Lemma 9.11.3E7163CA56764

Proof. In a local embedding, choose an ambient smooth strictly plurisubharmonic function $r$ with $\mathrm{dd}^{\mathrm{c}}r\geq \theta $ on a smaller neighborhood. For a disk $f$ and a local potential $a$ of $f^*\theta $, the function $r\circ f-a$ is subharmonic. Hence $(u+r)\circ f$ is subharmonic or identically $-\infty $. The Fornaess–Narasimhan criterion, together with the assumption that $u$ is not identically $-\infty $ on any open set, shows that $u+r$ is plurisubharmonic. The smooth case now follows from lemma 5.43.2. $\square$