Duality between the pseudoeffective and the movable cone on a projective manifold

TL;DR

This paper proves a duality conjecture between pseudoeffective and movable cones on projective manifolds, establishing a transcendental Morse inequality and linking the movable cone to balanced metrics.

Contribution

It proves the duality conjecture between pseudoeffective and movable cones and establishes a transcendental Morse inequality for nef classes.

Findings

Duality between pseudoeffective and movable cones confirmed

Movable cone equals the closure of balanced metrics

Volume function is differentiable on the big cone

Abstract

We prove a conjecture of Boucksom-Demailly-P\u{a}un-Peternell, namely that on a projective manifold $X$ the cone of pseudoeffective classes in $H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in $H^{n-1,n-1}_{\mathbb{R}}(X)$ via the Poincar\'e pairing. This is done by establishing a conjectured transcendental Morse inequality for the volume of the difference of two nef classes on a projective manifold. As a corollary the movable cone is seen to be equal to the closure of the cone of balanced metrics. In an appendix by Boucksom it is shown that the Morse inequality also implies that the volume function is differentiable on the big cone, and one also gets a characterization of the prime divisors in the non-K\"ahler locus of a big class via intersection numbers.