Proposition 18.10.13B87EFC632CF

Proof. The coordinate $d_ O$ distinguishes different ordinary envelopes. Conversely, potentials with equal ordinary envelopes have equal multiplier ideals at every weight by corollary 18.4.4. Their ideal envelopes and all their divisorial coefficients therefore agree, so every generating entourage has identical sections at the two classes.

To produce distinct classes of this kind, choose a cutoff logarithmic pole $w\leq 0$ supported in a coordinate ball, plurisubharmonic near its pole and smooth on the annulus where the cutoff varies. After multiplication by a small positive constant, $w\in \mathrm{PSH}(X,\omega )$. Put $\chi (s)=1-\sqrt{1-s}$ for $s\leq 0$ and $u=\chi (w)$. The inequalities $0<\chi '\leq 1/2$ and $\chi ''\geq 0$, applied first to bounded truncations, give $u\in \mathrm{PSH}(X,\omega )$. In local coordinates $z$ centered at the pole, $u$ has size comparable to $\sqrt{-\log |z|^2}$, so it is unbounded and $\mathrm e^{-tu}$ is integrable for every $t>0$. Hence all its multiplier ideals are trivial and $P_\omega [u]_{\mathcal I}=0$.

For every $0<\delta \leq 1$, the function $\delta s-\chi (s)$ is bounded above on $(-\infty ,0]$. Hence $\delta w\preceq u$, and $\delta w$ is a competitor for the ordinary envelope of $u$. Letting $\delta \downarrow 0$ off the pole gives $P_\omega [u]=0$. Thus $[u]$ and $[0]$ are distinct classes with the same neighborhoods. $\square$