The approximation in theorem 18.6.4 loses all non-pluripolar mass. For the nonconstant smooth $u$ at the end of its proof, the approximants $q_ j$ have zero non-pluripolar measure, whereas $\mathrm{MA}_\theta (u)=\theta +\mathrm{dd}^{\mathrm{c}}u>0$.
After normalization to mean zero, the same sequence converges in the Gauduchon trace metric of corollary 11.31.6. Nevertheless, the sequence $q_ j$ does not converge to $u$ in capacity. Otherwise, since $q_ j,u\leq 0$, theorem 13.15.9 would give
\[ 0=\mathrm{e}^{q_ j}\mathrm{MA}_\theta (q_ j) \rightharpoonup \mathrm{e}^{u}(\theta +\mathrm{dd}^{\mathrm{c}}u), \]
contradicting the strict positivity of the limiting measure. Thus weak convergence of traces and $L^1$ convergence do not imply convergence in capacity, even on a curve.
The same sequence shows that ideal projection is not continuous for the $L^1$ topology: $P_\theta [q_ j]_{\mathcal I}=q_ j\to u$, whereas $P_\theta [u]_{\mathcal I}=0$. The completion therefore distinguishes smooth normalized potentials that the quotient by multiplier-ideal equivalence identifies.
The loss of mass also occurs along sequences of fixed positive mass.
Proposition 18.6.6.8B21A468DD06 In the curve setting above, for every rational $0<\tau <1$ there are ideal models $r_ j$ with analytic singularities such that
\[ \begin{gathered} d_{\mathcal I,\theta ,\mathrm{d}V}([r_ j]_{\mathcal I},[0]_{\mathcal I})\longrightarrow 0,\\ \mathrm{MA}_\theta (r_ j)(Y)=\tau , \qquad \mathrm{MA}_\theta (r_ j)\rightharpoonup \tau \theta . \end{gathered} \]
In particular, non-pluripolar mass is not continuous for this metric, even on a family whose masses have a fixed positive lower bound.
Proof.
Choose finitely supported probability measures $\nu _ j$ with rational weights converging weakly to $\theta $, and let $q_ j$ be their normalized Green potentials; the preceding proof gives $q_ j\to 0$ in $L^1$. Put $a=1-\tau $ and $r_ j=P_\theta [a q_ j]_{\mathcal I}$. The scaled potential lies in the same background, since
\[ \theta +\mathrm{dd}^{\mathrm{c}}(aq_ j)=(1-a)\theta +a\nu _ j\geq 0. \]
Since $aq_ j$ is a competitor, $aq_ j\leq r_ j\leq 0$, and hence $r_ j\to 0$ in $L^1$. By projection, $r_ j$ is an ideal model. By proposition 18.4.8, it has exactly the logarithmic coefficients of $aq_ j$, which are rational, with bounded local remainders, and it is locally bounded away from the finite pole set. On the curve, removing the polar part of the curvature therefore gives
\[ \mathrm{MA}_\theta (r_ j)=\theta +\mathrm{dd}^{\mathrm{c}}r_ j-a\nu _ j. \]
By distributional Stokes, this measure has total mass $1-a=\tau $. The positive curvature measures $\theta +\mathrm{dd}^{\mathrm{c}}r_ j$ have mass one and converge as distributions, hence weakly, to $\theta $. Since $\nu _ j\rightharpoonup \theta $, the last identity gives $\mathrm{MA}_\theta (r_ j)\rightharpoonup \tau \theta $. Finally, $P_\theta [0]_{\mathcal I}=0$, so the distance in the statement is $\int _ Y|r_ j|\, \mathrm{d}V$, which tends to zero. The limiting potential $0$ has non-pluripolar measure $\theta $, of mass one.
$\square$
For comparison, let $d_{S,\theta }$ denote the singularity pseudometric $d_ S$ recalled at the beginning of the chapter, computed in the background $\theta $. Since $r_ j\leq 0$, the ordered-distance formula of Lemma 16.9.7 applies and gives
\[ d_{S,\theta }(r_ j,0) =\frac12\left(\int _ Y\theta -\int _ Y\mathrm{MA}_\theta (r_ j)\right) =\frac{1-\tau }{2}>0. \]
Thus $d_{S,\theta }$ keeps the fixed mass gap that disappears in the elementary ideal metric.
Pome
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