The structure of projective algebraic manifolds is to a large extent governed by the geometry of its cones of divisors or curves. In the case of divisors, two cones are of primary importance: the cone of ample divisors and the cone of effective divisors (and the closure of these cones as well). These cones have natural transcendental analogues on any compact Kähler manifold, namely the cone of Kähler classes (called the Kähler cone) and the cone of pseudoeffective (1,1)-classes (called the pseudoeffective cone).
A conjecture of Boucksom-Demailly-Paun-Peternell says that the pseudoeffective cone is dual to the cone of movable classes. I will discuss my recent proof of the conjecture in the case when the manifold is projective