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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Threefolds with vanishing Hodge cohomology

Author: Jing Zhang
Journal: Trans. Amer. Math. Soc. 357 (2005), 1977-1994
MSC (2000): Primary 14J30, 14B15, 14C20
Published electronically: October 7, 2004
MathSciNet review: 2115086
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Abstract: We consider algebraic manifolds $Y$ of dimension 3 over $\mathbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$to $C$ such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\infty$ and the $D$-dimension of $X$ is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of $Y$.

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  • [Ab] Abhyankar, Shreeram S., Local analytic geometry, Pure and Applied Mathematics, vol. 14, Academic Press, 1964. MR 31:173
  • [AK] Altman, A.; Kleiman, S., Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics, Springer-Verlag, 1970. MR 43:224
  • [Ara] Arapura, D., Complex Algebraic Varieties and their Cohomology, Lecture Notes, 2003.
  • [Art] Artin, M., Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math. 84(1962), 485-496. MR 26:3704
  • [AtM] Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass. (1969). MR 39:4129
  • [B1] Bourbaki, N., Commutative Algebra, Springer-Verlag, 1989. MR 90a:13001
  • [B2] Bourbaki, N., General Topology, Chapters 5-10, Springer-Verlag, 1989. MR 90a:54001b
  • [C] Cutkosky, S. D., Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53(1986), no. 1, 149-156. MR 87f:14004
  • [FQ] Friedman, Robert; Qin, Zhenbo On Complex surfaces diffeomorphic to rational surfaces. Invent. Math. 120(1995), no.1, 81-117. MR 96d:57032
  • [G] Grothendieck, A., On the De Rham cohomology of algebraic varieties, Inst. Hautes Etudes Sci. Publ. Math. 29(1966), 95-103. MR 33:7343
  • [GH] Goodman, J., Hartshorne, R., Schemes with finite-dimensional cohomology groups, American Journal of Mathematics, v. 91, Issue 1, 258-266, 1969. MR 39:2772
  • [GrH] Griffiths, P. and Harris, J., Principals of Algebraic Geometry, John Wiley $\&$ Sons, Inc., 1994. MR 95d:14001
  • [H1] Hartshorne, R., Algebraic Geometry, Springer-Verlag, 1997.
  • [H2] Hartshorne, R., Ample Subvarieties of Algebraic Varieties, Lecture Notes in Mathematics, 156, Springer-Verlag, 1970. MR 44:211
  • [H3] Hartshorne, R., Local Cohomology, Lecture Notes in Math., 41, Springer-Verlag, 1967. MR 37:219
  • [H4] Hartshorne, R., On the De Rham cohomology of algebraic varieties, Publ. Math. IHES 45(1976), 5-99. MR 55:5633
  • [Hi] Hirzebruch, F., Topological Methods in Algebraic Geometry, Springer-Verlag, 1966. MR 34:2573
  • [HP] Hodge, W. V. D. and Pedoe, D., Methods of Algebraic Geometry, II, Cambridge University Press, 1952. MR 13:972c
  • [I1] Iitaka, S., Birational Geometry for Open Varieties, Les Presses de l'Universite de Montreal, 1981. MR 83j:14011
  • [I2] Iitaka, S., Birational Geometry of Algebraic Varieties, ICM, 1983.
  • [I3] Iitaka, S., Birational geometry and logarithmic forms, Recent Progress of Algebraic Geometry in Japan, North-Holland Mathematics Studies 73, 1-27. MR 85g:14041
  • [I4] Iitaka, S., Deformation of compact complex surfaces I, Global Analysis, papers in honor of K. Kodaira, Princeton Univ. Press, 1969, 267-272. MR 40:8086
  • [I5] Iitaka, S., Deformation of compact complex surfaces II, J. Math. Soc. Japan 22, 1970, 247-261. MR 41:6252
  • [I6] Iitaka, S., Deformation of compact complex surfaces III, J. Math. Soc. Japan 23, 1971, 692-705. MR 44:7598
  • [K] Katz, Nicholas M., Nilpotent connection and the monodromy theorem: applications of a result of Turrittin, Publications Mathematiques, 39(1970), 175-232. MR 45:271
  • [Ka1] Kawamata, Y., Characterization of Abelian varieties, Comp. Math. 43(1981), 253-276. MR 83j:14029
  • [Ka2] Kawamata, Y., Kodaira dimension of algebraic fibre spaces over curves, Invent. Math. 66(1982), 57-71. MR 83h:14025
  • [Ka3] Kawamata, Y., Addition formula of logarithmic Kodaira dimension for morphisms of relative dimension one. Proc. Internat. Symp. on algebraic geometry at Kyoto (1977), 207-217. Tokyo: Kinokuniya, 1978. MR 82d:14019
  • [Ka4] Kawamata, Y., On the extension problem of pluricanonical forms, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 193-207, Contemp. Math., 241. Amer. Math. Soc., Providence, RI, 1999. MR 2000i:14053
  • [Ka5] Kawamata, Y., Deformations of canonical singularities. J. Amer. Math. Soc. 12(1999), no. 1, 85-92. MR 99g:14003
  • [Kl] Kleiman, S.L., On the vanishing of $H^n(X, F)$ for an $n$-dimensional variety, Proceedings of AMS, vol. 18, No. 5, 940-944, 1967. MR 35:4233
