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

2004/05/14 by Sébastien Boucksom, Jean-Pierre Demailly, Boucksom, Sébastien +5 · 9 citations
Mathematics · #14C17 #14C30 #32J27 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C17 #msc:14C30 #msc:32J27

paper · pdf · doi:10.48550/arxiv.math/0405285

39 pages

arxiv created 2004/05/14 · arxiv updated 2009/12/01

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 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 covering family, has non negative Kodaira dimension.

Cited by

• Holomorphic p-Contact and s-Symplectic Line Bundles
• Hyperbolicity of adjoint linear series on varieties with positive tangent bundle
• On the Kobayashi-Hitchin correspondence for Kähler currents
• Isoresidual fibrations and the birational geometry of moduli of pointed stable curves
• Generalized Bogomolov Inequalities
• Two Calabi-Yau theorems for degenerations of compact Kähler manifolds
• Degree two unirational parametrizations over the real field
• Singularity of non-pluripolar cohomology classes
• Nonvanishing results for Kähler varieties

Related

• Pseudo-effective line bundles on compact Kähler manifolds
• On compact Kähler manifold with strongly pseudo-effective tangent bundle
• Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds
• Geometric stability of the cotangent bundle and the universal cover of a projective manifold
• Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic