a compact complex manifold, such that $\mu^{-1}{\mathcal{I}}$ is just ${\mathcal{O}}(-D)$ for some effective divisor $D$ upstairs. The pull-back $\mu^{\star}T$ clearly has analytic singularities along $V(\mu^{-1}{\mathcal{I}})=D$, thus its Siu decomposition writes $$\mu^{\star}T=\theta+cD$$ where $\theta$ is a smooth $(1,1)$-form. If $T\geq\gamma$ for some smooth form $\gamma$, then $\mu^{\star}T\geq\mu^{\star}\gamma$, and thus $\theta\geq\mu^{\star}\gamma$. This operation we call a resolution of the singularities of $T$.

### Regularization(s) of currents

We will need two basic types of regularizations (inside a fixed cohomology class) for closed $(1,1)$-currents, both due to J.-P.Demailly.

Let $T$ be a closed almost positive $(1,1)$-current on a compact complex manifold $X$, and fix a Hermitian form $\omega$. Suppose that $T\geq\gamma$ for some smooth real $(1,1)$-form $\gamma$ on $X$. Then:\
(i) There exists a sequence of smooth forms $\theta_k$ in $\{T\}$ which converges weakly to $T$, and such that $\theta_k\geq\gamma-C\lambda_k\omega$ where $C>0$ is a constant depending on the curvature of $(T_X,\omega)$ only, and $\lambda_k$ is a decreasing sequence of continuous functions such that $\lambda_k(x)\to\nu(T,x)$ for every $x\in X$.\
(ii) There exists a sequence $T_k$ of currents with analytic singularities in $\{T\}$ which converges weakly to $T$, such that $T_k\geq\gamma-{\varepsilon}_k\omega$ for some sequence ${\varepsilon}_k>0$ decreasing to $0$, and such that $\nu(T_k,x)$ increases to $\nu(T,x)$ uniformly 