Proposition 5.13.17726DDFDDA8F2

Proof. Work on one connected component $Y_0$ of $Y$. In coordinate neighborhoods of $Y$ and $X$, holomorphic pullback says that $\varphi\circ f$ is either plurisubharmonic or identically $-\infty$. A plurisubharmonic function cannot be identically $-\infty$ on a nonempty open subset of a connected domain, by its local integrability. Thus a neighborhood on which the pullback is identically $-\infty$ cannot overlap a neighborhood on which it is plurisubharmonic. The two types form disjoint open subsets of $Y_0$; connectedness leaves only one type. If the first type never occurs on any component, the pullback is plurisubharmonic on $Y$. Zero-dimensional charts are covered by the finite-value convention.