Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T00:56:18.008Z Has data issue: false hasContentIssue false
  • English
  • Français

The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base

Part of: Families, fibrations Abelian varieties and schemes Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  26 April 2016

Behrouz Taji
Behrouz Taji*
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1031, 805 Sherbrooke Street West, Montreal, QC, CanadaH3A 0B9 email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.

Keywords

familiesisotrivialitymoduliminimal model programspecial varietiesKodaira dimension

MSC classification

Primary: 14D22: Fine and coarse moduli spaces 14D23: Stacks and moduli problems 14E30: Minimal model program (Mori theory, extremal rays) 14J10: Families, moduli, classification 14K10: Algebraic moduli, classification
Secondary: 14K12: Subvarieties
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 7 , July 2016 , pp. 1421 - 1434
Copyright
© The Author 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Birkar, C., Cascini, P., Hacon, C. D. and Mckernan, J., Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), 405–468.CrossRefGoogle Scholar
Boucksom, S., Demailly, J.-P., Păun, M. and Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension , J. Algebraic Geom. 22 (2013), 201–248, doi:10.1090/S1056-3911-2012-00574-8.Google Scholar
Campana, F., Orbifolds, special varieties and classification theory , Ann. Inst. Fourier (Grenoble) 54 (2004), 499–630.Google Scholar
Campana, F., Orbifoldes spéciales et classification biméromorphe des variétés Kählériennes compactes , J. Inst. Math. Jussieu 10 (2011), 809–934.Google Scholar
Campana, F. and Păun, M., Orbifold generic semi-positivity: an application to families of canonically polarized families, version 4 , Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861, doi:10.5802/aif.2945.Google Scholar
Iitaka, S., Algebraic geometry, Graduate Texts in Mathematics, vol. 76 (Springer, New York, 1982).Google Scholar
Jabbusch, K. and Kebekus, S., Positive sheaves of differentials coming from coarse moduli spaces , Ann. Inst. Fourier (Grenoble) 61 (2011), 2277–2290.Google Scholar
Jabbusch, K. and Kebekus, S., Families over special base manifolds and a conjecture of Campana , Math. Z. 269 (2011), 847–878.Google Scholar
Kebekus, S. and Kovács, S., The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties , Duke Math. J. 155 (2010), 1–33.Google Scholar
Lazarsfeld, R., Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 48 (Springer, Berlin, 2004).Google Scholar
Lu, S. S. Y., A refined Kodaira dimension and its canonical fibration, Preprint (2002),arXiv:math/0211029.Google Scholar
Miyaoka, Y., The Chern classes and Kodaira dimension of a minimal variety , in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 449–476; MR 89k:14022.Google Scholar
Miyaoka, Y., Deformations of a morphism along a foliation and applications , in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 245–268; MR 927960 (MR 89e:14011).Google Scholar
Viehweg, E., Weal 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.Google Scholar
Viehweg, E., Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 30 (Springer, Berlin, 1995).Google Scholar
Viehweg, E. and Zuo, K., Base spaces of non-isotrivial families of smooth minimal models , in Complex geometry (Göttingen, 2000) (Springer, Berlin, 2002), 279–328.Google Scholar