PomeLemma 7.3.6.4286535DCE99 Let $S_ j$, $j\ge 1$, and $S$ be currents of the same degree on an open set $\Omega \subset \mathbb C^ n$. If every point of $\Omega $ has an open neighborhood $U\subset \Omega $ such that $S_ j\to S$ weakly on $U$, then $S_ j\to S$ weakly on $\Omega $.
Lemma 7.3.64286535DCE99
Proof. Let $\varphi $ be a test form on $\Omega $ of complementary degree. Its support is compact, so it is covered by finitely many open sets $U_1,\ldots ,U_ m$ as in the hypothesis. By lemma 2.5.2, there are smooth functions $\eta _1,\ldots ,\eta _ m$ with compact supports in $U_1,\ldots ,U_ m$, respectively, such that $\sum _ i\eta _ i=1$ on a neighborhood of $\operatorname{Supp}\varphi $. Then $\varphi =\sum _ i\eta _ i\varphi $, where $\eta _ i\varphi $ is a test form supported in $U_ i$, and therefore $S_ j(\varphi )=\sum _ iS_ j(\eta _ i\varphi )\to \sum _ iS(\eta _ i\varphi )=S(\varphi )$. $\square$
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