PomeE6AC80354EB9E6AC80354EB9
On a curve the preceding theory becomes explicit. The part of the curvature current carried by the pole set determines the ordinary envelope, and its atomic part determines the ideal envelope. Comparing these two measures describes both the zero-volume boundary and the failure of analytic approximation for diffuse polar singularities. Total variation of the polar measures keeps this singularity data and at the same time controls convergence of the model potentials in capacity.
In this section, $Y$ is a compact connected Riemann surface, $\theta $ is a smooth positive $(1,1)$-form on $Y$, and $A=\int _ Y\theta $. By Stokes' theorem every bounded $\theta $-plurisubharmonic function has mass $A$; thus $Y$ has the bounded mass property, $\theta $ is big, and the preceding results apply with $Y$ in place of $X$. For $u\in \mathrm{PSH}(Y,\theta )$ write $T_ u=\theta +\mathrm{dd}^{\mathrm{c}}u$, and let $\mu _ u=\mathbf1_{\{ u=-\infty \} }T_ u=T_ u^{\mathrm{pol}}$ be the polar curvature of $u$ (subsection 5.27.13); in this and the following sections, $\mu _ u$ no longer denotes $\mathrm{MA}_\theta (u)$. A positive measure carried by a polar set will be called a polar measure. Since $\theta $ is closed, the two singularity volumes of $u$ coincide with
Fix a nonpositive Green kernel $G$ for $\eta =\theta /A$, as in subsection 20.3.9, so that $\mathrm{dd}^{\mathrm{c}}_ xG(x,y)=\delta _ y-\eta $, where $\delta _ y$ is the Dirac measure at $y$. We write $\| \cdot \| _{\mathrm{TV}}$ for the total variation norm of finite signed measures.
Lemma 13.28.1.2422DD1AB3AB For a polar measure $\mu $ on $Y$ with $\mu (Y)\leq A$, put
Then $p_\mu \sim w_\mu $, $\mu _{p_\mu }=\mu $ and $U(p_\mu )=A-\mu (Y)$. Every $u\in \mathrm{PSH}(Y,\theta )$ satisfies
Consequently $\mu \mapsto p_\mu $ is a bijection from the polar measures of mass at most $A$ onto the set of all ordinary models on $Y$.
Proof. The bounds for the Green kernel give $w_\mu \in L^1(Y)$ and
In particular $w_\mu $ is $\theta $-plurisubharmonic, and its polar curvature is $\mu $. We claim that, for every $v\in \mathrm{PSH}(Y,\theta )$,
C63C9A28E474
13.28.3
The forward implication follows from corollary 5.27.15, which gives $T_ v\geq \mu _ v\geq \mu _{w_\mu }=\mu $. Conversely, if $T_ v\geq \mu $, the distribution $v-w_\mu $ satisfies $\mathrm{dd}^{\mathrm{c}}(v-w_\mu )\geq -(A-\mu (Y))\eta $, so its quasi-subharmonic representative is bounded above on the compact curve $Y$. By lemma 5.24.3, $v\preceq w_\mu $.
Thus $p_\mu $ is the upper envelope of all $v\leq 0$ with $T_ v\geq \mu $. This family is closed under maxima by (5.27.16). By Choquet's lemma the envelope is the limit of an increasing sequence of such maxima, which converges in $L^1$; hence the current inequality $T_{p_\mu }\geq \mu $ passes to the limit. Also $p_\mu \geq w_\mu -\sup _ Yw_\mu $. Hence $b=p_\mu -w_\mu $ has a quasi-subharmonic representative that is bounded below, and compactness bounds it above. Therefore $p_\mu \sim w_\mu $. The positive measure
charges no polar set because $b$ is bounded. This proves $\mu _{p_\mu }=\mu $ and $U(p_\mu )=A-\mu (Y)$, and ordinary equivalence gives $P_\theta [p_\mu ]=P_\theta [w_\mu ]=p_\mu $.
