Pome5DCB508EA9ADProposition 5.13.9.5DCB508EA9AD Let $n'\in \mathbb {N}$ and $\Omega '\subseteq \mathbb {C}^{n'}$ be a domain. Given any holomorphic map $f\colon \Omega \rightarrow \Omega '$ and any $\varphi \in \mathrm{PSH}(\Omega ')$, exactly one of the following cases occurs:
11DBE4FA0035$f^*\varphi \equiv -\infty $;
EEDAC90E09ED$f^*\varphi \in \mathrm{PSH}(\Omega )$.
Proof. For positive source and target dimensions, apply the holomorphic pullback theorem. Its conclusion is precisely that $\varphi\circ f$ is plurisubharmonic or identically $-\infty$ on each connected component. Here the source is connected, so there is just one alternative to consider. They are disjoint because the definition of plurisubharmonicity excludes the function identically $-\infty$.
If the source has dimension zero, it is a point: the pullback is either a finite number or $-\infty$. If the target has dimension zero, $\varphi$ is finite and constant, and so is its pullback. Thus these cases satisfy the same assertion.
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