Theorem 8.10.5.0241EC758621 For two bounded admissible potentials $u,v$, the recursive current $D_2(u,v)=\mathrm{dd}^{\mathrm{c}}(v\mathrm{dd}^{\mathrm{c}}u)$ is symmetric in its arguments. If $u_ j\downarrow u$ and $v_ j\downarrow v$ are bounded admissible sequences with a common uniform bound, then
\[ D_2(u_ j,v_ j)\longrightarrow D_2(u,v) \]
in distributions. Their shifted positive currents
\[ Q_2(u,v)=\bigl (\omega ^2+\omega \wedge \mathrm{dd}^{\mathrm{c}}u +\omega \wedge \mathrm{dd}^{\mathrm{c}}v+D_2(u,v)\bigr )\wedge \omega ^{n-m} \]
converge weakly as positive currents. For $m=2$ these are positive Radon measures. For $m>2$, the unwedged $D_2$ also converges weakly as a current with measure coefficients.
Proof.
For smooth admissible approximations $u_ k\downarrow u$ and $v_ k\downarrow v$, the weighted convergence proposition gives
\[ \mathrm{dd}^{\mathrm{c}}(v_ k\mathrm{dd}^{\mathrm{c}}u_ k)\longrightarrow \mathrm{dd}^{\mathrm{c}}(v\mathrm{dd}^{\mathrm{c}}u). \]
The smooth current on the left is $\mathrm{dd}^{\mathrm{c}}v_ k\wedge \mathrm{dd}^{\mathrm{c}}u_ k$ and is symmetric. Applying the same argument in the opposite order proves $D_2(u,v)=D_2(v,u)$. For the bounded sequences in the statement, weighted convergence followed by $\mathrm{dd}^{\mathrm{c}}$ gives the asserted simultaneous distributional limit.
The displayed subset expansion is the current of 8.9.6 with $p=2$. Its lower-degree terms converge weakly, so its whole distributional limit is $Q_2(u,v)$. Positivity and the uniform local mass bounds turn this into weak convergence of currents of order zero. When $m>2$, the same conclusion for $D_2$ follows from the uniform coefficient bounds below degree $m$ in 8.9.3.
$\square$
Pome
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