For arbitrary $u$, take $\mu =\mu _ u$. Then $T_ u\geq \mu $, so (13.28.3) gives $u\preceq p_\mu $, and $U(u)=A-\mu (Y)=U(p_\mu )$. If this mass is positive, corollary 13.19.8 gives (13.28.2). If it is zero, then $\mu (Y)=A$ and $T_ u=\mu =T_{w_\mu }$. The difference $u-w_\mu $ is then harmonic on $Y$ and hence constant, which gives the same conclusion. Finally, every ordinary model $p$ equals $P_\theta [p]=p_{\mu _ p}$, and distinct polar measures give distinct models because $\mu _{p_\mu }=\mu $. $\square$
Theorem 13.28.4.74646DD62F50 For every $u,v\in \mathrm{PSH}(Y,\theta )$,
FB9F740F1579
13.28.5
In particular, $d_\theta $ is a metric on the set of ordinary models on $Y$. For positive atomic measures $D,E$ of mass at most $A$, let $q_ D,q_ E$ be the ideal models of lemma 20.3.20, with polar curvatures $D$ and $E$. Then
Proof. By (5.27.16), the polar curvature of $\max (u,v)$ is the minimum $\mu _ u\wedge \mu _ v$ of the two measures. The formula $U=A-\mu (Y)$ therefore gives
The classification in lemma 13.28.1 shows that $d_\theta $ separates ordinary models, and the triangle inequality follows from that of the total variation norm. By lemma 20.3.20, the ideal models are exactly the models whose polar measures are atomic, which gives (13.28.6). $\square$
Corollary 13.28.7.FB89848D2500 The ordinary models on $Y$ form a complete geodesic metric space for $d_\theta $. The ideal models form a closed geodesically convex subspace, which is the metric completion of the analytic ordinary models of positive mass.
Proof. By lemma 13.28.1, theorem 13.28.4, the space of ordinary models is isometric to the set of polar measures of mass at most $A$ with the total variation norm. This set is closed in total variation: a limit is positive, satisfies the mass bound, and is carried by the countable union of polar carriers of the sequence. Completeness therefore follows from that of finite signed measures. For two such measures $\mu ,\nu $, the measures $\mu _ t=(1-t)\mu +t\nu $, $0\leq t\leq 1$, remain in this set and satisfy
Their models give a constant-speed geodesic.
The atomic measures form a closed convex subset, since a total-variation limit is carried by the countable union of the sets of atoms of the sequence. Restricting to finitely many atoms and approximating the coefficients from below by rational numbers, with an arbitrarily small further decrease if needed, approximates every such measure by finite atomic measures with rational coefficients and mass strictly less than $A$. Their models have analytic singularities by lemma 20.3.20, and (13.28.6) gives the density. $\square$
The models of positive mass do not form a complete space for $d_\theta $: for a fixed $x\in Y$, the models of $tA\delta _ x$ form a Cauchy family as $t\uparrow 1$, with a limit of zero mass. For the complete metric of theorem 13.27.4, this family is not Cauchy, because its reciprocal lower volumes diverge. The two metrics induce the same topology on models of positive mass but have different Cauchy sequences near the boundary.
The metric $d_\theta $ also controls convergence of the potentials at the zero-mass boundary. There the polar part exhausts the limiting curvature current, so its total variation controls the whole current.
Theorem 13.28.8.3F90EA1D636A Let $p_ j,p$ be ordinary models on $Y$. If $d_\theta (p_ j,p)\to 0$, then $p_ j\to p$ in $L^1$ and in capacity. If $U(p)=0$, then in addition
E5C7D16CF970
13.28.9
Proof. Write $\mu _ j=\mu _{p_ j}$ and $\mu =\mu _ p$. By (13.28.5), $\mu _ j\to \mu $ in total variation. Thus $U(p_ j)=A-\mu _ j(Y)\to U(p)$. If $U(p)>0$, the models lie in a common $\mathcal M_\delta (\theta )$ for all large $j$, and proposition 13.22.4, theorem 13.23.3 apply.
Suppose that $U(p)=0$. Then $T_ p=\mu $ and $\mu (Y)=A$, while $T_{p_ j}-\mu _ j$ is a positive measure of mass $A-\mu _ j(Y)=\mu (Y)-\mu _ j(Y)$. Consequently
which proves (13.28.9). By normalized compactness, each subsequence has a further subsequence converging in $L^1$ to a function $q$ with supremum zero. Then $T_{p_ j}\to T_ q$ weakly, so $T_ q=T_ p$ and $\mathrm{dd}^{\mathrm{c}}(q-p)=0$. Hence $q-p$ is constant, and the normalization gives $q=p$. This proves $L^1$ convergence of the whole sequence. Finally, apply lemma 5.27.11 on finitely many nested coordinate disks: both terms on its right-hand side tend to zero, and the comparison between local and global capacities in section 5.29 gives convergence in capacity. $\square$
For potentials with supremum zero, weak convergence $T_{p_ j}\rightharpoonup T_ p$ is equivalent to $p_ j\to p$ in $L^1$, by theorem 11.31.2 and normalized compactness. It need not imply convergence in capacity, even when the sequence and the limit are ideal models. Indeed, scaling the area-one construction in theorem 18.6.4 gives zero-mass ideal models converging to zero in $L^1$. Convergence in capacity would contradict theorem 13.15.9, since their damped measures vanish while the limiting measure is $\theta $.
