3A638A8288E0
On a curve, the polar curvature separates the singularity data of a model from the area of the background. A smooth gauge change reduces any background of positive area to a positive form. Keeping the polar measure fixed preserves distances whenever both backgrounds admit it, while rescaling by the ratio of the areas identifies the entire model spaces; the gauge function controls the normalized representatives. At zero area the whole model space collapses to the unique normalized potential.
We keep the curve $Y$ of section 13.28, and we fix a smooth positive $(1,1)$-form $\eta $ on $Y$ with $\int _ Y\eta =1$, which we use for $L^1$ norms and capacities. In this section any smooth real $(1,1)$-form $\sigma $ on $Y$ with $A_\sigma =\int _ Y\sigma \geq 0$ is allowed as a background. By lemma 5.29.22, these are exactly the forms for which $\mathrm{PSH}(Y,\sigma )$ is nonempty, and $\sigma =A_\sigma \eta +\mathrm{dd}^{\mathrm{c}}h$ for a smooth real function $h$. Let $\mathcal M(\sigma )$ be the set of ordinary models in $\mathrm{PSH}(Y,\sigma )$, and put $U_\sigma (p)=\int _ Y\mathrm{MA}_\sigma (p)$. The volume comparison $d_\sigma $ is defined by the formula of definition 13.22.1; at zero area it vanishes identically. Polar curvatures, their atomic parts and ideal models are defined as before, with $\sigma $ in place of $\theta $.
Proposition 13.29.1.C9C98321BEFC Polar curvature identifies $\mathcal M(\sigma )$ with the set of polar measures of mass at most $A_\sigma $. Write $p_\sigma (\mu )$ for the model with polar measure $\mu $. Then
\[ U_\sigma (p_\sigma (\mu ))=A_\sigma -\mu (Y),\qquad d_\sigma (p_\sigma (\mu ),p_\sigma (\nu ))=\| \mu -\nu \| _{\mathrm{TV}}, \]
and ideal projection corresponds to taking the atomic part. The space $(\mathcal M(\sigma ),d_\sigma )$ is complete and geodesic, and convergence for $d_\sigma $ implies convergence in $L^1$ and in capacity. If $A_\sigma =0$, its only member is the smooth potential $u_\sigma $ of lemma 5.29.22.
Proof.
Suppose first that $A=A_\sigma >0$, and write $\sigma =A\eta +\mathrm{dd}^{\mathrm{c}}h$. For $v\in \mathrm{PSH}(Y,A\eta )$, the inequalities for ordinary envelopes in proposition 18.5.1 give
\[ P_{A\eta }[v]+\min _ Yh \leq P_\sigma [v-h]+h \leq P_{A\eta }[v]+\max _ Yh. \]
For a model $p$ in background $A\eta $, the potential $q=P_\sigma [p-h]$ therefore satisfies $q+h\sim p$, and since the envelope depends only on the ordinary type, $P_\sigma [q]=P_\sigma [p-h]=q$. Conversely, for $q\in \mathcal M(\sigma )$, the same bounds with $v=q+h$ show that $P_{A\eta }[q+h]\sim q+h$, and taking the envelope back recovers $q$. Thus $p\mapsto P_\sigma [p-h]$ is a bijection from $\mathcal M(A\eta )$ onto $\mathcal M(\sigma )$.
Smooth gauge changes preserve curvature currents, and ordinary equivalence preserves their polar parts by corollary 5.27.15. Consequently the bijection preserves polar curvature. It also preserves singularity volumes, and subtracting the same smooth function commutes with maxima. The classification and the distance formula of lemma 13.28.1, theorem 13.28.4, applied to the positive form $A\eta $, now give the first assertions. The same gauge bounds for ideal envelopes show that the transported ideal projections are ordinarily equivalent. Both are ordinary models by corollary 18.4.4, and ordinary models of the same type are equal. Thus the bijection commutes with ideal projection, which still takes the atomic part. Completeness and the geodesics follow from the identification with polar measures, as in corollary 13.28.7.
For convergence to a model of positive mass, note that $\sigma $ is big because $\sigma +\mathrm{dd}^{\mathrm{c}}(-h)=A\eta >0$, and that the convergence of the polar masses gives a common positive lower-volume bound; then apply proposition 13.22.4, theorem 13.23.3. For a limit of zero mass, the current estimate (13.28.9) follows from the same polar decomposition as before. Normalized compactness and the uniqueness of harmonic functions give $L^1$ convergence, and lemma 5.27.11 on a finite cover gives convergence in capacity.
