Proposition 15.31.13.BE532EB6E98C If $\mu $ is non-polar, then $\nu $ is non-pluripolar and
\[ u=(Q_\mu -\sup _ YQ_\mu )\circ \pi _ Y,\qquad \mathrm{MA}^{\mathrm{ext}}_ H(u)=c\nu ,\qquad \int _ X\mathrm{MA}^{\mathrm{ext}}_ H(u)=NV_0. \]
This is the unique supremum-zero solution among pullbacks of $\kappa $-plurisubharmonic functions. It is bounded exactly when $Q_\mu $ is bounded, and continuous exactly when $Q_\mu $ is continuous. If $v_-(h_0)=0$, then $v_-(H)=0$ and $H$ is not closed.
Proof.
A nontrivial local plurisubharmonic function on a product chart has a nontrivial subharmonic restriction to the curve for smooth-volume-almost every point of $X_0$, by local integrability and Fubini's theorem. Non-polarity of $\mu $ and a second application of Fubini give its pole set product measure zero. A countable cover proves that $\nu $ is non-pluripolar.
For bounded $p\in \mathrm{PSH}(Y,\kappa )$, bounded multilinearity and lemma 5.28.4 give
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15.31.14
\begin{equation} \mathrm{MA}_ H(p\circ \pi _ Y) =N h_0^ m\otimes (\kappa +\mathrm{dd}^{\mathrm{c}}p). \end{equation}
Only one curve factor survives in the expansion. For arbitrary $p$, apply this to $p_ j=\max (p,-j)$, restrict to the pullback of $\{ p>-j\} $ and increase $j$. The defining extended limit gives the same formula with $\mathrm{MA}^{\mathrm{ext}}_\kappa (p)$ on the right. Lemma 15.31.3 now proves the equation, finite mass and separated uniqueness. Boundedness and continuity follow from the same constant-difference identity.
If $v_-(h_0)=0$, pull back bounded potentials whose $h_0$-masses tend to zero. Their $H$-masses are $N$ times those masses, because $\int _ Y\kappa =1$, so $v_-(H)=0$. A closed $h_0$ would give every bounded potential mass $V_0>0$ by Stokes. Hence $h_0$ is not closed; restricting $H$ to a factor slice shows that $H$ is not closed either.
$\square$
Pome
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