Proposition 10.30.7A8BE1F55208F

Proof. Resolve the analytic singularity ideal as in the proof of theorem 20.4.18. On a suitable model $\pi :Z\to Y$ we obtain

where $D$ is a finite effective rational divisor, $\beta $ is smooth with $\mathrm{d}\beta =\pi ^*\mathrm{d}\theta $, and $v$ is globally bounded after the choice of metrics on the line bundles of the components of $D$. Positivity of $R$ follows from divisorial removal.

Fix a Hermitian form $\gamma $ on $Z$ and apply proposition 5.31.2 to $v$ in the smooth background $\beta $. It gives smooth functions $v_ j$ with $\beta +\mathrm{dd}^{\mathrm{c}}v_ j\geq -\epsilon _ j\gamma $ and $\epsilon _ j\downarrow 0$. These forms represent the same affine class and have exterior derivative $\pi ^*\mathrm{d}\theta $, which proves nefness.

On every further modification $g:Z'\to Z$, the current $g^*\beta +\mathrm{dd}^{\mathrm{c}}(v\circ g)$ is positive by lemma 5.43.2. Its bounded potential has zero Lelong numbers, so the divisorial part of the full pullback is exactly $g^*D$. The residual affine class on $Z'$ is therefore $g^*\{ R\} $, which proves the Cartier realization. $\square$