Lemma 5.27.2.2A60C2B3B4C3 Let $a_ j\downarrow a$ and $b_{i,j}\downarrow b_ i$, $1\leq i\leq r\leq n$, be locally uniformly bounded plurisubharmonic functions on a domain in $\mathbb C^ n$. Then
\[ a_ j\, \mathrm{dd}^{\mathrm{c}}b_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}b_{r,j} \longrightarrow a\, \mathrm{dd}^{\mathrm{c}}b_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}b_ r \]
weakly as currents. The same holds for quasi-plurisubharmonic functions that become plurisubharmonic after fixed smooth corrections, and after wedging with smooth forms that converge locally in $C^\infty $.
Proof.
First let $r=n$, and write $\mu _ j=\bigwedge _ i\mathrm{dd}^{\mathrm{c}}b_{i,j}$ and $\mu =\bigwedge _ i\mathrm{dd}^{\mathrm{c}}b_ i$. Work over a relatively compact ball on which $-M\leq a_ j,a\leq M$. On the product of this ball with a disk of radius $R$ in an auxiliary variable $w$, where $\log R^2>M+1$, set
\[ v_ j(z,w)=\max \{ a_ j(z),\log |w|^2\} ,\qquad v(z,w)=\max \{ a(z),\log |w|^2\} . \]
These are locally bounded plurisubharmonic functions, also at $w=0$, and $v_ j\downarrow v$. Pulling back the functions $b_{i,j}$ and $b_ i$ to the product and applying theorem 5.27.1 in dimension $n+1$, we obtain
\[ T_ j=\mu _ j\wedge \mathrm{dd}^{\mathrm{c}}v_ j\longrightarrow T=\mu \wedge \mathrm{dd}^{\mathrm{c}}v. \]
Since $\mu $ is closed, $T=\mathrm{dd}^{\mathrm{c}}(v\mu )$. As $\mu $ has top degree in the base variables, only derivatives in $w$ contribute. Testing against product test functions and applying Fubini's theorem therefore identifies $T$ with $\mu (\mathrm dz)$ times the circle measure
\[ \mathrm{dd}^{\mathrm{c}}_ w\max \{ a(z),\log |w|^2\} =\sigma _{\exp (a(z)/2)}. \]
Here $\sigma _ t$ is the uniform probability measure on the circle $|w|=t$: the radial derivative jumps from zero to $2/t$ there, so the Euclidean Laplacian has total mass $4\pi $, and our normalization of $\mathrm{dd}^{\mathrm{c}}$ divides that mass by $4\pi $. This distributional computation is valid for every bounded Borel function $a$; no continuity in $z$ is needed. The same description holds for $T_ j$.
Choose a smooth radial compactly supported function $\vartheta (w)$, supported away from zero and from the circle $|w|=R$, that equals $\log |w|^2$ on all circles with $-M\leq \log |w|^2\leq M$. For every smooth compactly supported test function $\xi $ on the base,
\[ \langle T_ j,\xi \vartheta \rangle =\int \xi a_ j\, \mathrm d\mu _ j, \qquad \langle T,\xi \vartheta \rangle =\int \xi a\, \mathrm d\mu . \]
The convergence $T_ j\to T$ therefore proves the top-degree assertion.
For $r<n$, wedge with $n-r$ constant positive $(1,1)$-forms, each the Hessian of a positive quadratic function, and apply the top-degree case. By polarization, the products of positive constant $(1,1)$-forms span the real constant forms of bidegree $(n-r,n-r)$, and multiplying them by smooth test functions recovers every coefficient of the currents. For quasi-plurisubharmonic functions, add the fixed smooth corrections to the decreasing sequences, apply the result to the corrected functions, and expand; the term involving the correction to $a_ j$ converges by the ordinary decreasing continuity of the products. Finally, the local Chern–Levine–Nirenberg bounds show that these finitely many products have uniformly bounded variation on compact sets. This allows smooth forms converging locally in $C^\infty $ to replace fixed forms in the tests.
$\square$
Pome
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