small enough. Finally, $\alpha$ is a Kähler class iff it contains a Kähler form (note that a smooth Kähler current is the same thing as a Kähler form).\
Since any two Hermitian forms $\omega_1$, $\omega_2$ are commensurable (i.e. $C^{-1}\omega_2\leq\omega_1\leq C\omega_2$ for some $C>0$), these definitions do not depend on the choice of $\omega$.\
The set of pseudo-effective classes is a closed convex cone ${\mathcal{E}}\subset{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$, called the pseudo-effective cone. It has compact base, because so is the case of the cone of closed positive $(1,1)$-currents. Similarly, one defines the nef cone ${\mathcal{N}}$ (a closed convex cone), the big cone ${\mathcal{B}}$ (an open convex cone), and the Kähler cone ${\mathcal{K}}$ (an open convex cone). We obviously have the inclusions $${\mathcal{K}}\subset{\mathcal{B}}\subset{\mathcal{E}}$$ and $${\mathcal{K}}\subset{\mathcal{N}}\subset{\mathcal{E}}.$$ By definition, $X$ is a Kähler manifold iff its Kähler cone ${\mathcal{K}}$ is non-empty. Similarly (but this is a theorem, cf. \[DP01\]) $X$ is a Fujiki manifold (i.e. bimeromorphic to a Kähler manifold) iff its big cone ${\mathcal{B}}$ is non-empty (see also the proof of proposition 2.3 below). If $X$ is Kähler, ${\mathcal{K}}$ is trivially the interior ${\mathcal{N}}^0$ of the nef cone. Similarly, if $X$ is Fujiki, ${\mathcal{B}}$ is trivially the interior ${\mathcal{E}}^0$ of the pseudo-effective cone.\
We will now 