We prove a conjecture of Boucksom-Demailly-Păun-Peternell, namely that on a projective manifold
X
X
the cone of pseudoeffective classes in
H
R
1
,
1
(
X
)
H^{1,1}_{\mathbb {R}}(X)
is dual to the cone of movable classes in
H
R
n
−
1
,
n
−
1
(
X
)
H^{n-1,n-1}_{\mathbb {R}}(X)
via the Poincaré 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-Kähler locus of a big class via intersection numbers.