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

Sébastien Boucksom (1) , Jean-Pierre Demailly (2) , Mihai Paun (3, 4) , Thomas Peternell (5)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 IF - Institut Fourier
3 IECL - Institut Élie Cartan de Lorraine
4 IRMA - Institut de Recherche Mathématique Avancée
5 Mathematisches Institut [Bayreuth]

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.

Domaines: Géométrie algébrique [math.AG]

Sébastien Boucksom, Jean-Pierre Demailly, Mihai Paun, Thomas Peternell. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. Journal of Algebraic Geometry, 2013, 22 (2), pp.201-248. ⟨10.1090/S1056-3911-2012-00574-8⟩. ⟨hal-01289055⟩