Effective Iitaka fibrations

Authors:
Eckart Viehweg and De-Qi Zhang

Journal:
J. Algebraic Geom. **18** (2009), 711-730

DOI:
https://doi.org/10.1090/S1056-3911-09-00515-3

Published electronically:
March 23, 2009

MathSciNet review:
2524596

Full-text PDF

Abstract | References | Additional Information

Abstract: For every -dimensional projective manifold of Kodaira dimension we show that is birational to an Iitaka fibration for a *computable* positive integer , where is minimal with for a general fibre of an Iitaka fibration of , and where is the Betti number of a smooth model of the canonical -cover of the -fold . In particular, is a universal constant if the dimension .

**[AM]**V. Alexeev and S. Mori, Bounding singular surfaces of general type. Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000) 143-174, Springer, Berlin, 2004. (also available at: http://www.math.princeton.edu/ kollar/) MR**2037085 (2005f:14077)****[Ar]**C. Araujo, The cone of effective divisors of log varieties, after Batyrev. Preprint, 2005, arXiv:math/0502174.**[Ba]**V. V. Batyrev, The cone of effective divisors of threefolds. Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), Contemp. Math.**131**(1992) 337-352. MR**1175891 (94f:14035)****[BCHM]**C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type. Preprint, 2006, arXiv:math/0610203.**[8aut]**T. Bauer, F. Campana, T. Eckl, S. Kebekus, T. Peternell, S. Rams, T. Szemberg and L. Wotzlaw, A reduction map for nef line bundles. Complex geometry (Göttingen, 2000) 27-36, Springer, Berlin, 2002. MR**1922095 (2003h:14023)****[CC]**J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type. preprint, 2007, arXiv:0706.2987.**[FM]**O. Fujino and S. Mori, A canonical bundle formula. J. Differential Geom. 56 (2000) 167-188. MR**1863025 (2002h:14091)****[GZ]**R. V. Gurjar and D. -Q. Zhang, of smooth points of a log del Pezzo surface is finite. I. J. Math. Sci. Univ. Tokyo 1 (1994) 137-180. MR**1298542 (95m:14015)****[HM]**C. D. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166 (2006) 1-25. MR**2242631 (2007e:14022)****[Ii]**S. Iitaka, Deformations of compact complex surfaces. II. J. Math. Soc. Japan 22 (1970) 247-261. MR**0261639 (41:6252)****[KU]**T. Katsura and K. Ueno, On elliptic surfaces in characteristic . Math. Ann. 272 (1985) 291-330. MR**799664 (87g:14040)****[Ka]**Y. Kawamata, On effective non-vanishing and base-point-freeness. Kodaira's issue, Asian J. Math. 4 (2000) 173-181. MR**1802918 (2002b:14010)****[KMM]**Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem.*Algebraic geometry, Sendai, 1985*(T. Oda ed.) 283-360. Adv. Stud. Pure Math.,**10**, Kinokuniya and North-Holland, 1987. MR**946243 (89e:14015)****[Ko86]**J. Kollár, Higher direct images of dualizing sheaves. I. Ann. of Math. 123 (1986) 11-42. MR**825838 (87c:14038)****[Ko94]**J. Kollár, Log surfaces of general type; some conjectures. Classification of algebraic varieties (L'Aquila, 1992) 261-275, Contemp. Math., 162, Amer. Math. Soc., Providence, RI, 1994. MR**1272703 (95c:14042)****[KM]**J. Kollár and S. Mori,*Birational geometry of algebraic varieties*. Cambridge Tracts in Mathematics,**134**, Cambridge University Press, 1998. MR**1658959 (2000b:14018)****[Ko]**J. Kollár et al, Flips and abundance for algebraic threefolds. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991. Asterisque No. 211 (1992). Soc. Mathamatique de France, Paris, 1992. MR**1225842 (94f:14013)****[La]**A. Langer, Adjoint linear systems on normal log surfaces. Compositio Math. 129 (2001) 47-66. MR**1856022 (2002h:14008)****[Mo]**S. Mori, Classification of higher-dimensional varieties. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) 269-331, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. MR**927961 (89a:14040)****[Na]**N. Nakayama, On Weierstrass models. Algebraic geometry and commutative algebra, Vol. II, 405-431, Kinokuniya, Tokyo, 1988. MR**977771 (90m:14030)****[Pa]**G. Pacienza, On the uniformity of the Iitaka fibration. preprint, 2007, arXiv:0709.0310.**[Ri]**A. Ringler, On a conjecture of Hacon and McKernan in dimension three. preprint, 2007, arXiv:0708.3662.**[Ta]**S. Takayama, Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165 (2006) 551 - 587. MR**2242627 (2007m:14014)****[Vi83]**E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces. Algebraic varieties and analytic varieties (Tokyo, 1981) 329-353, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983. MR**715656 (85b:14041)****[Vi06]**E. Viehweg: Compactifications of smooth families and of moduli spaces of polarized manifolds. Ann. of Math., to appear, arXiv:math/0605093.**[Z]**D. -Q. Zhang, Logarithmic del Pezzo surfaces of rank one with contractible boundaries. Osaka J. Math. 25 (1988) 461-497. MR**957874 (89k:14059)**

Additional Information

**Eckart Viehweg**

Affiliation:
Universität Duisburg-Essen, Fachbereich Mathematik, 45117 Essen, Germany

Email:
viehweg@uni-due.de

**De-Qi Zhang**

Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore

Email:
matzdq@nus.edu.sg

DOI:
https://doi.org/10.1090/S1056-3911-09-00515-3

Received by editor(s):
June 14, 2007

Received by editor(s) in revised form:
February 23, 2008

Published electronically:
March 23, 2009

Additional Notes:
This work has been supported by the DFG-Leibniz program and by the SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”. The second author is partially supported by an academic research fund of NUS