Corollary 9.10.162DA8FD7B7A99

Proof. Injectivity follows from the surjectivity of $\pi $. For surjectivity, let $u\in \mathrm{PSH}(Y,\pi ^*\theta )$. The assertion is local on $X$, so after subtracting a local potential of $\theta $ we may assume that $\theta =0$.

By theorem 9.10.5, the fibers of $\pi $ are connected. The restriction of $u$ to any compact fiber is psh, hence is constant by the maximum principle. Thus $u$ descends to a function $v$ on $X$. Since $\pi $ is proper, $v$ is upper semicontinuous.

It remains to check that $v$ is psh. By theorem 9.10.13, it suffices to test on holomorphic disks. Let $f\colon \Delta \rightarrow X$ be a holomorphic disk such that $f^*v\not\equiv -\infty $. As in the proof of theorem 9.10.14, after a finite ramified cover $\rho \colon \Delta \rightarrow \Delta $, the germ of $f\circ \rho $ lifts to $Y$. The pull-back $(f^*v)\circ \rho $ is therefore subharmonic, so $f^*v$ is subharmonic. Hence $v\in \mathrm{PSH}(X)$, and the general $\theta $-psh case follows by adding back the local potentials. $\square$