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Threefolds with vanishing Hodge cohomology. (English) Zbl 1067.14033
Summary: 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 $$Y$$.

##### MSC:
 14J30 $$3$$-folds 14B15 Local cohomology and algebraic geometry 14F40 de Rham cohomology and algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
local cohomology; fibration; higher direct images
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##### References:
 [1] Shreeram Shankar Abhyankar, Local analytic geometry, Pure and Applied Mathematics, Vol. XIV, Academic Press, New York-London, 1964. · Zbl 0205.50401 [2] Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. · Zbl 0215.37201 [3] Arapura, D., Complex Algebraic Varieties and their Cohomology, Lecture Notes, 2003. [4] Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485 – 496. · Zbl 0105.14404 · doi:10.2307/2372985 · doi.org [5] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601 [6] Nicolas Bourbaki, Commutative algebra. Chapters 1 – 7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. · Zbl 0666.13001 [7] Nicolas Bourbaki, General topology. Chapters 1 – 4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. Nicolas Bourbaki, General topology. Chapters 5 – 10, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. [8] Steven D. Cutkosky, Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53 (1986), no. 1, 149 – 156. · Zbl 0604.14002 · doi:10.1215/S0012-7094-86-05309-3 · doi.org [9] Robert Friedman and Zhenbo Qin, On complex surfaces diffeomorphic to rational surfaces, Invent. Math. 120 (1995), no. 1, 81 – 117. · Zbl 0823.14022 · doi:10.1007/BF01241123 · doi.org [10] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95 – 103. · Zbl 0145.17602 [11] Jacob Goodman and Robin Hartshorne, Schemes with finite-dimensional cohomology groups, Amer. J. Math. 91 (1969), 258 – 266. · Zbl 0176.18303 · doi:10.2307/2373281 · doi.org [12] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. · Zbl 0836.14001 [13] Hartshorne, R., Algebraic Geometry, Springer-Verlag, 1997. [14] Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. · Zbl 0208.48901 [15] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. · Zbl 0237.14008 [16] Robin Hartshorne, On the De Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5 – 99. · Zbl 0326.14004 [17] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0138.42001 [18] Hodge, W. V. D. and Pedoe, D., Methods of Algebraic Geometry, II, Cambridge University Press, 1952. · Zbl 0048.14502 [19] Shigeru Iitaka, Birational geometry for open varieties, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 76, Presses de l’Université de Montréal, Montreal, Que., 1981. · Zbl 0491.14005 [20] Iitaka, S., Birational Geometry of Algebraic Varieties, ICM, 1983. · Zbl 0587.14020 [21] Shigeru Iitaka, Birational geometry and logarithmic forms, Recent progress of algebraic geometry in Japan, North-Holland Math. Stud., vol. 73, North-Holland, Amsterdam, 1983, pp. 1 – 27. · Zbl 0514.14006 · doi:10.1016/S0304-0208(09)70004-0 · doi.org [22] Shigeru Iitaka, Deformations of compact complex surfaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 267 – 272. [23] Shigeru Iitaka, Deformations of compact complex surfaces. II, J. Math. Soc. Japan 22 (1970), 247 – 261. · Zbl 0188.53401 · doi:10.2969/jmsj/02220247 · doi.org [24] Shigeru Iitaka, Deformations of compact complex surfaces. III, J. Math. Soc. Japan 23 (1971), 692 – 705. · Zbl 0219.32012 · doi:10.2969/jmsj/02340692 · doi.org [25] Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175 – 232. · Zbl 0221.14007 [26] Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253 – 276. · Zbl 0471.14022 [27] Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982), no. 1, 57 – 71. · Zbl 0461.14004 · doi:10.1007/BF01404756 · doi.org [28] Yujiro Kawamata, Addition formula of logarithmic Kodaira dimensions for morphisms of relative dimension one, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 207 – 217. · Zbl 0437.14018 [29] Yujiro Kawamata, On the extension problem of pluricanonical forms, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 193 – 207. · Zbl 0972.14005 · doi:10.1090/conm/241/03636 · doi.org [30] Yujiro Kawamata, Deformations of canonical singularities, J. Amer. Math. Soc. 12 (1999), no. 1, 85 – 92. · Zbl 0906.14001 [31] Steven L. Kleiman, On the vanishing of \?$$^{n}$$(\?,\cal\?) for an \?