The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension

TL;DR

This paper characterizes pseudo-effective line bundles on projective manifolds via their degrees on covering families of curves, establishes a duality between pseudo-effective divisors and movable curves, and relates these to the Kodaira dimension and uniruledness.

Contribution

It introduces a duality between pseudo-effective divisors and movable curves, linking geometric properties to positivity conditions and Kodaira dimension.

Findings

A line bundle is pseudo-effective iff its degree on all covering curves is non-negative.

A projective manifold has pseudo-effective canonical bundle iff it is not uniruled.

A 4-fold with pseudo-effective canonical bundle and zero class on covering curves has non-negative Kodaira dimension.

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.