Theorem 10.30.8250F831D6E96

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,

which proves the smooth convergence of the exterior derivatives. $\square$