PomeE04E4D6C502ELemma 5.38.7.E04E4D6C502E Let $u_ j,u$ be quasi-plurisubharmonic functions on a complex manifold with a common smooth lower curvature bound, that is, $\mathrm{dd}^{\mathrm{c}}u_ j\geq -\beta $ and $\mathrm{dd}^{\mathrm{c}}u\geq -\beta $ for one smooth real $(1,1)$-form $\beta $. If $u_ j\to u$ in $L^1_{\mathrm{loc}}$, then at every point $x$,
Proof. On a coordinate ball about $x$, add one smooth function to all the potentials to make them plurisubharmonic, and write $p_ j,p$ for the resulting functions. Let $n$ be the dimension and $\kappa $ the Euclidean form of the chart. Distributional convergence gives weak convergence of the positive trace measures $\mathrm{dd}^{\mathrm{c}}p_ j\wedge \kappa ^{n-1}$ to $\mathrm{dd}^{\mathrm{c}}p\wedge \kappa ^{n-1}$. For every sufficiently small radius $r$ such that the sphere of radius $r$ about $x$ has zero mass for the limit measure, the masses of the balls converge. The Lelong monotonicity formula bounds each $\nu (p_ j,x)$ by the normalized trace mass of $p_ j$ on such a ball. Taking the upper limit in $j$, and then letting these radii decrease to zero, gives $\limsup _ j\nu (p_ j,x)\leq \nu (p,x)$. Since smooth additions do not change Lelong numbers, this proves the lemma. $\square$
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