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Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 307–316 | Cite as

Quasi-Frobenius splitting and lifting of Calabi–Yau varieties in characteristic p

  • Fuetaro YobukoEmail author
Article
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Abstract

Generalizing the notion of Frobenius-splitting, we prove that every finite height Calabi–Yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of Witt vectors of length two.

Notes

Acknowledgements

The author would like to express his sincere gratitude to his advisor Professor Nobuo Tsuzuki. He thanks Professor Kirti Joshi for informing him the conjecture on the lifting of Calabi–Yau threefolds and explaining him the relation between the conjecture and this work. He also thanks Professor Yukiyoshi Nakkajima and the referee for their careful readings of the manuscript and useful suggestions. He was supported by Grant-in-Aid for JSPS Fellow 15J05073.

References

  1. 1.
    Artin, M., Mazur, B.: Formal groups arising from algebraic varieties. Ann. Sci. Éc. Norm. Sup. 10, 87–132 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bogomolov, F.: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243, 1101–1104 (1978)MathSciNetGoogle Scholar
  3. 3.
    Cynk, S., van Straten, D.: Small resolutions and non-liftable Calabi–Yau threefolds. Manuscr. Math. 130, 233–249 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Deligne, P.: Relèvement des surfaces \(K3\) en caractéristique nulle. Lecture Note in Math, pp. 58–79. Springer, New York (1981)Google Scholar
  5. 5.
    Deligne, P., Illusie, L.: Relèvements modulo \(p^2\) et décomposition du complexe de de Rham. Invent. Math. 89, 247–270 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ekedahl, T.: Diagonal Complexes and F-gauge Structures. Hermann, Paris (1986). Travaux ex CourszbMATHGoogle Scholar
  7. 7.
    Ekedahl, T.: On non-liftable Calabi–Yau threefolds. arXiv:0306435v2
  8. 8.
    Ekedahl, T., Shepherd-Barron, N.I.: Tangent lifting of deformations in mixed characteristic. J. Algebra 291, 108–128 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hirokado, M.: A non-liftable Calabi–Yau threefold in characteristic \(3\). Tohoku Math. J. 51(2), 479–487 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hirokado, M., Ito, H., Saito, N.: Calabi–Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, I. Ark. Mat. 45, 279–296 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hirokado, M., Ito, H., Saito, N.: Calabi–Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, II. IManuscr. Math. 125, 325–343 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. sci. ecole. norm. s. 12, 501–661 (1979)CrossRefzbMATHGoogle Scholar
  13. 13.
    Joshi, K.: Exotic torsion, Frobenius splitting and the slope spectral sequence. Can. Math. Bull. 50, 567–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Joshi, K.: Crystalline aspects of geography of low dimensional varieties I: numerology. arXiv:1403.6402
  15. 15.
    Kawamata, Y.: Unobstructed deformations, a remark on a paper of Z. Ran: Deformation of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1, 183–190 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for schubert varieties. Ann. Math. 122, 27–40 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mehta, V.B., Srinivas, V.: Varieties in positive characteristic with trivial tangent bundle. Compos. Math. 64, 191–212 (1987)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ran, Z.: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1, 279–291 (1992)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rudakov, A.N., Shafarevich, I.R.: Inseparable morphisms of algebraic surfaces. Izv. Akad. Nauk SSSR 40, 1269–1307 (1976)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Schoen, C.: Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number. Compos. Math. 145, 89–111 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schröer, S.: The \(T^1\)-lifting theorem in positive characteristic. J. Algebraic Geom. 12, 699–714 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schröer, S.: Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors. Compos. Math. 140, 1579–1592 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Serre, J.-P. : Sur la topologie des variétés algébriques en caractéristique. \(p\), In: Symposium internacional de topología algebraica, 24–53, Universidad nacional Autonoma de Mexico and UNESCO. City, Mexico (1958)Google Scholar
  24. 24.
    Srinivas, V.: Decomposition of the de Rham complex. Proc. Indian Acad. Sci. 100, 103–106 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tian, G.: Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Peterson–Weil metric, In: Yau S. (ed.), Mathematical Aspects of String Theory, pp. 629-646. Adv. Ser. Math. Phys. 1. World Sci. Publishing, Singapore (1987)Google Scholar
  26. 26.
    Todorov, A.: Applications of the Kähler-Einstein–Calabi–Yau metric to moduli of K3 surfaces. Invent. Math. 61, 251–265 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    van der Geer, G., Katsura, T.: On the height of Calabi–Yau varieties in positive characteristic. Doc. Math. 8, 97–113 (2003)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan
  2. 2.Graduate School of Mathematics, Nagoya UniversityNagoyaJapan

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