Proposition 17.16.16.0290ABD0E348 Let $S$ be a strip and $V$ a genuine quasi-plurisubharmonic function on $Y\times S$. Then
\[ \operatorname {pr}_ X^*\omega +\mathrm{dd}^{\mathrm{c}}(\pi \times \mathrm{id})^*V\geq 0 \quad \Longleftrightarrow \quad \operatorname {pr}_ Y^*\kappa +\mathrm{dd}^{\mathrm{c}}V\geq 0. \]
For every $v\in \mathrm{PSH}(Y,\kappa )$, the extended non-pluripolar measure satisfies
4A3521DF9F72
17.16.17
\begin{equation} \operatorname{MA}^{\mathrm{ext}}_\omega (\pi ^*v) =n\beta _ n\wedge \pi ^*\operatorname{MA}_\kappa (v). \end{equation}
In particular, for every nonnegative Borel function $F$,
6D09A14331E6
17.16.18
\begin{equation} \int _ X\pi ^*F\, \operatorname{MA}^{\mathrm{ext}}_\omega (\pi ^*v) =2n\int _ YF\, \operatorname{MA}_\kappa (v). \end{equation}
If the base Perron family between $v_0,v_1$ is nonempty, the full Hermitian Perron envelope between $\pi ^*v_0,\pi ^*v_1$ is the pullback of the base envelope.
Proof.
Write $\alpha =\alpha _ n$ and $\beta =\beta _ n$. Locally $\beta =(\mathrm i/(2\pi ))\partial r\wedge \bar\partial r$, and $\partial r$ has a nonzero vertical component. Every vector on $Y\times S$ therefore has a lift annihilated by $\partial r$. Testing the current on these lifts removes $\beta $; integrating the resulting distributional inequality against a nonnegative compact fiber density gives positivity on the base. Conversely, pullback of a positive base current plus $\beta $ is positive. An invariant genuine potential descends canonically by restricting to local holomorphic sections, so the argument also applies to singular potentials.
For bounded $v$, the local product with smooth corrections from section 13.14 gives
\[ (\omega +\mathrm{dd}^{\mathrm{c}}\pi ^*v)^ n =(\beta +\pi ^*(\kappa +\mathrm{dd}^{\mathrm{c}}v))^ n =n\beta \wedge \pi ^*(\kappa +\mathrm{dd}^{\mathrm{c}}v)^ d. \]
All other terms vanish by $\beta ^2=0$ and the dimension of $Y$. Apply this identity to $\max (v,-j)$ and restrict to $\{ v>-j\} $. Increasing finite-level limits give (17.16.17), without assigning mass to the pole set. The identity $\int _{\pi ^{-1}(y)}\beta =2$ from subsection 23.1.32 gives (17.16.18).
The full Perron competitor family is preserved by $E$. Its supremum and upper semicontinuous regularization are invariant. The chord bound and a pulled-back base competitor make this envelope a genuine subgeodesic by proposition 17.16.6. Positivity descent makes its descended envelope a base competitor with the prescribed joint boundary upper bounds. Conversely, every base competitor pulls back to a full competitor. The two envelopes consequently agree; in submersion charts, upper semicontinuous regularization commutes with pullback.
$\square$
Pome
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