Proposition 5.17.8AF2A3F35959C

Proof. Injectivity holds because $\pi$ is an isomorphism over a dense open subset, and plurisubharmonic representatives are determined by their values almost everywhere. For surjectivity, descent across a smooth-target modification proves that every $\pi^*\theta$-plurisubharmonic function on $Y$ descends to a $\theta$-plurisubharmonic function on $X$. Locally, the descended function is defined off the exceptional image using the inverse map, extended by its upper semicontinuous regularization, and the removable-singularity argument in that proposition proves positivity and equality of its pullback with the original function. Thus every function on the right is a pullback, uniquely.