Диссертация (1136188), страница 30
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First, we do it by hand.X121Denote by o1 , o2 the fundamental weights of SL3 . Applying Proposition 4.1(2) to¼ X we getLðlÞ ¼ LðX21 Þ ðl; g2 Þ n LðX12 Þ ðl; g1 Þ :Hence, c1 Lðo1 Þ ¼ X12 and c1 Lðo2 Þ ¼ X21 . (Note that if we instead applied Proposition4.1(1) to R121 , we would obtain the more complicated expressionLðlÞ ¼ LðR21 Þ ðl; g2 Þ n LðR11 Þ ðl; g1 þg2 Þ n LðR12 Þ ðl; g1 Þ ;r121which does not allow us to express X12 ¼ R12 as the Chern class of the line bundle LðlÞ onR121 .)Hence, Lðo1 ÞZ12 Z21 ¼ c1 Lðo1 Þ Z21 ¼ r21 c1 r21by the projection formula: c1 LðlÞ ZI ¼ rI c1 rI LðlÞ : We now apply Proposition 4.1(1)to R21 and r21 Lðo1 Þ ¼ LðR11 Þ n LðR2 Þ: Using the formal group law we com Lðo1 Þ and getpute c1 LðR1 Þ n LðR2 Þ ¼ R1 þ R2 ½P Re : Finally, we use that rJ ½RJ ¼ ZJ and getthatZ12 Z21 ¼ Z1 þ Z2 ½P 1 Ze :Similarly, we can easily compute the following products:Z12 Z12 ¼ Z2 ;Z12 Z1 ¼ Z21 Z2 ¼ Ze ;Z21 Z21 ¼ Z1 ;Z12 Z2 ¼ Z21 Z1 ¼ 0;which in particular gives us another way to compute polynomials RI .So the only product that di¤ers from the analogous product in the Chow ring case isthe product Z12 Z21 .Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism83We now compute the product Z12 Z21 using formula (5.1).
We have Z12 ¼ c1 Lðo1 Þby Proposition 4.1(1). Hence, according to formula (5.1)Z12 Z21 ¼ c1 Lðo1 Þ Z21 ¼ b1 ðo1 ÞZ2 þ b2 ðo1 ÞZ1 þ b21 ðo1 ÞZe ; where b1 , b2 andb21 are the constant terms in A1 c1 Lðo1 Þ , ½A2 s1 c1 Lðo1 Þ and½A2 A1 c1 Lðo1 Þ , respectively. We already know thatb1 ðlÞ ¼ ðl; g1 Þand b2 ðlÞ ¼ ðl; s1 g2 Þ:It remains to compute b21 ðlÞ.
First, by using that LðlÞ ¼ Lðs1 lÞ n Lðg1 Þ ðl; g1 Þ and the formal group law we writeA1 c1 LðlÞ c1 Lðs1 lÞc1 LðlÞ ¼c1 Lðg1 Þ ðl; g1 Þ 1 c1 Lðg1 Þ þ terms of deg f 2:¼ ðl; g1 Þ þ a11 ðl; g1 Þ c1 Lðs1 lÞ þ2Hence,½A2 A1 and b21 ðl; g1 Þ 1 c1 Lðg1 Þ þ terms of deg f 1c1 LðlÞ ¼ a11 ðl; g1 ÞA2 c1 Lðs1 lÞ þ2ðl; g1 Þ 1¼ a11 ðl; g1 Þ ðl; s1 g2 Þ þ terms of deg f 1;2ðl; g1 Þ 1. We get¼ a11 ðl; g1 Þ ðl; s1 g2 Þ 2ðl; g1 Þ 1Ze :c1 LðlÞ Z21 ¼ ðl; g1 ÞZ2 þ ðl; s1 g2 ÞZ1 þ a11 ðl; g1 Þ ðl; s1 g2 Þ 2In particular, c1 Lðo1 Þ Z21 ¼ Z2 þ Z1 þ a11 Ze (which coincides with the answer we havefound above by hand).Finally, note that it takes more work to compute c1 LðlÞ Z21 using the algorithm in[4] because apart from certain formal group law calculations (which are more involved thanthe calculations we used to find b21 ) one has also to compute the products R12 and R22 inCH ðR21 Þ.6. Appendix.
