The distance calculation extends to all multiplier-ideal envelopes $P_\beta [\cdot ]_{\mathcal I}$ of definition 18.4.1 on the curve $Y$. Such an envelope is determined by its atomic curvature, and the maximum of two of them has as atomic curvature the common part of the two atomic measures. The formula for the distance of ordered potentials can therefore compare envelopes whose poles are unrelated. Throughout this subsection, fix a smooth positive form $\beta $ on $Y$, put $A=\int _ Y\beta $, and choose a nonpositive Green kernel $G$ for $\eta =\beta /A$. All atomic measures below have countable support and finite total mass.
For a positive atomic measure $D$ with $D(Y)\leq A$, set
\[ w_ D(x)=\int _ YG(x,p)\, \mathrm{d}D(p),\qquad q_ D=P_\beta [w_ D]_{\mathcal I}. \]
As in the proof of theorem 20.3.10, $w_ D$ is a quasi-plurisubharmonic function, not identically $-\infty $, with $\mathrm{dd}^{\mathrm{c}}w_ D=D-D(Y)\eta $.
Lemma 20.3.20.A374F6B4426E The function $w_ D$ is $\beta $-plurisubharmonic, and
\[ q_ D\sim w_ D,\qquad \int _ Y\mathrm{MA}_\beta (q_ D) =\int _ Y\mathrm{MA}_\beta (w_ D)=A-D(Y). \]
The map $D\mapsto q_ D$ is a bijection from the positive atomic measures of mass at most $A$ onto the set of all envelopes $P_\beta [a]_{\mathcal I}$, $a\in \mathrm{PSH}(Y,\beta )$, and $q_ D$ has positive mass exactly when $D(Y)<A$. The measure $D$ is the atomic part of $\beta +\mathrm{dd}^{\mathrm{c}}q_ D$.
Proof.
One has
\[ \beta +\mathrm{dd}^{\mathrm{c}}w_ D=D+(A-D(Y))\eta \geq 0. \]
Every atom of $D$ is a pole of $w_ D$, and the smooth measure $\eta $ charges no polar set, including any additional poles caused by accumulation of the atoms. Hence $\mathrm{MA}_\beta (w_ D)=(A-D(Y))\eta $.
Put $q=q_ D$. By theorem 18.4.2, $q$ has the same multiplier ideals, and hence the same Lelong numbers at points, as $w_ D$. Applied to $\mathrm{dd}^{\mathrm{c}}w_ D=D-D(Y)\eta $, lemma 5.38.5 gives $q\preceq w_ D$, and the envelope inequality $q\geq w_ D-\sup _ Yw_ D$ proves $q\sim w_ D$. By corollary 5.27.15, the polar part of $\beta +\mathrm{dd}^{\mathrm{c}}q$ is therefore also $D$, and Stokes' theorem gives the non-pluripolar mass $A-D(Y)$.
For any $a\in \mathrm{PSH}(Y,\beta )$, its Lelong numbers at points give an atomic measure $D_ a$ of mass at most $A$. On a smooth curve these numbers determine all multiplier ideals, by theorem 5.37.3. Thus $a\sim _{\mathcal I}w_{D_ a}$, so $P_\beta [a]_{\mathcal I}=q_{D_ a}$. If $D_ a(Y)=A$, the atoms exhaust the whole curvature measure, and the non-pluripolar mass is zero. This proves surjectivity and the description of the potentials of positive mass. Distinct atomic measures give distinct Lelong numbers, which proves injectivity.
$\square$
Theorem 20.3.21.2432A5A2AE14 For positive atomic measures $D,E$ with masses less than $A$,
\[ \frac1{96}\| D-E\| _{\mathrm{TV}} \leq d_{S,\beta }(q_ D,q_ E) \leq \frac12\| D-E\| _{\mathrm{TV}}. \]
Thus the singularity distance is a metric on these ideal envelopes, and convergence for it is exactly convergence of the atomic measures in total variation.
Proof.
Write $d_ S=d_{S,\beta }$ and $a\lor b=\max (a,b)$. For potentials $u,v\in \mathrm{PSH}(Y,\beta )$ of positive mass, besides the formula for ordered potentials we use the inequality [Lemma 6.2.1, SGPT]
\[ d_ S(u,v)\leq d_ S(u,u\lor v)+d_ S(v,u\lor v) \leq 48d_ S(u,v) \]
on a curve; here $48=3(1+1)2^{1+2}$ is the constant of that lemma in dimension one.
Put $z=\max (w_ D,w_ E)$. By lemma 20.3.20, the polar parts of the curvature measures of $w_ D$ and $w_ E$ are $D$ and $E$. By (5.27.16), the polar part of the curvature measure of $z$ is $D\wedge E$, the atomic measure with the smaller coefficient at each point. Stokes' theorem therefore gives
\[ \mathrm{MA}_\beta (z)(Y)=A-(D\wedge E)(Y). \]
The formula for ordered potentials yields
\[ \begin{aligned} d_ S(w_ D,z)+d_ S(w_ E,z) & =\frac12\bigl (D(Y)+E(Y)-2(D\wedge E)(Y)\bigr )\\ & =\frac12\| D-E\| _{\mathrm{TV}}. \end{aligned} \]
Combining this equality with the two inequalities above proves both bounds, since replacing $w_ D,w_ E$ by the ordinarily equivalent $q_ D,q_ E$ does not change the singularity distance. The lower bound and the injectivity in lemma 20.3.20 make the pseudometric a metric on these envelopes.
$\square$
Corollary 20.3.22.D416A3AB0845 For $0<\varepsilon <A$, the normalized ideal envelopes satisfying $\mathrm{MA}_\beta (q)(Y)\geq \varepsilon $ form a complete space for $d_{S,\beta }$. Analytic ideal envelopes are dense in this space. The space is not totally bounded.
Proof.
By lemma 20.3.20, the space corresponds to the positive atomic measures of mass at most $A-\varepsilon $. This set is complete for total variation. Indeed, a Cauchy sequence has as limit a positive finite measure with the same mass bound. The union of the countable supports of the sequence is countable, and total-variation convergence shows that the limit gives zero mass to the complement of that union, so the limit is again atomic. The two metric estimates of theorem 20.3.21 transfer completeness to the ideal envelopes.
Finite truncation, followed by rational approximation of each coefficient from below, preserves the mass bound and converges in total variation. The resulting Green potentials have finitely many logarithmic poles with rational coefficients, and their ideal envelopes have the same ordinary types by lemma 20.3.20. These envelopes therefore have analytic singularities, and the upper metric estimate proves density.
Finally, fix $0<a<A-\varepsilon $. The measures $a\delta _ p$, for distinct $p\in Y$, have pairwise total-variation distance $2a$, so their ideal envelopes have pairwise $d_ S$ distance at least $a/48$. This infinite separated family rules out total boundedness.
$\square$
This topology differs from the one given by the $L^1$ metric on ideal envelopes of proposition 18.6.1, since that metric is totally bounded. The sequence in proposition 18.6.6 makes the difference quantitative: it converges for that $L^1$ metric while remaining at a fixed positive $d_ S$ distance from its limit.
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