Conversely, convergence in capacity does not imply convergence of the polar parts in total variation. Moving a single pole gives an example, even within the zero-mass boundary.
Example 13.28.10.98D940133E3B On $\mathbb P^1$ with the Fubini–Study form $\theta $ of area one, there are zero-mass ideal models $p_ j,p$ with $p_ j\to p$ in capacity but $d_\theta (p_ j,p)=2$ for every $j$.
Proof. In the affine coordinate $z$, where $\theta =\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2)$, put for $a\in \mathbb C$
This function extends to $\mathbb P^1$, has supremum zero by the Cauchy–Schwarz inequality, and satisfies $T_{p_ a}=\delta _ a$. Hence it is the ideal model with polar measure $\delta _ a$, by lemma 20.3.20. Choose $a_ j\ne 0$ tending to zero and set $p_ j=p_{a_ j}$, $p=p_0$. On the complement of any fixed disk about zero the convergence is uniform, including near infinity. Small disks about zero lie in sublevel sets $\{ p_0<-t\} $ with $t$ large, so their capacities tend to zero as the radius tends to zero, by (5.29.4). Therefore $p_ j\to p$ in capacity. Since $\| \delta _{a_ j}-\delta _0\| _{\mathrm{TV}}=2$, (13.28.5) gives the asserted distance. $\square$
Return to the general curve $Y$. Write $\mu _ u=D_ u+\sigma _ u$, where $D_ u=\sum _{x\in Y}\nu (u,x)\delta _ x$ is the atomic part and $\sigma _ u$ has no atoms. Set $\tau (u)=\sigma _ u(Y)=A-U(u)-D_ u(Y)$, the diffuse polar mass of $u$, and let $\mathcal A_+(Y,\theta )$ denote the set of analytic ordinary models of positive mass.
Proposition 13.28.11.03E01F83CA7A For every $u\in \mathrm{PSH}(Y,\theta )$,
2C63A2647FAC
13.28.12
On ordinary models, $p\mapsto P_\theta [p]_{\mathcal I}$ is a $1$-Lipschitz retraction onto the ideal models, and $P_\theta [p]_{\mathcal I}$ is the unique ideal model nearest to $p$. There are $a_ j\in \mathcal A_+(Y,\theta )$ with $d_\theta (u,a_ j)\to 0$ if and only if $P_\theta [u]=P_\theta [u]_{\mathcal I}$.
Proof. The ideal envelope of $u$ is $q_{D_ u}$, by lemma 20.3.20. For any positive atomic measure $E$ of mass at most $A$, the nonatomic measure $\sigma _ u$ and the signed atomic measure $D_ u-E$ are mutually singular. Thus (13.28.5) gives
57E1FB41A4EB
13.28.13
Among ideal models, the minimum is attained exactly at $E=D_ u$. Since the analytic models of positive mass are dense among the ideal models (corollary 13.28.7), this gives (13.28.12). Taking the atomic part does not increase total variation, which proves the Lipschitz assertion. Finally, lemma 13.28.1 shows that $P_\theta [u]=p_{\mu _ u}$ equals $q_{D_ u}=p_{D_ u}$ if and only if $\mu _ u=D_ u$, that is, $\tau (u)=0$. $\square$
Example 13.28.14.730582377A86 There is an ordinary model $p$ on $Y$ with zero mass whose multiplier ideals $\mathcal I(tp)$ are trivial for every $t>0$. It satisfies
Proof. Choose a compact real interval inside a coordinate disk. Construct a binary Cantor set $K$ in it: at level $k$ there are $2^ k$ pairwise disjoint intervals of length $r_ k=\exp (-4^{k+1})$, each containing its two children of level $k+1$, and $K$ is the intersection of the unions over all levels. Let $\sigma $ be the probability measure on $K$ assigning mass $2^{-k}$ to each interval of level $k$. It has no atoms. Put $u(x)=A\int _ YG(x,y)\, \mathrm{d}\sigma (y)$, so that $T_ u=A\sigma $. On the chosen compact coordinate disk, $G(x,y)\leq \log |x-y|^2+C$ for a fixed constant $C$. For $x\in K$, integrating over the interval of level $k$ containing $x$, and using $G\leq 0$ elsewhere, gives
Thus $u=-\infty $ on $K$, while $u$ is smooth off $K$, since $\mathrm{dd}^{\mathrm{c}}u=-\theta $ there. Hence $K$ is polar, $\mu _ u=A\sigma $ and $U(u)=0$. By lemma 13.28.1, $p=P_\theta [u]$ has the same polar curvature. Since this measure already has mass $A$, we get $T_ p=T_ u$, and therefore $p=u-\sup _ Yu$. The Lelong numbers of $p$ vanish because $\sigma $ has no atoms, so all its multiplier ideals are trivial and its ideal envelope is zero. Finally $\tau (p)=A$, which gives the distance by (13.28.12). $\square$
7E4C501C1CA6We now compare the polar metric with the topology on ordinary singularity types introduced in section 18.8. That topology also uses the ordinary envelope, but it measures the total variation only of the atomic part of the curvature. On ideal models the atomic part is the whole polar measure, so the two topologies agree there. On arbitrary ordinary models, diffuse polar mass can concentrate without being detected by the atomic total variation.
