PomeProposition 18.10.1.3B87EFC632CF Two classes have the same $\tau _{\omega ,0}$ neighborhoods if and only if their ordinary envelopes coincide. This topology is not $T_0$ and cannot be induced by a metric on $\mathfrak S(X,\omega )$.
18.10 Separation and mass85FB0412C472
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The topology $\tau _{\omega ,0}$ sees an ordinary class only through its envelopes, its multiplier ideals and its divisorial Lelong numbers. Distinct ordinary types can have the same ordinary envelope, and such types cannot be separated.
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. $\square$
By theorem 13.14.3, the two envelopes have the extended masses
\[ \int _ X\mathrm{MA}_\omega (P_\omega [u]),\qquad \int _ X\mathrm{MA}_\omega (P_\omega [u]_{\mathcal I}). \]
In the present positive background, both are at most $\int _ X\omega ^ n$. For the ideal envelope this follows from corollary 18.4.4, proposition 16.28.7. For the ordinary envelope, normalize $u\leq 0$ and put $p_ j=P_\omega (u+j,0)$ and $p=P_\omega [u]$, with the obstacle envelopes of section 5.40. Then $p_ j\to p$ in capacity by lemma 5.29.19. For $k>0$, set $\eta _ k=\min \{ 1,(u+k)_+\} $. If $j>k$, the contact set of $p_ j$ with $u+j$ lies outside $\{ \eta _ k>0\} $, so proposition 16.28.6 gives $\eta _ k\mathrm{MA}_\omega (p_ j)\leq \eta _ k\omega ^ n$. Testing against a nonnegative continuous function and applying lemma 13.14.8, we obtain the same inequality for $p$. Letting $k\to \infty $ proves $\mathrm{MA}_\omega (p)\leq \omega ^ n$, since the remaining set $\{ u=-\infty \} $ is pluripolar.
The potential itself can nevertheless have infinite mass: proposition 18.5.5 gives such a $u$ whose two envelopes are both zero. Its ordinary class belongs to $\mathcal G_\omega $. Thus the topology and its analytic-closure criterion need no mass condition on the representative, whereas the two canonical envelopes never have infinite mass in a Hermitian background.
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