identically $-\infty$ on $D$. Thus the restriction $(\varphi_{{\varepsilon}})_{|D}$ is pluri-subharmonic, and $(T_{{\varepsilon}}+{\varepsilon}Cdd^c|z|^2)_{|D}$ is a well defined closed positive current. It follows that $(T_{{\varepsilon}})_{|D}$ is a well defined almost positive current on $D$, with $(T_{{\varepsilon}})_{|D}\geq-{\varepsilon}C\omega_{|D}$. This certainly implies that $\alpha_{|D}$ is pseudo-effective. The case $\alpha\in{\mathcal{MK}}$ is treated similarly.

Currents with minimal singularities
-----------------------------------

Let $\varphi_1$, $\varphi_2$ be two almost pluri-subharmonic functions on a compact complex manifold $X$. Then, following \[DPS00\], we say that $\varphi_1$ is less singular than $\varphi_2$ (and write $\varphi_1\preceq\varphi_2$) if we have $\varphi_2\leq\varphi_1+C$ for some constant $C$. We denote by $\varphi_1\approx\varphi_2$ the equivalence relation generated by the pre-order relation $\preceq$. Note that $\varphi_1\approx\varphi_2$ exactly means that $\varphi_1=\varphi_2$ mod $L^{\infty}$.\
When $T_1$ and $T_2$ are two closed almost positive $(1,1)$-currents on $X$, we can also compare their singularities in the following fashion: write $T_i=\theta_i+dd^c\varphi_i$ for $\theta_i\in\{T_j\}$ a smooth form and $\varphi_i$ an almost pluri-subharmonic function. Since any $L^1_{loc}$ function $f$ with $dd^cf$ smooth is itself smooth, it is easy to check that $\varphi_i$ does not depend on the choices made up to equivalence of singularities, and we compare the singularities of the $T_i$’s by comparing those of the $\varphi_i$’s.\
Let now $\alpha$ be a class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ and $\gamma$ 