Pome504A5958272FExample 18.7.3.504A5958272F On a compact connected curve $Y$ with a positive form $\theta $ of total mass one, there are normalized $a_ j\in \mathrm{PSH}(Y,\theta )$ with analytic singularities and a smooth $u\in \mathrm{PSH}(Y,\theta )$ such that
but $P_\theta [a_ j]=P_\theta [a_ j]_{\mathcal I}\to u\ne P_\theta [u]=P_\theta [u]_{\mathcal I}$ in $L^1$.
Proof. Choose $u$ smooth, nonconstant and normalized, with $\theta +\mathrm{dd}^{\mathrm{c}}u>0$, and apply the construction in theorem 18.6.4 to the smooth probability measure $\nu =\theta +\mathrm{dd}^{\mathrm{c}}u$. Since $\nu $ has no atoms, the largest mass of a cell tends to zero along partitions of vanishing diameter. Choose the rational approximations of the weights so that their total absolute error tends to zero. The resulting atomic probability measures $\nu _ j$ converge weakly to $\nu $, and their largest atoms tend to zero.
By the same theorem, their normalized Green potentials $a_ j$ have analytic singularities, satisfy $P_\theta [a_ j]=P_\theta [a_ j]_{\mathcal I}=a_ j$, and converge to $u$ in $L^1$. At every point $y\in Y$,
Since local modifications of a smooth curve are isomorphisms, this gives all the required divisorial limits. As $u$ is bounded and $\theta >0$, both of its envelopes are zero, and since $u$ is not constant, the envelopes do not converge to them. Each $a_ j$ has a logarithmic pole, so $u\preceq _{\mathcal I}a_ j$ fails. $\square$
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