We keep $Y$, $\theta $, $A$ and the decomposition $\mu _ u=D_ u+\sigma _ u$, and we use $\eta =\theta /A$ for $L^1$ norms. Let $\mathfrak S(Y,\theta )=\mathrm{PSH}(Y,\theta )/{\sim }$ be the space of ordinary singularity types, and write $[u]$ for the class of $u$. Recall from proposition 18.8.4 that $\tau _{\theta ,0}$ is the uniform topology on $\mathfrak S(Y,\theta )$ generated by the $L^1$ distances between ordinary envelopes and between ideal envelopes, by the total variation of divisorial Lelong numbers on each fixed modification, and by the controlled ideal relations $\mathcal U(t,a)$ of (18.8.3). For ordinary classes put
D1D81F67E5DB
13.28.16
Theorem 13.28.17.1BAAC809D02C The finite pseudometric $d^{\mathrm{ot}}_\theta $ induces $\tau _{\theta ,0}$ on $\mathfrak S(Y,\theta )$. For $0\leq s\leq A$, the classes of ideal models
form a complete metric space both for $d^{\mathrm{ot}}_\theta $ and for the distance $\| D-E\| _{\mathrm{TV}}$ between $[q_ D]$ and $[q_ E]$, and these two metrics induce the same topology. The space $\mathcal C_ s$ is not totally bounded if $s>0$, and $\mathcal C_0=\{ [0]\} $.
Proof. A smooth compact modification of a smooth curve is an isomorphism, and the divisor weight of a point is one. Hence the divisorial distance in proposition 18.8.4 is exactly the atomic total variation, and the two summands of (13.28.16) are among the coordinates defining $\tau _{\theta ,0}$.
Conversely, suppose that a net $[u_ j]$ converges to $[u]$ for $d^{\mathrm{ot}}_\theta $. The ideal envelope of $u$ is $q_{D_ u}$, by lemma 20.3.20. Convergence of the atomic parts in total variation implies $L^1$ convergence of these envelopes, by theorem 13.28.4, theorem 13.28.8. These results also apply to nets: the map $D\mapsto q_ D$ goes between metric spaces, so sequential continuity is equivalent to continuity. It remains to check the controlled ideal relations.
At a point $x$ with local coordinate $z$, the multiplier ideals on the curve are
This follows from theorem 5.37.3, lemma 5.37.1: on a smooth curve the Lelong number at $x$ determines the ideal, and the logarithmic germ with that Lelong number has the displayed ideal. Write $D_ j=D_{u_ j}$ and $D=D_ u$, and fix $t>0$ and $a>1$. The set $F=\{ x:D(\{ x\} )\geq 1/(2t)\} $ is finite. At its points, $aD_ j(\{ x\} )\geq D(\{ x\} )$ and $aD(\{ x\} )\geq D_ j(\{ x\} )$ eventually. Outside $F$, the bound
makes both $\mathcal I(tu)_ x$ and $\mathcal I(tu_ j)_ x$ unit ideals for large $j$. Hence both inclusions in (18.8.3) hold eventually at every point. This proves that $d^{\mathrm{ot}}_\theta $ induces $\tau _{\theta ,0}$.
