A characterization of counterexamples to the kodaira-ramanujam vanishing theorem on surfaces in positive characteristic

Abstract

The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic. More precisely, it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic, then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type. Furthermore, it is proved that up to blow-ups, X admits a fibration to a smooth projective curve, such that each fiber is a singular curve.

Keywords

Characterization / Counterexample / Kodaira-Ramanujam vanishing

References

[1] Bombieri E., Mumford D.. Enriques’ classification of surfaces in char. p, II, Complex Analysis and Algebraic Geometry, A Collection of Papers Dedicated to K. Kodaira, 1977, Cambridge: Cambridge University Press 23-42

[2] Bombieri E., Mumford D.. Enriques’ classification of surfaces in char. p, III. Invent. Math., 1976, 36: 197-232

[3] Deligne P., Illusie L.. Relèvements modulo p2 et décomposition du complexe de de Rham. Invent. Math., 1987, 89: 247-270

[4] Ekedahl T.. Canonical models of surfaces of general type in positive characteristic. Publ. Math. IHES, 1988, 67: 97-144

[5] Hartshorne R.. Algebraic Geometry, 1977, New York: Springer-Verlag

[6] Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, Alg. Geom. Sendai, 1985, Adv. Stud. Pure Math., 10, 1987, 283–360.

[7] Ménégaux R.. Un théorème d’annulation en caractéristique positive, Séminaire sur les pinceaux de courbes de genre au moins deux by L. Szpiro. Astérisque, 1981, 86: 35-43

[8] Mukai, S., On counterexamples to the Kodaira vanishing theorem and the Yau inequality in positive characteristic (in Japanese), Symposium on Algebraic Geometry, Kinosaki, 1979, 9–23.

[9] Oort, F., Finite group schemes, local moduli for abelian varieties and lifting problems, Proc. 5th Nordic Summer School, Walter-Noordhoff, Groningen, 1970, 223–254.

[10] Ramanujam C. P.. Remarks on the Kodaira vanishing theorem. J. Indian. Math. Soc., 1972, 36: 41-51

[11] Raynaud M.. Contre-exemple au “vanishing theorem” en caractéristique p > 0, C. P. Ramanujam-A tribute. Studies in Math., 1978, 8: 273-278

[12] Serre J.-P.. Corps Locaux, 1962, Paris: Hermann

[13] Tango H.. On the behavior of extensions of vector bundles under the Frobenius map. Nagoya Math. J., 1972, 48: 73-89

[14] Tango H.. On the behavior of cohomology classes of vector bundles under the Frobenius map (in Japanese). Res. Inst. Math. Sci., Kôkyûroku, 1972, 144: 93-102

[15] Terakawa H.. The d-very ampleness on a projective surface in positive characteristic. Pacific J. Math., 1999, 187: 187-199

[16] Xie Q. H.. Counterexamples to the Kawamata-Viehweg vanishing on ruled surfaces in positive characteristic. J. Algebra, 2010, 324: 3494-3506

[17] Xie Q. H.. Kawamata-Viehweg vanishing on rational surfaces in positive characteristic. Math. Zeit., 2010, 266: 561-570

[18] Xie Q. H.. Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic. Math. Res. Lett., 2010, 17: 563-572