The pseudo-effective cone of a compact K\"ahler manifold and varieties   of negative Kodaira dimension
  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. 
S\'ebastien Boucksom (Paris), Jean-Pierre Demailly (Grenoble), Mihai   Paun (Strasbourg), Thomas Peternell (Bayreuth)