$T$ can also be constructed, and writes $T=R+\sum\nu(T,D)[D]$, where $D$ ranges over the prime divisors of $X$, and $R$ is a closed almost positive $(1,1)$-current. In fact, we have $R\geq\gamma$ as soon as $T\geq\gamma$ for a smooth form $\gamma$.\

### Pull-back of a current

When $f:Y\to X$ is a $surjective$ holomorphic map between compact complex manifolds and $T$ is a closed almost positive $(1,1)$-current on $X$, it is possible to define its pull back $f^{\star}T$ by $f$ using the analogue of local equations for divisors: write $T=\theta+i{\partial\overline{\partial}}\varphi$ for some smooth form $\theta\in\{T\}$. $\varphi$ is then an almost pluri-subharmonic function, thus locally a pluri-subharmonic function modulo $\mathcal{C}^{\infty}$. One defines $f^{\star}T$ to be $f^{\star}\theta+i{\partial\overline{\partial}}f^{\star}\varphi$, as this is easily seen to be independent of the choices made. Of course, we then have $\{f^{\star}T\}=f^{\star}\{T\}$.

### Gauduchon metrics and compactness

On any compact complex $n$-fold $X$, there exists a Hermitian metric $\omega$ such that $\omega^{n-1}$ is ${\partial\overline{\partial}}$-closed. Such a metric is called a Gauduchon metric. As a consequence, for every smooth real $(1,1)$-form $\gamma$, the quotient map $T\mapsto\{T\}$ from the set of closed $(1,1)$-currents $T$ with $T\geq\gamma$ to ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is proper. Indeed, the mass of the positive current $T-\gamma$ is controled by $\int(T-\gamma)\wedge\omega^{n-1}$, and $\int 