If $A_\sigma =0$, lemma 5.29.22 leaves only $u_\sigma $ after normalization. Its curvature current vanishes, and both envelope operations fix it. All assertions reduce to statements about the zero measure and the one-point metric space.
$\square$
The criterion for equality of the ordinary and ideal envelopes also depends only on the polar data, including at zero area.
Corollary 13.29.2.02F73F083FCE For $u\in \mathrm{PSH}(Y,\sigma )$ put
\[ q=P_\sigma [u]_{\mathcal I},\qquad \mu _ u=\mathbf1_{\{ u=-\infty \} }(\sigma +\mathrm{dd}^{\mathrm{c}}u),\qquad D_ u=\sum _{y\in Y}\nu (u,y)\delta _ y. \]
The following are equivalent:
\[ \begin{gathered} u\in \mathcal E_{\mathrm{env}}(Y,\sigma ,q),\qquad \mu _ u=D_ u,\\ P_\sigma [u]=P_\sigma [u]_{\mathcal I},\\ P_{\sigma ,F}[u]=P_{\sigma ,F}[u]_{\mathcal I} \quad \text{for every smooth real }F. \end{gathered} \]
If $A_\sigma =0$, all four conditions hold for every $u\in \mathrm{PSH}(Y,\sigma )$.
Proof.
Suppose first that $A=A_\sigma >0$, and write $\sigma =A\eta +\mathrm{dd}^{\mathrm{c}}h$. Put $v=u+h$ and $q_+=P_{A\eta }[v]_{\mathcal I}$. The gauge change preserves the curvature current and its pole set, hence both $\mu _ u$ and $D_ u$. By proposition 18.5.1, $q+h\sim q_+$. The gauge invariance of the relative envelope class and its invariance under equivalent references (proposition 16.28.62) give
\[ u\in \mathcal E_{\mathrm{env}}(Y,\sigma ,q) \quad \Longleftrightarrow \quad v\in \mathcal E_{\mathrm{env}}(Y,A\eta ,q_+). \]
By corollary 13.28.23, this is equivalent to $\mu _ u=D_ u$.
The gauge bounds for the ordinary and ideal envelopes, corollary 5.27.15 and the classifications in the positive background $A\eta $ identify the polar curvatures of $P_\sigma [u]$ and $P_\sigma [u]_{\mathcal I}$ with $\mu _ u$ and $D_ u$, respectively. Both envelopes are ordinary models, the ideal one by corollary 18.4.4. The classification of models in proposition 13.29.1 therefore shows that they are equal if and only if $\mu _ u=D_ u$.
For a smooth real $F$, the bijection $a\mapsto a+h$ between competitors gives
\[ P_{\sigma ,F}[u]+h=P_{A\eta ,F+h}[v], \]
and the same identity for the ideal envelopes. Indeed, this bijection preserves the obstacle bound, the ordinary order and multiplier-ideal equivalence, and addition of $h$ commutes with upper regularization. Since $F\mapsto F+h$ is a bijection of the smooth obstacles, corollary 13.28.23 gives the remaining equivalence.
If $A_\sigma =0$, lemma 5.29.22 gives $u=-h+c$ for a constant $c$, and the curvature current of $u$ vanishes. Thus $\mu _ u=D_ u=0$. All potentials in this background are smooth and ordinarily equivalent, so their ordinary and ideal competitor families below every obstacle coincide, and they are nonempty after a constant translation. The reference $q$ is ordinarily equivalent to $u$, and proposition 16.28.62 gives $u\in \mathcal E_{\mathrm{env}}(Y,\sigma ,q)$.