Complex realization for cellular varietiesWe will now prove the following result stated in Section 2:Theorem 6.1. For any smooth cellular variety X over k and any embeddingk ! C,the complex geometric realization functor of L-algebras r : W ðX Þ ! MU X ðCÞ an is anisomorphism.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4684Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismProof. Recall (see above) that the geometric realization functor coincides with themap given by the universal property of W , and that both sides are freely generated by (resolutions of the closures of) the cells.
Thus itNsu‰ces to show that it is an isomorphism if wepass to the induced morphism after takingZ on both sides, which we denote by r 0 . NowLby a theorem of Totaro [20], Theorem 3.1 (compare also [14],1.2.21), for cellular Remarkvarieties the classical cycle class map c : CH ðX Þ ! H X ðCÞ an (which is an isomorphism for cellular varieties X ) defined using fundamentalclassesand Poincaréduality(seee.g. [16], Section A.3) factors as CH ðX Þ ! MU X ðCÞ an nL Z G H X ðCÞ an , and theleft arrow in this factorization is given by first taking any resolution of singularities of thealgebraic cycle and then applying ðCÞ an . We also have a morphism q : W ðX Þ ! CH ðX Þwhich induces an isomorphism q 0 : W ðX Þ nL Z ! CH ðX Þ by Levine–Morel [14], Theorem 1.2.19, and corresponds to resolution of singularities [14], Section 4.5.1.
Putting everything together, we obtain a commutative squarer0W ðX Þ nL Z ! MU X ðCÞ an nL Z????0?0q yGG?yfGCH ðX Þ ! H X ðCÞ ancwith the vertical maps and c being isomorphisms, which finishes the proof. rReferences[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells, and the cohomology of the spaces G=P,Russian Math. Surveys 28 (1973), no. 3, 1–26.A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Liecompacts, Ann.
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Levine and F. Morel, Algebraic cobordism, Springer Monogr. Math., Berlin 2007.M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), 63–130.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism85[16] L.
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Soc. 139(2005), no. 2, 243–260.Fachgruppe Mathematik und Informatik, Bergische Universität Wuppertal, 42119 Wuppertal, Germanye-mail: hornbostel@mathematik.uni-wuppertal.deFaculty of Mathematics, Higher School of Economics, Vavilova 7, Moscow, Russiaand Institute for Information Transmission Problems, B. Karetnyi 19, Moscow, Russiae-mail: vkiritch@hse.ruEingegangen 9.
April 2009, in revidierter Fassung 10. Februar 2010Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Приложение E.Статья 5.В.А.Кириченко, Е.Ю.Смирнов, В.А.Тиморин “Исчисление Шуберта имногогранники Гельфанда-Цетлина”Успехи Математических Наук, 2012, том 67, выпуск 4(406), 89–128Разрешение на копирование: Согласноhttp://www.mathnet.ru/supplement/authoragreement/license_agreement_rm_rus_v0318.pdf автор статьи может включать материалы статьи в обзоры, монографии,учебные материалы с указанием ссылки на выходные данные cтатьи.Math-Net.RuОбщероссийский математический порталВ.
А. Кириченко, Е. Ю. Смирнов, В. А. Тиморин, Исчисление Шуберта и многогранники Гельфанда–Цетлина,УМН, 2012, том 67, выпуск 4(406), 89–128DOI: https://doi.org/10.4213/rm9492Использование Общероссийского математического портала Math-Net.Ru подразумевает, что вы прочитали и согласны с пользовательским соглашениемhttp://www.mathnet.ru/rus/agreementПараметры загрузки:IP: 80.73.162.1297 декабря 2018 г., 18:13:242012 г. июль — августт.
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