The map $D\mapsto q_ D$ is injective, so $d^{\mathrm{ot}}_\theta $ is a metric on $\mathcal C_ s$. The positive atomic measures of mass at most $s$ are closed in total variation: a limit is positive, satisfies the mass bound, and is carried by the countable union of the sets of atoms of the sequence. Completeness of the space of finite signed measures therefore gives completeness for total variation. A $d^{\mathrm{ot}}_\theta $-Cauchy sequence in $\mathcal C_ s$ is Cauchy in total variation; its limit is again atomic, and theorem 13.28.8 gives convergence of the envelope coordinate. This proves completeness for $d^{\mathrm{ot}}_\theta $ as well as the equality of the two topologies.
If $s>0$, fix $0<b\leq s$ and infinitely many distinct points $x_ j$. The measures $b\delta _{x_ j}$ have pairwise total-variation distance $2b$, so $\mathcal C_ s$ is not totally bounded for either metric. For $s=0$ only the zero measure remains. $\square$
By lemma 20.3.20, for $0\leq \delta \leq A$ the set $\mathcal C_{A-\delta }$ consists exactly of the classes of ideal models of mass at least $\delta $. The whole set $\mathcal C_ A$, including the models of zero mass, is complete. Its subset of positive mass is incomplete: for a point $x$, the measures $D_ j=A(1-1/j)\delta _ x$ form a Cauchy sequence whose limit $A\delta _ x$ has zero mass.
For general ordinary classes, the corresponding statement concerns a pseudometric. Put
For $0\leq \delta \leq A$ this set is complete for $d^{\mathrm{ot}}_\theta $, since every class $[u]$ in it is at distance zero from $[q_{D_ u}]\in \mathcal C_{A-\delta }$; limits need not be unique as ordinary classes. The set is not totally bounded when $\delta <A$. At $\delta =A$ the pseudometric vanishes identically on it, although it contains distinct classes by proposition 18.10.1. The union of these sets over all $\delta >0$ is incomplete, by the same one-atom sequence.
0F287C888CF7The atomic total variation cannot be omitted from theorem 13.28.17. On $\mathbb P^1$ with its Fubini–Study form $\theta $ of area one, fix a rational $0<\tau <1$ and take the atomic probability measures $\nu _ j$ and the ideal models $r_ j=P_\theta [(1-\tau )q_ j]_{\mathcal I}$ of proposition 18.6.6, choosing the rational approximations so that the largest atom of $\nu _ j$ tends to zero. The ordinary and ideal envelopes of $r_ j$ both tend to zero in $L^1$, and for each $t>0$ and $a>1$, the pair $([r_ j],[0])$ eventually lies in $\mathcal U(t,a)$. The currents $\theta +\mathrm{dd}^{\mathrm{c}}r_ j$ converge weakly to $\theta $. Nevertheless,
Thus these weaker conditions do not imply convergence for $\tau _{\theta ,0}$, even at a fixed positive mass.
The opposite difficulty concerns diffuse polar mass: the pseudometric controls the ordinary envelope in $L^1$, but not the total variation of the diffuse measure itself.
Proposition 13.28.19.4651CC70BDA5 For the Fubini–Study form $\theta $ of area one on $\mathbb P^1$, the space $(\mathfrak S(\mathbb P^1,\theta ),d^{\mathrm{ot}}_\theta )$ is incomplete. In fact, there is a $d^{\mathrm{ot}}_\theta $-Cauchy sequence without limit consisting of ordinary models of one fixed positive non-pluripolar mass, all with trivial multiplier ideals.
Proof. Choose a nonatomic polar probability measure $\sigma $ with compact support in an affine chart, using the Cantor construction in example 13.28.14. Fix $0<c<1$ and a point $x$ in that chart. Translating and shrinking the support towards $x$ gives nonatomic polar probability measures $\sigma _ j$ supported in disks about $x$ with radii tending to zero. Let $p_ j=p_{c\sigma _ j}$ be the ordinary models of lemma 13.28.1. Their polar measures are $c\sigma _ j$, their masses are $1-c$, and $D_{p_ j}=0$. Hence their Lelong numbers vanish, their multiplier ideals are trivial, and their ideal envelopes are zero.
We claim that $p_ j\to q_{c\delta _ x}$ in $L^1$. Write $q_ y=q_{c\delta _ y}$. Rotations of $\mathbb P^1$ preserving $\theta $ and sending $x$ to $y$ can be chosen to depend continuously on $y$ near $x$. They transport $q_ x$ to $q_ y$, so continuity of translations in $L^1$ gives $q_ y\to q_ x$ in $L^1$ as $y\to x$. Set
The common $L^1$ bound for normalized potentials makes this integral well defined. The function $w_ j$ is nonpositive, its curvature current dominates $c\sigma _ j$, and $w_ j\to q_ x$ in $L^1$. By (13.28.3), $p_ j$ is the upper envelope of all $v\leq 0$ with $\theta +\mathrm{dd}^{\mathrm{c}}v\geq c\sigma _ j$, so $w_ j\leq p_ j$.