$\square$
Theorem 13.29.3.151BAA2FA1AB Let $\theta _0,\theta _1$ be smooth real $(1,1)$-forms on $Y$ with positive areas $A_ i=A_{\theta _ i}$, and put $\lambda =A_1/A_0$. The map
\[ F_{10}:\mathcal M(\theta _0)\longrightarrow \mathcal M(\theta _1), \qquad F_{10}(p_{\theta _0}(\mu ))=p_{\theta _1}(\lambda \mu ), \]
is a bijection and satisfies
7ADEC26E190A
13.29.4
\begin{equation} \begin{split} d_{\theta _1}(F_{10}(p),F_{10}(q))& =\lambda d_{\theta _0}(p,q),\\ U_{\theta _1}(F_{10}(p))& =\lambda U_{\theta _0}(p),\\ P_{\theta _1}[F_{10}(p)]_{\mathcal I} & =F_{10}(P_{\theta _0}[p]_{\mathcal I}). \end{split} \end{equation}
For a third smooth background $\theta _2$ of positive area, the same construction gives $F_{21}F_{10}=F_{20}$.
Proof.
Multiplication by $\lambda $ is a bijection from the polar measures of mass at most $A_0$ onto those of mass at most $A_1$. By proposition 13.29.1, it gives the asserted bijection, and the distances satisfy
\[ \| \lambda \mu -\lambda \nu \| _{\mathrm{TV}} =\lambda \| \mu -\nu \| _{\mathrm{TV}}. \]
The volume identity follows from $A_1-\lambda \mu (Y)=\lambda (A_0-\mu (Y))$. Taking the atomic part commutes with multiplication by $\lambda $, which gives the ideal-envelope identity. The area ratios multiply, proving the composition rule.
$\square$
Thus, for $A_\sigma >0$, the rescaled distances $d_\sigma /A_\sigma $ make all these model spaces isometric. The unscaled spaces have diameter $2A_\sigma $: this is an upper bound for the total variation of the difference of two positive measures of mass at most $A_\sigma $, and the models of $A_\sigma \delta _ x$ and $A_\sigma \delta _ y$, $x\ne y$, attain it. Consequently the unscaled spaces can be isometric only when their areas agree.
Alternatively, $p_{\theta _0}(\mu )\mapsto p_{\theta _1}(\mu )$ is an isometry on the polar measures with $\mu (Y)\leq \min (A_0,A_1)$. The two models have the same ordinary type. Indeed, the Green representatives have the same polar term, so their difference has smooth $\mathrm{dd}^{\mathrm{c}}$ and is smooth by elliptic regularity, while lemma 13.28.1, proposition 13.29.1 show that each model differs from its Green representative, corrected by the smooth gauge function, by a bounded function. Under this correspondence the non-pluripolar volume changes by $A_1-A_0$.
The rescaling map preserves analytic models when $\lambda $ is rational, by the convention of section 5.34: on a curve, the polar measures of analytic models are finite atomic measures with rational coefficients. If $\lambda $ is irrational, an atom $a\delta _ x$ with rational $0<a<A_0$ acquires an irrational Lelong number, so its image is not analytic in this convention. Ideal models are preserved for every $\lambda $, by (13.29.4).
Proposition 13.29.5.D21EE85B9F1F Choose a smooth real $h$ with $\theta _1=\lambda \theta _0+\mathrm{dd}^{\mathrm{c}}h$. For $p\in \mathcal M(\theta _0)$ and $q=F_{10}(p)$,
C555907478C9
13.29.6
\begin{equation} \begin{gathered} q=P_{\theta _1}[\lambda p-h],\\ \lambda p+\min _ Yh\leq q+h\leq \lambda p+\max _ Yh. \end{gathered} \end{equation}
If $U_{\theta _0}(p)=0$, then
\[ q=\lambda p-h-\sup _ Y(\lambda p-h). \]
Proof.
The form $\theta _1-\lambda \theta _0$ has integral zero, so the scalar Poisson equation gives $h$, as in lemma 5.29.22. Write $p=p_{\theta _0}(\mu )$. Scaling the competitors in the ordinary envelope shows that $\lambda p$ is an ordinary model for $\lambda \theta _0$. The gauge construction in the proof of proposition 13.29.1 therefore makes $P_{\theta _1}[\lambda p-h]$ an ordinary model in background $\theta _1$. The curvature current of $\lambda p-h$ in this background is $\lambda (\theta _0+\mathrm{dd}^{\mathrm{c}}p)$, so its polar measure is $\lambda \mu $. The construction preserves this measure, which gives the first identity in (13.29.6).
