PomeProposition 10.30.7.A8BE1F55208F If $u\in \mathrm{PSH}(Y,\theta )$ has analytic singularities, then $\mathbb D_\theta (u)$ has a nef Cartier realization on a smooth compact modification.
10.30.6 Nef realizations and residual-current limitsAFB10A9AB20B
On a resolution of an analytic singularity, the bounded residual potential has smooth approximants with arbitrarily small curvature error, and these prove nefness within the same affine class. Boundedness also prevents new divisorial terms on further modifications, which gives the Cartier realization.
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
\[ \pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )=[D]+R, \qquad R=\beta +\mathrm{dd}^{\mathrm{c}}v\geq 0, \]
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. $\square$
For general $u$, add twice the curvature error of analytic regularization to the background. This gives positive currents with analytic singularities, to which the proposition applies, while the ideal orders identify the limiting residual family.
Theorem 10.30.8.250F831D6E96 Let $u\in \mathrm{PSH}(Y,\theta )$ and fix a Hermitian form $\omega $ on $Y$. There are potentials $u_ m$ with analytic singularities, converging to $u$ in $L^1(Y)$, and positive numbers $\epsilon _ m\downarrow 0$, indexed by sufficiently large positive integers $m$, such that
\[ \theta _ m=\theta +2\epsilon _ m\omega , \qquad \theta _ m+\mathrm{dd}^{\mathrm{c}}u_ m\geq \epsilon _ m\omega . \]
For each $m$, the family $\mathbb D_{\theta _ m}(u_ m)$ has a nef Cartier realization. On every fixed compact model $\pi :Z\to Y$, its class has the current representative
\[ C_{m,Z}=\pi ^*\theta +2\epsilon _ m\pi ^*\omega -D_{m,Z}, \qquad D_{m,Z}=\frac1m\sum _ E\operatorname {ord}_ E\mathcal I(mu)[E], \]
where $E$ ranges over the prime divisors on $Z$ and the order is computed after pulling the ideal back to $Z$. These representatives satisfy
\[ C_{m,Z}\longrightarrow \pi ^*\theta -N_ Z(u) \quad \hbox{weakly as currents}, \]
and the positive residual representatives satisfy
\[ R_{m,Z}:=C_{m,Z}+\mathrm{dd}^{\mathrm{c}}(u_ m\circ \pi ) \longrightarrow \pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )-N_ Z(u)\geq 0 \quad \hbox{weakly as currents}. \]
The exterior derivatives of both $C_{m,Z}$ and $R_{m,Z}$ converge smoothly to $\pi ^*\mathrm{d}\theta $.
Proof. Take $u_ m$ from corollary 5.36.2; they converge to $u$ in $L^1$. If $e_ m\to 0$ are their curvature errors, put $\epsilon _ m=\sup _{\ell \geq m}e_\ell +1/m$. These numbers are positive and decrease to zero, and the indices $m$ are unchanged. Adding $2\epsilon _ m\omega $ to the background turns the lower bound $-\epsilon _ m\omega $ into $\epsilon _ m\omega $. Since each $u_ m$ has analytic type $(1/m,\mathcal I(mu))$, proposition 10.30.7 gives a nef Cartier realization in the background $\theta _ m$.
On a fixed model, the generic divisorial Lelong numbers of $u_ m\circ \pi $ are exactly $m^{-1}\operatorname {ord}_ E\mathcal I(mu)$, so $N_ Z(u_ m)=D_{m,Z}$ and $C_{m,Z}=\pi ^*\theta _ m-N_ Z(u_ m)$ represents the class of $\mathbb D_{\theta _ m}(u_ m)$ on $Z$. The self-trace identity and theorem 20.4.2 give $D_{m,Z}\to N_ Z(u)$ in the trace mass of subsection 20.4.1, hence weakly as currents. Since the smooth error $2\epsilon _ m\pi ^*\omega $ tends to zero with all its derivatives, $C_{m,Z}$ converges as stated.
Divisorial removal gives $R_{m,Z}\geq 0$. All the $u_ m$ and $u$ have a common smooth lower curvature bound: choose $A>0$ with $\theta +\epsilon _ m\omega \leq A\omega $ for every index $m$ considered. By lemma 5.43.19, the convergence in $L^1(Y)$ implies $u_ m\circ \pi \to u\circ \pi $ in $L^1(Z)$. Applying $\mathrm{dd}^{\mathrm{c}}$ and adding the convergent currents $C_{m,Z}$ proves the convergence of the residual currents, and positivity persists in the weak limit. Finally, since each $D_{m,Z}$ is closed,
\[ \mathrm{d}R_{m,Z}=\mathrm{d}C_{m,Z} =\pi ^*\mathrm{d}\theta +2\epsilon _ m\pi ^*\mathrm{d}\omega , \]
which proves the smooth convergence of the exterior derivatives. $\square$
The classes $\mathbb D_{\theta _ m}(u_ m)_ Z$ lie in $\mathscr A_ Z(\theta _ m)$. Translation by $-2\epsilon _ m\pi ^*\omega $ identifies this space with $\mathscr A_ Z(\theta )$, but subtracting this form need not preserve nefness. Thus the convergence of currents above does not give an approximation by nef Cartier classes with exactly the prescribed exterior derivative.
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