PomeLemma 20.4.17.6370BDE30D1B On a connected compact complex manifold $Z$, let $\beta $ be a smooth real $(1,1)$-form and $v$ a bounded quasi-plurisubharmonic function. If $R=\beta +\mathrm{dd}^{\mathrm{c}}v\geq 0$ is closed, then $\{ R\} _{\mathrm{BC}}$ is nef.
20.4.16 Nef Cartier realizationsEB6B5025AAF9
If $T$ has analytic singularities, a log resolution splits its pullback into a finite divisor and a current with bounded potentials. The bounded part acquires no divisorial component on higher models, so its class determines the whole b-divisor class there, and smooth regularization of its bounded potential shows that this class is nef.
A class $a\in H^{1,1}_{\mathrm{BC}}(Z,\mathbb R)$ is nef if, for one Hermitian metric $\gamma $, it has for every $\epsilon >0$ a smooth representative $\beta _\epsilon \geq -\epsilon \gamma $. Since any two Hermitian metrics are comparable, this does not depend on the metric. A Cartier b-divisor class is called nef if it has a Cartier realization $(Z,a_ Z)$ with $a_ Z$ nef.
Proof. Since $R$ is closed, so is $\beta $. Fix a Hermitian metric $\gamma $. By proposition 5.31.2, there are smooth functions $v_ j$ and numbers $\epsilon _ j\downarrow 0$ such that
\[ \beta +\mathrm{dd}^{\mathrm{c}}v_ j\geq -\epsilon _ j\gamma . \]
These forms are closed representatives of $\{ R\} _{\mathrm{BC}}$. Given $\epsilon >0$, choosing $j$ with $\epsilon _ j\leq \epsilon $ gives the estimate required in the definition of nefness. $\square$
Theorem 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.
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. $\square$
For a general current, analytic regularization gives closed approximants bounded below by $-\epsilon _ m\gamma $. Adding $\epsilon _ m\gamma $ makes them positive but, when $\gamma $ is not closed, changes their exterior derivative. The approximation then takes place in the larger spaces of section 10.30, and does not by itself approximate the original Bott–Chern b-divisor class by nef Cartier classes.
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