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

Takayama, Yukihide

doi:10.1090/S0002-9939-2012-11271-7

Raynaud-Mukai construction and Calabi-Yau threefolds in positive characteristic
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Proc. Amer. Math. Soc. 140 (2012), 4063-4074 Request permission

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

Sławomir Cynk and Duco van Straten, Small resolutions and non-liftable Calabi-Yau threefolds, Manuscripta Math. 130 (2009), no. 2, 233–249. MR 2545516, DOI 10.1007/s00229-009-0293-0
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
Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, DOI 10.1007/BF01389078
T. Ekedahl, On non-liftable Calabi-Yau threefolds (preprint), math.AG/0306435
Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913, DOI 10.1007/978-3-0348-8600-0
Masayuki Hirokado, A non-liftable Calabi-Yau threefold in characteristic $3$, Tohoku Math. J. (2) 51 (1999), no. 4, 479–487. MR 1725623, DOI 10.2748/tmj/1178224716
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, DOI 10.1007/s11512-007-0041-1
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, DOI 10.1007/s00229-007-0151-x
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, DOI 10.1007/978-3-662-03276-3
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
S. Mukai, On counter-examples of Kodaira’s vanishing theorem and Yau’s inequality (in Japanese), Kinosaki Algebraic Geometry Symposium, 1979.
S. Mukai, Counterexamples of Kodaira’s vanishing and Yau’s inequality in higher dimensional variety of characteristic $p>0$, RIMS preprint, 2005.
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
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
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, DOI 10.1007/BFb0071778
Stefan Schröer, Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors, Compos. Math. 140 (2004), no. 6, 1579–1592. MR 2098403, DOI 10.1112/S0010437X04000545
N. I. Shepherd-Barron, Fano threefolds in positive characteristic, Compositio Math. 105 (1997), no. 3, 237–265. MR 1440723, DOI 10.1023/A:1000158618674
Yukihide Takayama, On non-vanishing of cohomologies of generalized Raynaud polarized surfaces, J. Pure Appl. Algebra 214 (2010), no. 7, 1110–1120. MR 2586990, DOI 10.1016/j.jpaa.2009.09.017
Yoshifumi Takeda, Pre-Tango structures and uniruled varieties, Colloq. Math. 108 (2007), no. 2, 193–216. MR 2291633, DOI 10.4064/cm108-2-4
H. Tango, On the behavior of cohomology classes of vector bundles with regard to Frobenius morphism (Japanese), Suurikaisekikenkyusho Kokyuroku, 1972.
Hiroshi Tango, On the behavior of extensions of vector bundles under the Frobenius map, Nagoya Math. J. 48 (1972), 73–89. MR 314851