  • [Ko1] Kollár, J., Higher Direct Images of Dualizing Sheaves I, Ann. of Math., v.123, 1(1986), 11-42. MR 87c:14038
  • [Ko2] Kollár, J., Higher Direct Images of Dualizing Sheaves, II, Ann. of Math., v.124, 1(1986), 171-202. MR 87k:14014
  • [KM] Kollár, J., Mori, S., Birational Geometry of Algebraic Varieties, Cambridge University Press, 1998. MR 2000b:14018
  • [Ku] Kumar, N. Mohan, Affine-Like Surfaces, Journal of Algebraic Geometry, 2(1993), 689-703. MR 94i:14039
  • [KuM] Kumar, N. Mohan; Murthy, M. Pavaman Algebraic cycles and vector bundles over affine threefolds, Ann. of Math. (2), no. 3, 579-591. MR 84d:14006
  • [L1] Luo, Tie, Global 2-forms on regular 3-folds of general type, Duke Math. J. 71 (1993), no.3, 859-869. MR 94k:14032
  • [L2] Luo, Tie, Global holomorphic forms 2-forms and pluricanonical systems on threefolds, Math. Ann. 318 (2000), no. 4, 707-730. MR 2002a:14043
  • [LZ] Luo, Tie; Zhang, Qi, Holomorphic forms on threefolds, preprint, 2003.
  • [M] Matsuki, Kenji, Introduction to the Mori program, Universitext. Springer-Verlag, New York, 2002. MR 2002m:14011
  • [Ma] Matsumura, H., Commutative Algebra, Second edition, W. A. Benjamin Co., New York, 1980. MR 82i:13003
  • [Mi] Miyanishi, M., Non-complete Algebraic Surfaces, Lecture Notes in Mathematics, 857, Springer-Verlag, 1981. MR 83b:14011
  • [Mo1] Mori, S., Birational Classification of Algebraic Threefolds, ICM, 1990. MR 92m:14043
  • [Mo2] Mori, S., Birational Classification of Algebraic Threefolds, Algebraic geometry and related topics (Inchon, 1992), 1-17, Conf. Proc. Lecture Notes Algebraic Geom., I, Internat. Press, Cambridge, MA, 1993. MR 95f:14026
  • [Mu] Mumford, D., Abelian Varieties, Oxford University Press, 1970. MR 44:219
  • [N] Nagata, M., Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2(1962), 1-10. MR 26:118
  • [Na1] Nakayama, N., Invariance of the plurigenera of algebraic varieties, Topology 25 (1986), 237-251. MR 87g:14034
  • [Na2] Nakayama, N., Zariski decomposition and abundance, RIMS preprint (June 1997).
  • [Na3] Nakayama, N., Invariance of the plurigenera of algebraic varieties, RIMS preprint (March 1998).
  • [NS] Norguet, F.; Siu, Y.T. Holomorphic convexity of spaces of analytic cycles. Bull. Soc. Math. France 105, 191-223(1977). MR 58:28677
  • [P] Peternell, T., Hodge-kohomolodie und Steinsche Mannigfaltigkeiten, Complex Analysis, Aspects of Mathematics, Vieweg-Verlag, 1990, 235-246. MR 92j:32038
  • [SaW] Sampson, J. H., Washnitzer, G., A Künneth formula for coherent algebraic sheaves, Illinois J. Math. 3, 389-402(1959). MR 21:5636
  • [Se] Serre, J. P., Quelques problèmes globaux relatifs aus variétés deStein, Collected Papers, Vol.1, Springer-Verlag(1985), 259-270.
  • [Sh] Shafarevich, I. R., Basic Algebraic Geometry 1, 2, Springer-Verlag, 1994.
  • [Sho1] Shokurov, V. V., 3-fold log flips. Izv. Russ. A. N. Ser. Mat., 56: 105-203, 1992. MR 93j:14012
  • [Sho2] Shokurov, V. V., 3-fold log models. Algebraic geometry, 4. J. Math. Sci. 81(1996), no. 3, 2667-2699. MR 97i:14015
  • [Si1] Siu, Y.-T., Analytic sheaf cohomology of dimension $n$ of $n$-dimensional complex spaces. Trans. Amer. Math. Soc. 143, 77-94(1969). MR 40:5902
  • [Si2] Siu, Y.-T., Invariance of plurigenera, Invent. Math. 134, 661-673(1998). MR 99i:32035
  • [U1] Ueno, K., Algebraic Geometry 1, 2, AMS, 1999. MR 2000g:14001 MR 2001j:14001
  • [U2] Ueno, K., Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Mathematics, v.439, 1975, Springer-Verlag. MR 58:22062
  • [V] Viehweg, E., Weak positivity and the additivity of the Kodaira dimension certain fibre spaces, Adv. Studies Pure Math. 1(1983), 329-353. MR 85b:14041
  • [Z] Zariski, O., The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2), 76(1962), 560-616. MR 25:5065
  • [Zh] Zhang, Qi, Global holomorphic one-forms on projective manifolds with ample canonical bundles, J. Algebraic Geometry 6 (1997), 777-787. MR 99k:14071

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Additional Information

Jing Zhang
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Address at time of publication: Department of Mathematics, University of Missouri–Columbia, Columbia, Missouri 65211

Keywords: 3-folds, Hodge cohomology, local cohomology, fibration, higher direct images
Received by editor(s): May 9, 2003
Received by editor(s) in revised form: November 21, 2003
Published electronically: October 7, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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