has no divisor component. We select a resolution of the singularities of $T_k$ $\mu:{\widetilde}{X}\to X$, and write $\mu^{\star}T_k=\theta+F$, where $\theta$ is a smooth form and $F$ is an effective ${\mathbf{R}}$-divisor. Since $T_k\geq{\varepsilon}\omega$ for ${\varepsilon}>0$ small enough, we get that $\theta\geq\mu^{\star}{\varepsilon}\omega$. Denoting by $E_1,...,E_r$ the $\mu$-exceptional prime divisors on ${\widetilde}{X}$, it is shown in \[DP01\], lemma 3.5, that one can find ${\delta}_1,...,{\delta}_r>0$ small enough and a closed smooth $(1,1)$-form $\tau$ in $\{{\delta}_1E_1+...+{\delta}_rE_r\}$ such that $\mu^{\star}{\varepsilon}\omega-\tau$ is positive definite everywhere. It follows that $\theta-\tau$ is a Kähler form upstairs. Now, we have $$\alpha=\mu_{\star}\{T_k\}=\mu_{\star}\{\theta-({\delta}_1E_1+...+{\delta}_rE_r)\}=\mu_{\star}\{\theta-\tau\},$$ since $E_j$ is $\mu$-exceptional and so is $F$ because $\mu_{\star}F$ is an effective divisor contained in the scheme $V_k$; this concludes the proof of proposition 2.3.

That a modified nef classe is somehow nef in codimension 1 is reflected in the following

If $\alpha$ is a modified Kähler (resp. nef) class, then $\alpha_{|D}$ is big (resp. pseudo-effective) for every prime divisor $D\subset X$.

$Proof$: if $\alpha$ is a modified nef class and ${\varepsilon}>0$ is given, choose a current $T_{{\varepsilon}}\geq-{\varepsilon}\omega$ in $\alpha$ with analytic singularities in codimension at least 2. Locally, we have $\omega\leq Cdd^c|z|^2$ for some $C>0$, thus $T_{{\varepsilon}}+{\varepsilon}Cdd^c|z|^2$ writes as $dd^c\varphi_{{\varepsilon}}$, where $\varphi_{{\varepsilon}}$ is pluri-subharmonic and is not 