Threefolds with vanishing Hodge cohomology

Zhang, Jing

doi:10.1090/S0002-9947-04-03582-2

Threefolds with vanishing Hodge cohomology
HTML articles powered by AMS MathViewer

by Jing Zhang PDF

Trans. Amer. Math. Soc. 357 (2005), 1977-1994 Request permission

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 $Y$.

References

Shreeram Shankar Abhyankar, Local analytic geometry, Pure and Applied Mathematics, Vol. XIV, Academic Press, New York-London, 1964. MR 0175897
Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
Arapura, D., Complex Algebraic Varieties and their Cohomology, Lecture Notes, 2003.
Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 146182, DOI 10.2307/2372985
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
Nicolas Bourbaki, General topology. Chapters 1–4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. MR 979294, DOI 10.1007/978-3-642-61703-4
Steven D. Cutkosky, Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53 (1986), no. 1, 149–156. MR 835801, DOI 10.1215/S0012-7094-86-05309-3
Robert Friedman and Zhenbo Qin, On complex surfaces diffeomorphic to rational surfaces, Invent. Math. 120 (1995), no. 1, 81–117. MR 1323983, DOI 10.1007/BF01241123
A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103. MR 199194, DOI 10.1007/BF02684807
Jacob Goodman and Robin Hartshorne, Schemes with finite-dimensional cohomology groups, Amer. J. Math. 91 (1969), 258–266. MR 241432, DOI 10.2307/2373281
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
Hartshorne, R., Algebraic Geometry, Springer-Verlag, 1997.
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. MR 0282977, DOI 10.1007/BFb0067839
Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620, DOI 10.1007/BFb0073971
Robin Hartshorne, On the De Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5–99. MR 432647
F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
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. MR 647148
Iitaka, S., Birational Geometry of Algebraic Varieties, ICM, 1983.
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. MR 722140, DOI 10.1016/S0304-0208(09)70004-0
Shigeru Iitaka, Deformations of compact complex surfaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 267–272. MR 0254879
Shigeru Iitaka, Deformations of compact complex surfaces. II, J. Math. Soc. Japan 22 (1970), 247–261. MR 261639, DOI 10.2969/jmsj/02220247
Shigeru Iitaka, Deformations of compact complex surfaces. III, J. Math. Soc. Japan 23 (1971), 692–705. MR 290417, DOI 10.2969/jmsj/02340692
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. MR 291177, DOI 10.1007/BF02684688
Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982), no. 1, 57–71. MR 652646, DOI 10.1007/BF01404756
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. MR 578860
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. MR 1718145, DOI 10.1090/conm/241/03636
Yujiro Kawamata, Deformations of canonical singularities, J. Amer. Math. Soc. 12 (1999), no. 1, 85–92. MR 1631527, DOI 10.1090/S0894-0347-99-00285-4
Steven L. Kleiman, On the vanishing of $H^{n}(X,\,{\cal F})$ for an $n$-dimensional variety, Proc. Amer. Math. Soc. 18 (1967), 940–944. MR 213369, DOI 10.1090/S0002-9939-1967-0213369-9
János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42. MR 825838, DOI 10.2307/1971351
János Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171–202. MR 847955, DOI 10.2307/1971390
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. MR 1658959, DOI 10.1017/CBO9780511662560
N. Mohan Kumar, Affine-like surfaces, J. Algebraic Geom. 2 (1993), no. 4, 689–703. MR 1227473
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. MR 678482, DOI 10.2307/2007024
Tie Luo, Global $2$-forms on regular $3$-folds of general type, Duke Math. J. 71 (1993), no. 3, 859–869. MR 1240606, DOI 10.1215/S0012-7094-93-07132-3
Tie Luo, Global holomorphic 2-forms and pluricanonical systems on threefolds, Math. Ann. 318 (2000), no. 4, 707–730. MR 1802507, DOI 10.1007/s002080000136
Luo, Tie; Zhang, Qi, Holomorphic forms on threefolds, preprint, 2003.
Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR 1875410, DOI 10.1007/978-1-4757-5602-9
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Masayoshi Miyanishi, Noncomplete algebraic surfaces, Lecture Notes in Mathematics, vol. 857, Springer-Verlag, Berlin-New York, 1981. MR 635930, DOI 10.1007/BFb0090459
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. MR 1127162
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. MR 1285373
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
Masayoshi Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 1–10. MR 142549, DOI 10.1215/kjm/1250524969
Noboru Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, Topology 25 (1986), no. 2, 237–251. MR 837624, DOI 10.1016/0040-9383(86)90042-X
Nakayama, N., Zariski decomposition and abundance, RIMS preprint (June 1997).
Nakayama, N., Invariance of the plurigenera of algebraic varieties, RIMS preprint (March 1998).
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). MR 590090, DOI 10.24033/bsmf.1849
Thomas Peternell, Hodge-Kohomologie und Steinsche Mannigfaltigkeiten, Complex analysis (Wuppertal, 1991) Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991, pp. 235–246 (German). MR 1122185
J. H. Sampson and G. Washnitzer, A Künneth formula for coherent algebraic sheaves, Illinois J. Math. 3 (1959), 389–402. MR 106906, DOI 10.1215/ijm/1255455260
Serre, J. P., Quelques problèmes globaux relatifs aus variétés deStein, Collected Papers, Vol.1, Springer-Verlag(1985), 259-270.
Shafarevich, I. R., Basic Algebraic Geometry 1, 2, Springer-Verlag, 1994.
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. MR 1162635, DOI 10.1070/IM1993v040n01ABEH001862
V. V. Shokurov, $3$-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667–2699. Algebraic geometry, 4. MR 1420223, DOI 10.1007/BF02362335
Yum-tong Siu, Analytic sheaf cohomology groups of dimension $n$ of $n-dimensional$ $complex$ $spaces.$, Trans. Amer. Math. Soc. 143 (1969), 77–94. MR 252684, DOI 10.1090/S0002-9947-1969-0252684-6
Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673. MR 1660941, DOI 10.1007/s002220050276
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. MR 1714917, DOI 10.1090/mmono/185
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. MR 0506253, DOI 10.1007/BFb0070570
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. MR 715656, DOI 10.2969/aspm/00110329
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. MR 141668, DOI 10.2307/1970376
Qi Zhang, Global holomorphic one-forms on projective manifolds with ample canonical bundles, J. Algebraic Geom. 6 (1997), no. 4, 777–787. MR 1487236