Theorem 13.10.4.032A3F745585 Let $(\varphi _ i^ j)_{j\in \mathbb {Z}_{>0}}$ be decreasing sequences (resp. increasing sequences) of locally bounded psh functions on $X$ converging (resp. converging a.e.) to locally bounded psh functions $\varphi _ i$'s, where $i=0,\ldots ,p$. Then
\[ \varphi _0^ j\, \mathrm{dd}^{\mathrm{c}}\varphi _1^ j\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p^ j\rightharpoonup \varphi _0\, \mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p \]
as $j\to \infty $. In particular, if $(\varphi _0^ j)_ j$ is the constant sequence $1$, we have
\[ \mathrm{dd}^{\mathrm{c}}\varphi _1^ j\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p^ j\rightharpoonup \mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p. \]
Proof. On a relatively compact coordinate ball, the decreasing case, including the factor $\varphi_0^j$, is weighted Bedford--Taylor decreasing continuity. The increasing case is weighted increasing continuity, which allows almost-everywhere convergence to the specified locally bounded representatives. The local bounds needed in these results hold on a slightly larger compact set: in the decreasing case they are bounded below by the limit and above by the first term; in the increasing case they are bounded below by the first term and above by the limit, using almost-everywhere order and uniqueness of plurisubharmonic representatives.
For a compactly supported test form on $X$, use a finite partition of unity subordinate to such balls and sum the local limits. This proves weak convergence of the currents globally. Taking the auxiliary factor constantly equal to $1$ gives the unweighted assertion.
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