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The Iitaka conjecture Cn,m in dimension six

Part of: Birational geometry

Published online by Cambridge University Press:  27 July 2009

Caucher Birkar
Caucher Birkar*
Affiliation:
DPMMS, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WB, UK (email: c.birkar@dpmms.cam.ac.uk)
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Abstract

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We prove that the Iitaka conjecture Cn,m for algebraic fibre spaces holds up to dimension six, that is, when n≤6.

Keywords

Iitaka conjectureminimal models

MSC classification

Secondary: 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009 , pp. 1442 - 1446
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Ambro, F., Nef dimension of minimal models, Math. Ann. 330(2) (2004), 309–322.CrossRefGoogle Scholar
[2]Bauer, T., Campana, F., Eckl, T., Kebekus, S., Peternell, T., Rams, S., Szemberg, T. and Wotzlaw, L., A reduction map for nef line bundles, in Complex geometry: collection of papers dedicated to Hans Grauert (Gottingen, 2000) (Springer, Berlin, 2002), 27–36.CrossRefGoogle Scholar
[3]Birkar, C., On existence of log minimal models, arXiv:0706.1792v2 (2008).Google Scholar
[4]Birkar, C., Cascini, P., Hacon, C. and McKernan, J., Existence of minimal models for varieties of log general type, arXiv:math/0610203v2 (2008).CrossRefGoogle Scholar
[5]Campana, F. and Peternell, T., Geometric stability of the cotangent bundle and the universal cover of a projective manifold (with an appendix by M. Toma), arXiv:math/0405093v4 (2007).Google Scholar
[6]Chen, J. A. and Hacon, C., On Ueno’s conjecture K, Math. Ann., to appear.Google Scholar
[7]Fujino, O. and Mori, S., A canonical bundle formula, J. Differential Geom. 56 (2000), 167–188.CrossRefGoogle Scholar
[8]Kawamata, Y., Characterization of abelian varieties, Compositio Math. 43 (1981), 253–276.Google Scholar
[9]Kawamata, Y., Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66(1) (1982), 57–71.CrossRefGoogle Scholar
[10]Kawamata, Y., Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46.Google Scholar
[11]Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), 567–588.CrossRefGoogle Scholar
[12]Kawamata, Y., Abundance theorem for minimal threefolds, Invent. Math. 108(2) (1992), 229–246.CrossRefGoogle Scholar
[13]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry (Sendai, 1985), Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283–360.CrossRefGoogle Scholar
[14]Kollár, J., Subadditivity of the Kodaira dimension: fibers of general type, in Algebraic geometry (Sendai, 1985), Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 361–398.CrossRefGoogle Scholar
[15]Miyaoka, Y., Abundance conjecture for 3-folds: case ν=1, Compositio Math. 68(2) (1988), 203–220.Google Scholar
[16]Mori, S., Classification of higher-dimensional varieties, in Algebraic geometry (Bowdoin, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46, part 1 (American Mathematical Society, Providence, RI, 1987), 269–331.CrossRefGoogle Scholar
[17]Shokurov, V. V., Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), 82–219; Proc. Steklov Inst. Math. 240 (2003), 75–213 (English transl.).Google Scholar
[18]Tsuji, H., Numerical trivial fibrations, arXiv:math/0001023v6 (2000).Google Scholar
[19]Viehweg, E., Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, in Algebraic varieties and analytic varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, vol. 1 (North-Holland, Amsterdam, 1983), 329–353.CrossRefGoogle Scholar