PomeProposition 5.16.22.DE693424352F Let $\pi \colon Y\rightarrow X$ be a smooth morphism between complex manifolds. Assume that $\varphi \in \mathrm{PSH}(X)$. Then
\[ \mathcal{I}(\pi ^*\varphi )=\pi ^*\mathcal{I}(\varphi ). \]
Proposition 5.16.22DE693424352F
Proof. It follows from [Théorème 3.10, SHC6] that locally $\pi $ can be written as the composition of an étale morphism and a projection. It suffices to handle the two cases separately.
Recall that in the complex analytic setting, an étale morphism is locally biholomorphic, so there is nothing to prove in this case.
Next, assume that $Y=X\times U$, where $U\subseteq \mathbb {C}^ n$ is a domain and $\pi $ is the natural projection. It follows from Fubini's theorem that
\[ \mathcal{I}(\pi ^*\varphi )\supseteq \pi ^*\mathcal{I}(\varphi ). \]
The reverse inequality is proved in [Proposition 14.3, Dem12]. $\square$
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