T\wedge\omega^{n-1}=\{T\}\cdot\{\omega\}$ only depends on the class of $T$. The result follows by the weak compactness of positive currents with bounded mass. Another consequence is that the kernel of $T\mapsto\{T\}$ meets the cone of closed positive $(1,1)$-currents at the origin only.\

### Cycles as currents

One can associate to any effective $p$-codimensional ${\mathbf{R}}$-cycle $Y=a_1Y_1+...+a_rY_r$ a closed positive $(p,p)$-current $[Y]=a_1[Y_1]+...+a_r[Y_r]$, called the integration current on $Y$. The map $Y\mapsto[Y]$ so defined is injective, and a result of Thie says that the Lelong number $\nu([Y],x)$ is just the multiplicity of $Y$ at $x$. Consequently, we shall drop the brackets in $[Y]$ when no confusion is to be feared, and write for instance $T=R+\sum\nu(T,D)D$ for a Siu decomposition, because this is more in the spirit of this work.

Cones in the ${\partial\overline{\partial}}$-cohomology
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We now assume that $X$ is compact, and fix some reference Hermitian form $\omega$ (i.e. a smooth positive definite $(1,1)$-form). A cohomology class\
$\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is said to be pseudo-effective iff it contains a positive current;\
$\alpha$ is nef (numerically effective) iff, for each ${\varepsilon}>0$, $\alpha$ contains a smooth form $\theta_{{\varepsilon}}$ with $\theta_{{\varepsilon}}\geq-{\varepsilon}\omega$;\
$\alpha$ is big iff it contains a Kähler current, i.e. a closed $(1,1)$-current $T$ such that $T\geq{\varepsilon}\omega$ for ${\varepsilon}>0$ 