-dimensional variety, Proc. Amer. Math. Soc. 18 (1967), 940 – 944. · Zbl 0165.24001 [32] János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11 – 42. · Zbl 0598.14015 · doi:10.2307/1971351 · doi.org [33] János Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171 – 202. · Zbl 0605.14014 · doi:10.2307/1971390 · doi.org [34] János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. · Zbl 0926.14003 [35] N. Mohan Kumar, Affine-like surfaces, J. Algebraic Geom. 2 (1993), no. 4, 689 – 703. · Zbl 0810.14015 [36] N. Mohan Kumar and M. Pavaman Murthy, Algebraic cycles and vector bundles over affine three-folds, Ann. of Math. (2) 116 (1982), no. 3, 579 – 591. · Zbl 0519.14009 · doi:10.2307/2007024 · doi.org [37] Tie Luo, Global 2-forms on regular 3-folds of general type, Duke Math. J. 71 (1993), no. 3, 859 – 869. · Zbl 0838.14032 · doi:10.1215/S0012-7094-93-07132-3 · doi.org [38] Tie Luo, Global holomorphic 2-forms and pluricanonical systems on threefolds, Math. Ann. 318 (2000), no. 4, 707 – 730. · Zbl 1005.14016 · doi:10.1007/s002080000136 · doi.org [39] Luo, Tie; Zhang, Qi, Holomorphic forms on threefolds, preprint, 2003. · Zbl 1216.14013 [40] Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. · Zbl 0988.14007 [41] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. · Zbl 0441.13001 [42] Masayoshi Miyanishi, Noncomplete algebraic surfaces, Lecture Notes in Mathematics, vol. 857, Springer-Verlag, Berlin-New York, 1981. · Zbl 0456.14018 [43] Shigefumi Mori, Birational classification of algebraic threefolds, ICM-90, Mathematical Society of Japan, Tokyo; distributed outside Asia by the American Mathematical Society, Providence, RI, 1990. A plenary address presented at the International Congress of Mathematicians held in Kyoto, August 1990. · Zbl 0751.14026 [44] Shigefumi Mori, Birational classification of algebraic threefolds, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 1 – 17. · Zbl 0840.14006 [45] 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. · Zbl 0223.14022 [46] Masayoshi Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 1 – 10. · Zbl 0109.39503 [47] Noboru Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, Topology 25 (1986), no. 2, 237 – 251. · Zbl 0596.14026 · doi:10.1016/0040-9383(86)90042-X · doi.org [48] Nakayama, N., Zariski decomposition and abundance, RIMS preprint (June 1997). [49] Nakayama, N., Invariance of the plurigenera of algebraic varieties, RIMS preprint (March 1998). [50] François Norguet and Yum Tong Siu, Holomorphic convexity of spaces of analytic cycles, Bull. Soc. Math. France 105 (1977), no. 2, 191 – 223 (English, with French summary). · Zbl 0382.32010 [51] Thomas Peternell, Hodge-Kohomologie und Steinsche Mannigfaltigkeiten, Complex analysis (Wuppertal, 1991) Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991, pp. 235 – 246 (German). · Zbl 0736.32012 [52] J. H. Sampson and G. Washnitzer, A Künneth formula for coherent algebraic sheaves, Illinois J. Math. 3 (1959), 389 – 402. · Zbl 0088.39402 [53] Serre, J. P., Quelques problèmes globaux relatifs aus variétés deStein, Collected Papers, Vol.1, Springer-Verlag(1985), 259-270. [54] Shafarevich, I. R., Basic Algebraic Geometry 1, 2, Springer-Verlag, 1994. · Zbl 0797.14001 [55] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105 – 203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95 – 202. · Zbl 0785.14023 · doi:10.1070/IM1993v040n01ABEH001862 · doi.org [56] V. V. Shokurov, 3-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667 – 2699. Algebraic geometry, 4. · Zbl 0873.14014 · doi:10.1007/BF02362335 · doi.org [58] Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661 – 673. · Zbl 0955.32017 · doi:10.1007/s002220050276 · doi.org [59] Kenji Ueno, Algebraic geometry. 1, Translations of Mathematical Monographs, vol. 185, American Mathematical Society, Providence, RI, 1999. From algebraic varieties to schemes; Translated from the 1997 Japanese original by Goro Kato; Iwanami Series in Modern Mathematics. Kenji Ueno, Algebraic geometry. 2, Translations of Mathematical Monographs, vol. 197, American Mathematical Society, Providence, RI, 2001. Sheaves and cohomology; Translated from the 1997 Japanese original by Goro Kato; Iwanami Series in Modern Mathematics. [60] Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. · Zbl 0299.14007 [61] Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329 – 353. · Zbl 0513.14019 [62] Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560 – 615. · Zbl 0124.37001 · doi:10.2307/1970376 · doi.org [63] Qi Zhang, Global holomorphic one-forms on projective manifolds with ample canonical bundles, J. Algebraic Geom. 6 (1997), no. 4, 777 – 787. · Zbl 0922.14008
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