Proposition 5.14.10372195E73A0

Proof. It suffices to work near a point $x$ in a positive-dimensional chart. The Lelong number $\nu(\varphi,x)$ is finite. Choose $c>0$ with $c\nu(\varphi,x)<1$. The upper bound in the multiplier-ideal order estimate gives \[0\leq\operatorname{ord}_x\mathcal I(c\varphi)\leq c\nu(\varphi,x)<1.\] This integral order is zero; hence the ideal contains a germ nonvanishing at $x$, so $\exp(-c\varphi)$ is integrable on a neighborhood of $x$. After subtracting a local upper bound for $\varphi$, assume $\varphi\leq0$. For every finite $p\geq1$, the elementary bound $t^p\leq C_{p,c}\exp(ct)$ for $t\geq0$ now proves local integrability of $|\varphi|^p$. Finitely many such neighborhoods cover each compact set. In dimension zero the function is finite at every point, so the assertion is immediate.