Proposition 18.35.1.609A9325CEC9 Let $\theta $ be a smooth closed real $(1,1)$-form on $X$ and let $\varphi \in \mathrm{PSH}(X,\theta )$. For $v\in C(X)$ let $E_ X(v)=E_{\theta ,X}[\varphi ]_{\mathcal I}(v)$ be the partial envelope with obstacle $v$ on the whole of $X$. Then
\[ E_{\pi ^*\theta ,Y}[\varphi \circ \pi ]_{\mathcal I}(v\circ \pi ) =E_ X(v)\circ \pi . \]
Moreover, for every $u\in \mathrm{PSH}(X,\theta )$ the non-pluripolar measures satisfy
\[ \pi _*\mathrm{MA}_{\pi ^*\theta }(u\circ \pi ) =\mathrm{MA}_\theta (u) \]
as positive Borel measures, possibly of infinite total mass.
Proof.
By proposition 5.43.18, $a\mapsto a\circ \pi $ is a bijection from $\mathrm{PSH}(X,\theta )$ onto $\mathrm{PSH}(Y,\pi ^*\theta )$. For a continuous obstacle on the whole manifold, the quasi-everywhere and the pointwise obstacle conditions are equivalent, as recalled in section 5.40, and since $\pi $ is surjective, $a\leq v$ on $X$ if and only if $a\circ \pi \leq v\circ \pi $ on $Y$. Together with lemma 5.43.23, this identifies the two families of competitors. By theorem 18.32.2, each partial envelope is the largest member of its family, which proves the envelope identity; no interchange of pullback and regularized supremum is needed.
The map $\pi $ is a biholomorphism from $Y\setminus D$ onto $X\setminus S$, so locality of the non-pluripolar product identifies the two measures there. Neither measure charges the analytic sets $D$ and $S$. Testing against nonnegative Borel functions, or exhausting the complements by increasing compact sets, proves the pushforward identity, including the case of infinite total mass.
$\square$
Pome
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