, it is high time for computations,which have been carried out only in a very small number of cases (see e.g. [22] and [23]).The present article focuses on cellular varieties X , for which the additive structure ofW ðX Þ is easy to describe: it is the free L-module generated by the cells (see the next sectionfor more precise definitions, statements, proofs and references). So, additively, algebraiccobordism for cellular varieties behaves exactly as Chow groups do. Of course, algebraicK-theory also behaves in a similar way, but we will restrict our comparisons here andThe second author would like to thank Jacobs University Bremen, the Hausdor¤ Center for Mathematicsand the Max Planck Institute for Mathematics in Bonn for hospitality and support.

She was also partially supported by the Dynasty Foundation fellowship and RFBR grant 10-01-00540-a.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4660Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismbelow to Chow groups. There is a ring homomorphism W ðX Þ ! MU 2 X ðCÞ an whichfor cellular varieties is an isomorphism (see Section 2.2 and the appendix). However, computations in W ðX Þ become more transparent and suitable for algebro-geometric applications if they are done by algebro-geometric methods rather than by a translation of the already existing results for MU 2 X ðCÞ an (e.g. those of Bressler and Evens, see [4] andbelow), especially if the latter were obtained by topological methods which do not havecounterparts in algebraic geometry.Let us concentrate on complete flag varieties X ¼ G=B where B is a Borel subgroupof a connected split reductive group G over k.

In the case where G ¼ GLn ðkÞ, the cobordism ring W ðX Þ may be described as the quotient of a free polynomial ring over L with generators xi being the first Chern classes of certain line bundles on X and explicit relations.More precisely, we show (see Theorem 2.6):Theorem 1.1. The cobordism ring W ðX Þ is isomorphic to the graded ring L½x1 ; .

. . ; xn of polynomials with coe‰cients in the Lazard ring L and deg xi ¼ 1, quotient by the ideal Sgenerated by the homogeneous symmetric polynomials of strictly positive degreeW ðX Þ F L½x1 ; . . . ; xn =S:This generalizes a theorem of Borel [2] on the Chow ring (or equivalently the singularcohomology ring) of a flag variety to its algebraic cobordism ring.The Chow ring of the flag variety has a natural basis given by the Schubert cycles.The central problem in Schubert calculus was to find polynomials (later called Schubertpolynomials) representing the Schubert cycles in the Borel presentation. This problem wassolved independently by Bernstein–Gelfand–Gelfand [1] and Demazure [10] using divideddi¤erence operators on the Chow ring (most of the ingredients were already contained in amanuscript of Chevalley [8], which for many years remained unpublished).

Explicit formulas for Schubert polynomials give an algorithm for decomposing the product of any twoSchubert cycles into a linear combination of other Schubert cycles with integer coe‰cients.The complex (as well as the algebraic) cobordism ring of the flag variety also hasa natural generating set given by the Bott–Samelson resolutions of the Schubert cycles(note that the latter are not always smooth and so, in general, do not define any cobordismclasses).

For the complex cobordism ring, Bressler and Evens described the cobordismclasses of Bott–Samelson resolutions in the Borel presentation using generalized divideddi¤erence operators on the cobordism ring [3], [4] (we thank Burt Totaro from whom wefirst learned about this reference). Their formulas for these operators are not algebraicand involve a passage to the classifying space of a compact torus in G and homotopy theoretic considerations (see [3], Corollary–Definition 1.9, Remark 1.11, and [4], Proposition3).

One of the goals of the present paper is to prove an algebraic formula for the generalized divided di¤erence operators (see Definition 2.2 and Corollary 2.3). This formula inturn implies explicit purely algebraic formulas for the polynomials (now with coe‰cientsin the Lazard ring L) representing the classes of Bott–Samelson resolutions.

Note thateach such polynomial contains the respective Schubert polynomial as the lowest degreeterm (but in most cases also has non-trivial higher-order terms). We also give an algorithmfor decomposing the product of two Bott–Samelson resolutions into a linear combinationof other Bott–Samelson resolutions with coe‰cients in L.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism61We now formulate our main theorem (compare Theorem 3.2), which can be viewedas an algebro-geometric analogue of the results of Bressler–Evens [4], Corollary 1, Proposition 3. Let I ¼ ða1 ; .

. . ; al Þ be an l-tuple of simple roots of G, and RI the correspondingBott–Samelson resolution of the Schubert cycle XI (see Section 3 for the precise definitions). Recall that there is an isomorphism between the Picard group of the flag varietyand the weight lattice of G such that very ample line bundles map to strictly dominantweights (see, for instance, [5], 1.4.3). We denote by LðlÞ the line bundle on X corresponding to a weight l, and by c1 LðlÞ its first Chern class in algebraic cobordism.

For each ai ,we define the operator Ai on W ðX Þ in a purely algebraic way (see Section 3.2 for the rigorous definition for arbitrary reductive groups). Informally, the operator Ai can be definedin the case G ¼ GLn by the formulaAi ¼ ð1 þ sai Þ1;c1 Lðai Þwhere sai acts on the variables ðx1 ; . . . ; xn Þ by the transposition corresponding to ai . Herewe use that the Weyl group of GLn can be identified with the symmetric group Sn so thatthe simple reflections sai correspondto elementary transpositions (see Section 2 for moredetails).

Note that the c1 Lðai Þ can be written explicitly as polynomials in x1 ; . . . ; xn usingthe formal group law (see Section 2).Theorem 1.2. For any complete flag variety X ¼ G=B and any tuple I ¼ ða1 ; . . . ; al Þof simple roots of G, the class of the Bott–Samelson resolution RI in the algebraic cobordismring W ðX Þ is equal toAl . . . A1 Re ;where Re is the class of a point.This theorem reduces the computation of the products of the geometric Bott–Samelson classes to the products in the polynomial ring given by the previous theorem.Note that in the cohomology case, analogously defined operators Ai coincide with thedivided di¤erences operators defined in [1], [10], so our theorem generalizes [1], Theorem4.1, and [10], Theorem 4.1, for Schubert cycles in cohomology and Chow ring, respectively,to Bott–Samelson classes in algebraic cobordism.Note that in the case of Chow ring, the theorem analogous to Theorem 1.2 has twodi¤erent proofs.

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