PomeTheorem 20.4.18.0AD88DBD3706 If the closed positive current $T$ has analytic singularities, then $\mathbb D(T)$ is Cartier and has a nef Cartier realization on a smooth compact modification.
Theorem 20.4.180AD88DBD3706
Proof. Resolve the analytic singularity ideal, using the log-resolution theorem in [section on analytic singularities, theorem on existence of log resolutions, SGPT]. On a suitable smooth compact model $\pi :Z\to Y$,
\[ \pi ^*T=[D]+R, \]
where $D$ is a finite effective rational divisor and $R$ is a closed positive current with locally bounded potentials. To obtain a global bounded potential, choose smooth metrics on the line bundles of the components of $D$ and subtract the corresponding logarithms of squared norms of the canonical sections from $w\circ \pi $. The remainder $v$ is globally bounded, and
\[ R=\beta _ Z+\mathrm{dd}^{\mathrm{c}}v \]
for a smooth closed form $\beta _ Z$. By the lemma, the class of $R$ is nef.
On every further compact modification $g:Z'\to Z$, the current $g^*R$ still has bounded local potentials and therefore zero Lelong numbers. Hence the divisorial part of $g^*\pi ^*T$ is exactly $[g^*D]$, and its regular part is $g^*R$. It follows that $\mathbb D(T)_{Z'}=g^*\{ R\} _{\mathrm{BC}}$. Thus $(Z,\{ R\} _{\mathrm{BC}})$ is a nef Cartier realization. $\square$
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