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Calabi Yau Hypersurfaces and SU-Bordism

  • Ivan Yu. Limonchenko
  • Zhi Lü
  • Taras E. Panov
Article
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Abstract

V. V. Batyrev constructed a family of Calabi–Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi–Yau manifolds whose SU-bordism classes generate the special unitary bordism ring \({\Omega ^{SU}}[\frac{1}{2}] \cong Z[\frac{1}{2}][{y_i}:i \geqslant 2]\). We also describe explicit Calabi–Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions.

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References

  1. 1.
    V. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties,” J. Algebr. Geom. 3 (3), 493–535 (1994).MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. M. Buchstaber, “Cobordisms, manifolds with torus action, and functional equations,” Proc. Steklov Inst. Math. 302, 48–87 (2018) [transl. from Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 302, 57–97 (2018)].Google Scholar
  3. 3.
    V. M. Buchstaber and T. E. Panov, Toric Topology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.CrossRefzbMATHGoogle Scholar
  4. 4.
    V. Buchstaber, T. Panov, and N. Ray, “Toric genera,” Int. Math. Res. Not. 2010 (16), 3207–3262 (2010).MathSciNetzbMATHGoogle Scholar
  5. 5.
    P. E. Conner and E. E. Floyd, Torsion in SU-Bordism (Am. Math. Soc., Providence, RI, 1966), Mem. AMS, No. 60.zbMATHGoogle Scholar
  6. 6.
    M. Kreuzer and H. Skarke, “Calabi–Yau data,” http://hep.itp.tuwien.ac.at/~kreuzer/CY/.Google Scholar
  7. 7.
    Z. Lü and T. Panov, “On toric generators in the unitary and special unitary bordism rings,” Algebr. Geom. Topol. 16 (5), 2865–2893 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. E. Mosley, “The greatest common divisor of multinomial coefficients,” arXiv: 1411.0706 [math.NT].Google Scholar
  9. 9.
    J. E. Mosley, “In search of a class of representatives for SU-cobordism using the Witten genus,” PhD Thesis (Univ. Kentucky, Lexington, 2016).Google Scholar
  10. 10.
    S. P. Novikov, “Homotopy properties of Thom complexes,” Mat. Sb. 57 (4), 407–442 (1962). Engl. transl. is available at http://www.mi-ras.ru/~snovikov/6.pdf.MathSciNetGoogle Scholar
  11. 11.
    R. E. Stong, Notes on Cobordism Theory (Princeton Univ. Press, Princeton, NJ, 1968), Math. Notes.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Ivan Yu. Limonchenko
    • 1
  • Zhi Lü
    • 1
  • Taras E. Panov
    • 2
    • 3
    • 4
  1. 1.School of Mathematical SciencesFudan UniversityShanghaiP.R. China
  2. 2.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  3. 3.Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of National Research Centre “Kurchatov Institute,”MoscowRussia
  4. 4.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of Sciences, Bol’shoi Karetnyi per. 19MoscowRussia

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