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Raynaud-Mukai construction and Calabi-Yau threefolds in positive characteristic

Author: Yukihide Takayama
Journal: Proc. Amer. Math. Soc. 140 (2012), 4063-4074
MSC (2010): Primary 14F17, 14J32; Secondary 14M99, 14J45
Published electronically: April 5, 2012
MathSciNet review: 2957196
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Abstract: In this article, we study the possibility of producing a Calabi-Yau threefold in positive characteristic which is a counterexample to Kodaira vanishing. The only known method to construct the counterexample is the so-called inductive method such as the Raynaud-Mukai construction or Russel construction. We consider Mukai's method and its modification. Finally, as an application of the Shepherd-Barron vanishing theorem of Fano threefolds, we compute $ H^1(X, H^{-1})$ for any ample line bundle $ H$ on a Calabi-Yau threefold $ X$ on which Kodaira vanishing fails.

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  • 1. Sławomir Cynk and Duco van Straten, Small resolutions and non-liftable Calabi-Yau threefolds, Manuscripta Math. 130 (2009), no. 2, 233–249. MR 2545516, 10.1007/s00229-009-0293-0
  • 2. P. Deligne, Relèvement des surfaces 𝐾3 en caractéristique nulle, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 58–79 (French). Prepared for publication by Luc Illusie. MR 638598
  • 3. Pierre Deligne and Luc Illusie, Relèvements modulo 𝑝² et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, 10.1007/BF01389078
  • 4. T. Ekedahl, On non-liftable Calabi-Yau threefolds (preprint), math.AG/0306435
  • 5. Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
  • 6. Masayuki Hirokado, A non-liftable Calabi-Yau threefold in characteristic 3, Tohoku Math. J. (2) 51 (1999), no. 4, 479–487. MR 1725623, 10.2748/tmj/1178224716
  • 7. Masayuki Hirokado, Hiroyuki Ito, and Natsuo Saito, Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces. I, Ark. Mat. 45 (2007), no. 2, 279–296. MR 2342606, 10.1007/s11512-007-0041-1
  • 8. Masayuki Hirokado, Hiroyuki Ito, and Natsuo Saito, Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces. II, Manuscripta Math. 125 (2008), no. 3, 325–343. MR 2373064, 10.1007/s00229-007-0151-x
  • 9. János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
  • 10. Masaki Maruyama, On a family of algebraic vector bundles, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 95–146. MR 0360587
  • 11. S. Mukai, On counter-examples of Kodaira's vanishing theorem and Yau's inequality (in Japanese), Kinosaki Algebraic Geometry Symposium, 1979.
  • 12. S. Mukai, Counterexamples of Kodaira's vanishing and Yau's inequality in higher dimensional variety of characteristic $ p>0$, RIMS preprint, 2005.
  • 13. David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985
  • 14. M. Raynaud, Contre-exemple au “vanishing theorem” en caractéristique 𝑝>0, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 273–278 (French). MR 541027
  • 15. Peter Russell, Factoring the Frobenius morphism of an algebraic surface, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 366–380. MR 749947, 10.1007/BFb0071778
  • 16. Stefan Schröer, Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors, Compos. Math. 140 (2004), no. 6, 1579–1592. MR 2098403, 10.1112/S0010437X04000545
  • 17. N. I. Shepherd-Barron, Fano threefolds in positive characteristic, Compositio Math. 105 (1997), no. 3, 237–265. MR 1440723, 10.1023/A:1000158618674
  • 18. Yukihide Takayama, On non-vanishing of cohomologies of generalized Raynaud polarized surfaces, J. Pure Appl. Algebra 214 (2010), no. 7, 1110–1120. MR 2586990, 10.1016/j.jpaa.2009.09.017
  • 19. Yoshifumi Takeda, Pre-Tango structures and uniruled varieties, Colloq. Math. 108 (2007), no. 2, 193–216. MR 2291633, 10.4064/cm108-2-4
  • 20. H. Tango, On the behavior of cohomology classes of vector bundles with regard to Frobenius morphism (Japanese), Suurikaisekikenkyusho Kokyuroku, 1972.
  • 21. Hiroshi Tango, On the behavior of extensions of vector bundles under the Frobenius map, Nagoya Math. J. 48 (1972), 73–89. MR 0314851

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Additional Information

Yukihide Takayama
Affiliation: Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan

Received by editor(s): October 18, 2010
Received by editor(s) in revised form: May 2, 2011, and May 21, 2011
Published electronically: April 5, 2012
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.