manuscripta mathematica

, Volume 130, Issue 1, pp 121–135 | Cite as

Correspondences between modular Calabi–Yau fiber products

  • Michał KapustkaEmail author


We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi–Yau varieties. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi–Yau threefolds.

Mathematics Subject Classification (2000)

14J32 14J27 14G35 


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  1. 1.
    Alekseev, V., Nikulin, V.: Del Pezzo and K3 surfaces. Mathematical Society of Japan Memoirs, vol. 15 (2006)Google Scholar
  2. 2.
    Alekseev V., Nikulin V.: Classification of del Pezzo surfaces with log-terminal singularities of index = 2 and involutions of K3 surfaces. Soviet Math. Dokl. 39(3), 507–511 (1989)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Barth W., Peters C., Van de Ven A.: Compact complex surfaces. Springer, Heidelberg (1984)zbMATHGoogle Scholar
  4. 4.
    Bouchard V., Donagi R.: On a class of non-simply connected Calabi–Yau threefolds. Comm. Numb. Theor. Phys. 2, 1–61 (2008)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Cynk S.: Defect of a nodal hypersurface. Manuscripta math. 104, 325–331 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cynk S.: Cohomologies of a double covering of a nonsingular algebraic 3-folds. Math. Z. 240(2), 731–743 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cynk S.: Double coverings of octic arrangements with isolated singularities. Adv. Theor. Math. Phys. 3, 217–225 (1999)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cynk S., Meyer C.: Geometry and arithmetic of certain double octic Calabi–Yau Manifolds. Can. Math. Bull. 48(2), 180–194 (2005)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Cynk S., Meyer C.: Modular Calabi–Yau threefolds of level eight. Int. J. Math. 18(3), 331–347 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cynk S., van Straten D.: Infinitesimal deformations of smooth algebraic varieties. Math. Nachr. 279(7), 716–726 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dieulefait, L., Manoharmayum, J.: Modularity of rigid Calabi–Yau threefolds over \({\mathbb{Q}}\) . In: Calabi–Yau varieties and mirror symmetry (Toronto, ON, 2001), Fields Inst. Commun. 38, Amer. Math. Soc., Providence, pp. 159–166 (2003)Google Scholar
  12. 12.
    Gouvea, F.Q., Yui, N.: Rigid Calabi–Yau Threefolds Over Q Are Modular: A Footnote to Serre, preprint (2009) arXiv:0902.1466v1 [math.NT]Google Scholar
  13. 13.
    Heckman, G., Looijenga, E.: The moduli space of rational elliptic surfaces. Algebraic geometry 2000, Azumino (Hotaka). Adv. Stud. Pure Math., vol. 36, pp. 185–248, Math. Soc. Japan, Tokyo (2002)Google Scholar
  14. 14.
    Herfurtner S.: Elliptic surfaces with four singular fibres. Math. Ann 291, 319–342 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hulek, K., Kloosterman, R., Schütt, M.: Modularity of Calabi–Yau varieties. Global Aspects of Complex Geometry, pp. 271–309. Springer, Berlin (2006)Google Scholar
  16. 16.
    Hulek, K., Verrill, H.: On the modularity of Calabi–Yau threefolds containing elliptic ruled surfaces. “Mirror Symmetry V”. In: Proceedings of the BIRS Workshop on Calabi–Yau varieties and mirror symmetry, December, pp. 6–11. AMS/IP Studies in Advanced Mathematics, preprint (2005), math.AG/0502158 (2003, to appear)Google Scholar
  17. 17.
    Hulek K., Verrill H.: On modularity of rigid and nonrigid Calabi–Yau varieties associated to the root lattice A 4. Nagoya Math. J. 179, 103–146 (2005)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Iskovskikh, V.A., Shafarevich, I.R.: Algebraic surfaces. Algebraic geometry, II, pp. 127–262. Encyclopaedia Math. Sci., 35, Springer, Berlin (1996)Google Scholar
  19. 19.
