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

Sébastien Boucksom·Jean-Pierre Demailly·Mihai Paun·Thomas Peternell
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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.

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Sébastien Boucksom, Jean-Pierre Demailly, Mihai Paun, Thomas Peternell. 2004-05-14. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. https://arxiv.org/abs/math/0405285

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