form. If $T_{\min}$ is Kähler, then so is ${\varepsilon}\theta+(1-{\varepsilon})T_{\min}=\theta+dd^c(1-{\varepsilon})\varphi$ for ${\varepsilon}>0$ small enough. We therefore get $\varphi\preceq(1-{\varepsilon})\varphi$ by minimality, that is: $(1-{\varepsilon})\varphi\leq\varphi+C$ for some constant $C$. But this shows that $\varphi$ is bounded, and thus $T_{\min}$ is a Kähler current with identically zero Lelong numbers. Using (i) of theorem 2.1, we can therefore regularize it into a Kähler form inside its cohomology class, qed.\
Finally, we remark that a positive current with minimal singularities in a pseudo-effective class is generally non-unique (as a current), as the example of a Kähler class already shows.

The divisorial Zariski decomposition
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In this section $X$ denotes a compact complex $n$-fold, and $\omega$ is a reference Hermitian form, unless otherwise specified.

Minimal multiplicities and non-nef locus
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When $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is a pseudo-effective class, we want to introduce minimal multiplicities $\nu(\alpha,x)$, which measure the obstruction to the numerical effectivity of $\alpha$. For each ${\varepsilon}>0$, let $T_{\min,{\varepsilon}}=T_{\min,{\varepsilon}}(\alpha)$ be a current with minimal singularities in $\alpha[-{\varepsilon}\omega]$ (cf. section 2.8 for the notation). We then introduce the following

The minimal multiplicity at $x\in X$ of the pseudo-effective class $\alpha\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is defined as $$\nu(\alpha,x):=\sup_{{\varepsilon}>0}\nu(T_{\min,{\varepsilon}},x).$$

The commensurability of any two Hermitian forms shows that the definition does not depend on 