    Kapustka G., Kapustka M.: Modularity of a nonrigid Calabi–Yau manifold with bad reduction at 13. Ann. Pol. Math. 90(1), 89–98 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kapustka, G., Kapustka, M.: Fiber products of elliptic surfaces with section and associated Kummer fibrations, preprint 2007, arXiv:0802.3760v1 [math.AG]. Int. J. Math. (to appear)Google Scholar
  21. 21.
    Khare C.: Serre’s modularity conjecture: the level one case. Duke Math. J. 134(3), 557–589 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture, preprint (2006).
  23. 23.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture (II), preprint (2006).
  24. 24.
    Kisin, M.: Modularity of 2-adic Barsotti-Tate representations, preprint (2007).
  25. 25.
    Livné, R.: Cubic exponential sums and Galois representations. In: Ribet, K. (ed.) Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif. 1985), Contemp. Math. vol. 67, pp. 247–261. Amer. Math. Soc., Providence R.I (1987)Google Scholar
  26. 26.
    Livné R.: Motivic orthogonal two-dimensional representations of Gal(\({\overline{\mathbb{Q}}/\mathbb{Q}}\)). Israel J. Math. 92, 149–156 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Livné R., Yui N.: The modularity of certain non-rigid Calabi–Yau threefolds. J. Math. Kyoto Univ. 45(4), 645–665 (2005)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Meyer, Ch.: Modular Calabi–Yau threefolds. Fields Institute Monograph 22, AMS, New York (2005)Google Scholar
  29. 29.
    Miranda, R.: The basic theory of elliptic surfaces. Notes of lecture, ETS Editrice PisaGoogle Scholar
  30. 30.
    Miranda R.: Persson’s list of singular fibers for a rational elliptic surface. Math. Z. 205(2), 191–211 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Miranda R., Persson U.: On extremal rational elliptic surfaces. Math. Z. 193, 537–558 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Miranda R., Persson U.: Torsion groups of elliptic surfaces. Composit. Math. 72(3), 249–267 (1989)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Oguiso K., Shioda T.: The Mordell–Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40(1), 83–99 (1991)MathSciNetGoogle Scholar
  34. 34.
    Persson U.: Configurations of Kodaira fibers on rational elliptic surfaces. Math. Z. 205(1), 1–47 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Reid, M.: Chapters on algebraic surfaces. In: Kollár, J. (ed.) Complex Algebraic Varieties, IAS/Park City Lecture Notes Series (1993 vol.), pp. 1–154. AMS, New York (1997)Google Scholar
  36. 36.
    Reid, M.: Young person’s guide to canonical singularities. In: Bloch, S. (ed.), Algebraic Geometry, Bowdoin 1985, Proc. of Symposia in Pure Math. 46, vol. 1, pp. 345–414. AMS, New York (1987)Google Scholar
  37. 37.
    Reid, M.: Surface cyclic quotient singularities and the Hirzebruch–Jung resolutions,
  38. 38.
    Schoen C.: On fiber products of rational elliptic surfaces with section. Math. Z. 197, 177–199 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Schoen C.: On the computation of the cycle class map. Ann. Sci. École Norm. Sup. 28(1), 1–50 (1995)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Schütt M.: Modularity of three Calabi–Yau threefolds with bad reduction at 11. Can. Math. Bull. 49(2), 296–312 (2006)zbMATHGoogle Scholar
  41. 41.
    Schütt M.: New examples of modular rigid Calabi–Yau threefolds. Collect. Math. 55, 219–228 (2004)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular functions of one variable, IV. In: Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972, Lecture Notes in Math., vol. 476, pp. 33–52. Springer, Berlin (1975)Google Scholar
  43. 43.
    Wilson P.M.H.: Elliptic ruled surfaces on Calabi–Yau threefolds. Math. Proc. Camb. Philos. Soc. 112, 45–52 (1992)zbMATHCrossRefGoogle Scholar

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsJagiellonian UniversityKrakówPoland

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