By normalized compactness, every subsequence of $p_ j$ has a further subsequence converging in $L^1$ to some $v\leq 0$. The inequalities $\theta +\mathrm{dd}^{\mathrm{c}}p_ j\geq c\sigma _ j$ pass to the limit and give $\theta +\mathrm{dd}^{\mathrm{c}}v\geq c\delta _ x$. Since $q_ x=p_{c\delta _ x}$, the same envelope description gives $v\leq q_ x$. The lower bound $w_ j\leq p_ j$ gives $v\geq q_ x$ almost everywhere, hence everywhere. This identifies every cluster point and proves the claim.
Since $P_\theta [p_ j]=p_ j$ and $D_{p_ j}=0$, the classes $[p_ j]$ form a $d^{\mathrm{ot}}_\theta $-Cauchy sequence. A limit $[v]$ would satisfy $D_ v=0$ and $P_\theta [v]=q_ x$. This is impossible: the ordinary envelope preserves all multiplier ideals and hence all atoms, whereas $q_ x$ has an atom of mass $c$ at $x$. The sequence therefore has no limit in $\mathfrak S(\mathbb P^1,\theta )$. $\square$
On ordinary models, $\tau _{\theta ,0}$ and the topology of the polar metric $d_\theta $ are also different. Choose the Cantor support of $\sigma $ inside a positive real interval in the affine coordinate $z$, away from zero. The rotations $R_ j(z)=\mathrm e^{\mathrm i\varepsilon _ j}z$, with $0<\varepsilon _ j\to 0$, move this support off itself. Put $p=p_{c\sigma }$ and $\widetilde p_ j=p\circ R_ j^{-1}$. By rotation invariance of the envelope, the $\widetilde p_ j$ are ordinary models with polar measures $c(R_ j)_*\sigma $, which have no atoms. Continuity of rotations in $L^1$ yields $d^{\mathrm{ot}}_\theta ([\widetilde p_ j],[p])\to 0$, whereas the polar measures have disjoint supports, so $d_\theta (\widetilde p_ j,p)=2c$ by theorem 13.28.4. Conversely, polar total variation controls both terms of (13.28.16), by theorem 13.28.8. Thus $\tau _{\theta ,0}$ is strictly weaker than the polar metric topology on ordinary models, although the two agree on ideal models. The failure of continuity under smooth gauge changes with a fixed zero obstacle (proposition 18.11.2) gives a further difference: the polar metric is invariant under such changes, by section 13.29.
70267E3AB91DWe keep the compact curve $Y$ and its positive form $\theta $ of area $A$. The topology $\tau ^{\mathrm{all}}_\theta $ of 18.12 is generated by the topologies $\tau _{\theta ,F}$ for all smooth obstacles $F$, in which the ordinary and ideal envelopes below $F$ replace those below zero. On a curve the envelopes below $F$ are still determined by the polar measure, and their ideal counterparts by its atomic part. This gives a description of $\tau ^{\mathrm{all}}_\theta $ by a single pseudometric and allows a comparison with polar total variation.
Choose, as in proposition 18.12.1, a countable family $\{ F_ k\} _{k\geq 1}\subset C^\infty (Y,\mathbb R)$ that is dense for the uniform norm, with $F_1=0$. Write $d_{O,F}$ and $d_{I,F}$ for the $L^1$ distances between the ordinary envelopes $P_{\theta ,F}[\cdot ]$ and between the ideal envelopes $P_{\theta ,F}[\cdot ]_{\mathcal I}$, which define $\tau _{\theta ,F}$ (section 18.8), and put
0B5C0EA6F405
13.28.21
Proposition 13.28.22.E7DAF71126F6 The pseudometric $d^{\mathrm{all}}_\theta $ induces the uniformity of $\tau ^{\mathrm{all}}_\theta $. Two classes have distance zero if and only if they have the same polar curvature. The closure of the analytic classes is $\{ [u]:\mu _ u=D_ u\} =\mathcal G_\theta $, where $\mathcal G_\theta =\{ [u]:P_\theta [u]=P_\theta [u]_{\mathcal I}\} $ as in (18.9.2). On each set $\mathcal C_ s$, $0\leq s\leq A$, the pseudometric $d^{\mathrm{all}}_\theta $ is a complete metric inducing the topology of total variation of the atomic measures. On ordinary models, polar total variation implies convergence for $d^{\mathrm{all}}_\theta $.
