Theorem 5.16.7 (Siu)6BAB6143467A

Proof. For every $c>0$, the analytic Lelong-level-set theorem proves that $\{x:\nu(\varphi,x)\geq c\}$ is an analytic subset. Since Lelong numbers are nonnegative, for $c\leq0$ this upper level set is all of $X$. These are exactly the upper level sets needed for upper semicontinuity in the analytic Zariski topology. The normalization is the same in both statements: $\mathrm{dd}^{\mathrm c}=\mathrm i\partial\bar\partial/(2\pi)$ and $\nu(\log|z|^2,0)=1$.