Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2012-11271-7
Published electronically: April 5, 2012
MathSciNet review: 2957196
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. S. Cynk and D. van Straten, Small resolutions and non-liftable Calabi-Yau threefolds. Manuscripta Math. 130 (2009), no. 2, 233-249. MR 2545516 (2010k:14074)
  • 2. P. Deligne, Rel $ \grave {{\rm e}}$vement des surfaces $ K3$ en caract $ \acute {{\rm e}}$ristique nulle. Prepared for publication by Luc Illusie. Lecture Notes in Math., 868, Algebraic surfaces (Orsay, 1976-78), pp. 58-79, Springer, Berlin-New York, 1981. MR 638598 (83j:14034)
  • 3. P. Deligne and L. Illusie, Relèvements modulo $ p\sp 2$ et décomposition du complexe de de Rham. Invent. Math. 89 (1987), no. 2, 247-270. MR 894379 (88j:14029)
  • 4. T. Ekedahl, On non-liftable Calabi-Yau threefolds (preprint), math.AG/0306435
  • 5. H. Esnault and E. Viehweg, Lectures on vanishing theorems. DMV Seminar, 20. Birkhäuser Verlag, Basel, 1992. MR 1193913 (94a:14017)
  • 6. M. Hirokado, A non-liftable Calabi-Yau threefold in characteristic $ 3$. Tohoku Math. J. (2) 51 (1999), no. 4, 479-487. MR 1725623 (2000m:14044)
  • 7. M. Hirokado, H. Ito and N. Saito, Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces. I. Ark. Mat. 45 (2007), no. 2, 279-296. MR 2342606 (2008j:14074)
  • 8. M. Hirokado, H. Ito and N. Saito, Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces. II. Manuscripta Math. 125 (2008), no. 3, 325-343. MR 2373064 (2008m:14078)
  • 9. J. Kollar, Rational Curves on Algebraic Geometry, Ergebnisse der Mathematik und ihre Grenzgebiete 3. Folge, Band 32, Springer, 1996. MR 1440180 (98c:14001)
  • 10. M. Maruyama, On a family of algebraic vector bundles, in Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Y. Akizuki, Kinokuniya, Tokyo, 1973, pp. 95-146. MR 0360587 (50:13035)
  • 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. D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5. Oxford University Press, London, 1970, published for the Tata Institute of Fundamental Research, Bombay. MR 0282985 (44:219)
  • 14. M. Raynaud, Contre-exemple au ``vanishing theorem'' en caractéristique $ p>0$ (French). C. P. Ramanujam--a tribute, pp. 273-278, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978. MR 541027 (81b:14011)
  • 15. P. Russell, Factoring the Frobenius morphism of an algebraic surface, Algebraic geometry, Bucharest 1982 (Bucharest, 1982), 366-380, Lecture Notes in Math., 1056, Springer, Berlin, 1984. MR 749947 (85m:14055)
  • 16. S. Schröer, Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors. Compos. Math. 140 (2004), no. 6, 1579-1592. MR 2098403 (2005i:14051)
  • 17. N. I. Shepherd-Barron, Fano threefolds in positive characteristic. Compositio Math. 105 (1997), no. 3, 237-265. MR 1440723 (98d:14054)
  • 18. Y. Takayama, On non-vanishing of cohomologies of generalized Raynaud polarized surfaces. J. Pure Appl. Algebra 214 (2010), no. 7, 1110-1120. MR 2586990 (2011g:14089)
  • 19. Y. Takeda, Pre-Tango structures and uniruled varieties. Colloq. Math. 108 (2007), no. 2, 193-216. MR 2291633 (2008g:14026)
  • 20. H. Tango, On the behavior of cohomology classes of vector bundles with regard to Frobenius morphism (Japanese), Suurikaisekikenkyusho Kokyuroku, 1972.
  • 21. H. Tango, On the behavior of extensions of vector bundles under the Frobenius map, Nagoya Math. J., 48 (1972), 73-89. MR 0314851 (47:3401)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14F17, 14J32, 14M99, 14J45

Retrieve articles in all journals with MSC (2010): 14F17, 14J32, 14M99, 14J45


Additional Information

Yukihide Takayama
Affiliation: Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan
Email: takayama@se.ritsumei.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2012-11271-7
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.

American Mathematical Society