Proof. Step 1: envelopes below an obstacle. For any smooth $F$, translating the competitors by $F$ gives
and likewise for the ideal envelope. The background $\theta +\mathrm{dd}^{\mathrm{c}}F$ on the right is a smooth real form of area $A$, though it need not be positive. We claim that these two envelopes are the models of $\mu _ u$ and of $D_ u$, respectively, in that background, in the sense of proposition 13.29.1. Set $\mu =\mu _ u$ and let $w_\mu $ be its Green potential, and put $r=P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[w_\mu -F]$. By the gauge bounds of proposition 18.5.1, $r+F$ differs from $P_\theta [w_\mu ]=p_\mu \sim w_\mu $ by a bounded function, so $r\sim w_\mu -F$. Hence $r=P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[r]$ is an ordinary model, and its polar measure is $\mu $. Moreover $u-F\preceq r$, and both have mass $A-\mu (Y)$. If this mass is positive, corollary 13.19.8, applied in the big background $\theta +\mathrm{dd}^{\mathrm{c}}F$, shows that $P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[u-F]=r$. If it is zero, the curvature measures of $u-F$ and $w_\mu -F$ both equal $\mu $, so these functions differ by a constant, and again $P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[u-F]=r$. The assertion for the ideal envelope follows in the same way from ideal projection and the classification of atomic models. It also follows that the envelope $P_{\theta ,F}[u]$ is the upper envelope of all $v\leq F$ with $\theta +\mathrm{dd}^{\mathrm{c}}v\geq \mu $.
Step 2: the uniformity. By proposition 13.29.1, polar total variation gives $L^1$ convergence of these models for every fixed $F$. The same assertion for atomic measures gives continuity of every ideal-envelope coordinate with respect to the total variation of the atomic parts. The controlled ideal relations are also implied, uniformly, by atomic total variation: for fixed $t>0$ and $a>1$, the bound $\| D-E\| _{\mathrm{TV}}<(a-1)/(at)$ guarantees both inclusions in (18.8.3), by the formula for multiplier ideals on a curve. Indeed, if $tE(\{ x\} )\geq 1$, the bound gives $aD(\{ x\} )>E(\{ x\} )$; otherwise the target ideal is the unit ideal. Interchange $D$ and $E$ for the other inclusion. Each divisorial coordinate is the atomic total variation. Therefore the envelope terms in the series, together with the atomic term, generate exactly the uniformity of $\tau ^{\mathrm{all}}_\theta $: finitely many coordinates control the sum up to its uniformly small tail, and the sum controls each fixed coordinate.
Step 3: zero distance and analytic closure. By Step 1 with $F=0$, distance zero forces equal polar measures; the classification at every obstacle proves the converse. The same classification shows that the ordinary and ideal envelopes agree at every $F$ if and only if $\mu _ u=D_ u$. Hence theorem 18.12.3 identifies the closure of the analytic classes.
Step 4: the sets $\mathcal C_ s$. For a $d^{\mathrm{all}}_\theta $-Cauchy sequence in $\mathcal C_ s$, the atomic measures converge in total variation to a measure of mass at most $s$, which is again atomic. The convergence of models in Step 2 gives convergence of every obstacle coordinate, and the summable tail then gives convergence for $d^{\mathrm{all}}_\theta $. Conversely, convergence for $d^{\mathrm{all}}_\theta $ implies convergence of the atomic parts, because of the first summand. This proves completeness and the equality of topologies on $\mathcal C_ s$. Finally, polar total variation controls the atomic part and all ordinary-envelope coordinates, which proves the last assertion. $\square$
On a curve, membership in the relative envelope class of the ideal envelope has the same polar criterion, including for ideal envelopes of zero mass.
Corollary 13.28.23.017BC9CAC896 For $u\in \mathrm{PSH}(Y,\theta )$ put $q=P_\theta [u]_{\mathcal I}$. The following are equivalent:
This also holds when $D_ u(Y)=A$, that is, when $q$ has zero mass.
Proof. The classifications of ordinary and ideal models in lemma 13.28.1, lemma 20.3.20 show that $P_\theta [u]=P_\theta [u]_{\mathcal I}$ if and only if $\mu _ u=D_ u$. Step 1 of the proof of proposition 13.28.22 shows that this is equivalent to equality of the envelopes at every smooth obstacle. Membership of $u$ in $\mathcal E_{\mathrm{env}}(Y,\theta ,q)$ implies the latter by proposition 18.12.2.
