The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

Boucksom, Sébastien; Demailly, Jean-Pierre; Păun, Mihai; Peternell, Thomas

doi:10.1090/S1056-3911-2012-00574-8

Authors: Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun and Thomas Peternell
Journal: J. Algebraic Geom. 22 (2013), 201-248
DOI: https://doi.org/10.1090/S1056-3911-2012-00574-8
Published electronically: May 31, 2012
MathSciNet review: 3019449
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Abstract | References | Additional Information

Abstract: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of “movable curves”, which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive $(1,1)$-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

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Additional Information

Sébastien Boucksom
Affiliation: Université de Paris VII, Institut de Mathématiques, 175 rue du Chevaleret, 75013 Paris, France
MR Author ID: 688226
Email: boucksom@math.jussieu.fr

Jean-Pierre Demailly
Affiliation: Université de Grenoble I, Institut Fourier, UMR 5582 du CNRS, BP 74, 100 rue des maths, 38402 Saint-Martin d’Hères, France
Email: demailly@fourier.ujf-grenoble.fr

Mihai Păun
Affiliation: Université de Strasbourg, Département de Mathématiques, 67084 Strasbourg, France
Address at time of publication: Université Henri Poincaré - Nancy I, Institut Élie Cartan, BP 239, F-54506 Vandœuvre lès Nancy, France
Email: mihai.paun@iecn.u-nancy.fr

Thomas Peternell
Affiliation: Universität Bayreuth, Mathematisches Institut, D-95440 Bayreuth, Deutschland
MR Author ID: 138450
Email: Thomas.Peternell@uni-bayreuth.de

Received by editor(s): April 29, 2010
Received by editor(s) in revised form: July 15, 2010
Published electronically: May 31, 2012
Additional Notes: The original version of the present paper was forwarded to arXiv (in May 2004), and has been revised several times since then. It had also been submitted to a journal in 2004, but then the submission was cancelled after the referee expressed concerns about certain parts of the paper, as they were written at that time. Although the results of sections 1-5 have been reproduced several times, e.g. in lecture notes of the second-named author or in Rob Lazarsfeld’s book \cite{La04}, a complete version never appeared in a refereed journal. We thank the Journal of Algebraic Geometry for suggesting to repair this omission.