Conversely, assume that $\mu _ u=D_ u$. By lemma 20.3.20, $\mu _ q=D_ u$. Ordinary equivalence preserves polar curvature by corollary 5.27.15, so every potential equivalent to $q$ has non-pluripolar mass $A-D_ u(Y)$, by Stokes' theorem. Hence
If $D_ u(Y)<A$, then $P_\theta [u]=P_\theta [u]_{\mathcal I}$, and proposition 13.24.2 puts $u$ in $\mathcal E_{\mathrm{env}}(Y,\theta ,q)$.
If $D_ u(Y)=A$, the positive curvature measures $T_ u$ and $T_ q$ both dominate $D_ u$ and both have total mass $A$. Hence $T_ u=T_ q=D_ u$. The distribution $u-q$ is harmonic, and therefore constant on the compact connected curve. Thus $u=q+C$ for a constant $C$, and proposition 16.28.62 gives $u\in \mathcal E_{\mathrm{env}}(Y,\theta ,q)$ also in this case. $\square$
Proposition 13.28.24.4805035AEC54 For the Fubini–Study form $\theta $ of area one on $\mathbb P^1$, the pseudometric $d^{\mathrm{all}}_\theta $ is incomplete on $\mathfrak S(\mathbb P^1,\theta )$: some Cauchy sequence of ordinary models of one fixed positive mass with trivial multiplier ideals has no limit. On ordinary models, its topology is strictly weaker than that of polar total variation and strictly stronger than $\tau _{\theta ,0}$.
Proof. Use the diffuse polar probability measures $\sigma _ j$ concentrating at a point $x$ and the models $p_ j=p_{c\sigma _ j}$ from the proof of proposition 13.28.19, with $0<c<1$. Fix a smooth obstacle $F$ and put $q_{y,F}=P_{\theta ,F}[q_{c\delta _ y}]$. Rotations preserving $\theta $ and sending $x$ to $y$ can be chosen to depend continuously on $y$ near $x$; they transport the obstacle $F$ to its composition with the rotation. Since envelopes below two obstacles differ by at most the uniform distance between the obstacles, continuity of rotations in $L^1$ gives $q_{y,F}\to q_{x,F}$ in $L^1$ as $y\to x$.
The uniform $L^1$ bound near $x$ makes the average $w_{j,F}=\int q_{y,F}\, \mathrm d\sigma _ j(y)$ well defined. It lies below $F$, its curvature current dominates $c\sigma _ j$, and it converges to $q_{x,F}$ in $L^1$. The description of envelopes below obstacles in Step 1 of the proof of proposition 13.28.22 gives $w_{j,F}\leq P_{\theta ,F}[p_ j]\leq F$. These bounds and normalized compactness give $L^1$ precompactness. Every cluster point of the envelopes has curvature at least $c\delta _ x$, so it is at most $q_{x,F}$, and the lower bound gives the reverse inequality. Hence $P_{\theta ,F}[p_ j]\to q_{x,F}$ for every $F$. The ideal envelopes of the $p_ j$ below $F$ all equal $P_\theta (F)$, since their multiplier ideals are trivial, and their atomic parts vanish. Thus $[p_ j]$ is a $d^{\mathrm{all}}_\theta $-Cauchy sequence. Since $F_1=0$, a limit would also be a $d^{\mathrm{ot}}_\theta $-limit, which proposition 13.28.19 excludes.
For the strict comparison with polar total variation, use the rotated diffuse models $\widetilde p_ j=p\circ R_ j^{-1}$ from the end of 13.28.15. For every smooth $F$,
The same obstacle bound and the $L^1$ continuity of rotations give convergence to $P_{\theta ,F}[p]$. All atomic parts vanish, so $d^{\mathrm{all}}_\theta ([\widetilde p_ j],[p])\to 0$, whereas the polar total-variation distance remains $2c$. Proposition 13.28.22 gives the implication in the other direction.
Finally, the sequence in proposition 18.11.2 converges for the zero obstacle but not for some smooth obstacle. By Step 1 of the proof of proposition 13.28.22, replacing its members by their ordinary models does not change any obstacle envelope, so $d^{\mathrm{all}}_\theta $ is strictly stronger than $\tau _{\theta ,0}$ already on ordinary models. $\square$
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