The function $\lambda p-h+\min _ Yh$ is a competitor for $q$, which gives the lower bound. If $v\leq 0$ is any competitor for $q$, then $v+h-\max _ Yh$ is $\lambda \theta _0$-plurisubharmonic, nonpositive and more singular than $\lambda p$, so it is at most $\lambda p$. Taking the upper envelope gives the upper bound. These inequalities hold also at the poles, and off the common pole set they imply $|q-\lambda p|\leq \operatorname {osc}_ Yh$.
At zero mass, the polar measures exhaust the curvature currents of both $p$ and $q$. Hence $\mathrm{dd}^{\mathrm{c}}(q-\lambda p+h)=0$, and the uniqueness of harmonic functions together with the normalization $\sup _ Yq=0$ gives the last formula.
$\square$
Corollary 13.29.7.21F64887A53E Suppose that smooth real $(1,1)$-forms $\theta _ j$ on $Y$ of positive area converge uniformly to a smooth form $\theta _0$ of positive area. Put $\lambda _ j=A_{\theta _ j}/A_{\theta _0}$, and let $F_{j0}:\mathcal M(\theta _0)\to \mathcal M(\theta _ j)$ be the correspondence of theorem 13.29.3. For every $t>0$,
0B648D2C5A55
13.29.8
\begin{equation} \begin{split} \sup _{p\in \mathcal M(\theta _0)} \| F_{j0}(p)-p\| _{L^1(Y)}& \longrightarrow 0,\\ \sup _{p\in \mathcal M(\theta _0)} \operatorname {Cap}_\eta (\{ |F_{j0}(p)-p|>t\} )& \longrightarrow 0. \end{split} \end{equation}
More generally, let $q_ j\in \mathcal M(\theta _ j)$ have polar measures $\nu _ j$, and let $p=p_{\theta _0}(\mu )$. If $\lambda _ j^{-1}\nu _ j\to \mu $ in total variation, then $q_ j\to p$ in $L^1$ and in capacity.
Proof.
The inverse of the scalar Laplacian for $\eta $ on functions of mean zero has a symmetric Green kernel $G$. Set
\[ h_ j(x)=\int _ YG(x,y)\, (\theta _ j-\lambda _ j\theta _0)(y). \]
The integrand form has integral zero, so $\mathrm{dd}^{\mathrm{c}}h_ j=\theta _ j-\lambda _ j\theta _0$, and elliptic regularity makes $h_ j$ smooth. Symmetry and the uniform $L^1$ bound from subsection 20.3.9 give
\[ \| h_ j\| _{L^\infty (Y)} \leq C\left\| \frac{\theta _ j-\lambda _ j\theta _0}{\eta }\right\| _{L^\infty (Y)} \longrightarrow 0. \]
Here $C$ depends only on $\eta $; the symmetric kernel may be made nonpositive by subtracting a constant, which does not change $h_ j$.
By (13.29.6), applied with $h=h_ j$, we have outside the common poles
\[ |F_{j0}(p)-p|\leq \operatorname {osc}_ Yh_ j+|\lambda _ j-1|\, |p|. \]
Normalized compactness bounds $\| p\| _{L^1}$ uniformly over $\mathcal M(\theta _0)$, which proves the first limit. Choose $C_0>0$ with $\theta _0\leq C_0\eta $. Then $p/C_0$ is $\eta $-plurisubharmonic with supremum zero, so the uniform sublevel estimate (5.29.4) bounds the capacity of $\{ |\lambda _ j-1|\, |p|>t/2\} $, uniformly in $p$; since pole sets have zero capacity, this proves the second limit.
For the last assertion put $p_ j=F_{0j}(q_ j)$, so that $q_ j=F_{j0}(p_ j)$. The polar measure of $p_ j$ is $\lambda _ j^{-1}\nu _ j$, so proposition 13.29.1 gives $p_ j\to p$ in $L^1$ and in capacity. The two uniform limits in (13.29.8), applied to $p_ j$, give $q_ j-p_ j\to 0$ in the same senses. The triangle inequality in $L^1$ and the subadditivity of capacity complete the proof.
$\square$
Finally, suppose that $\theta _ j\to \theta _0$ uniformly, where $A_{\theta _ j}\geq 0$ and $A_{\theta _0}=0$. The limiting model space is $\{ u_{\theta _0}\} $, and theorem 5.29.23 gives $L^1$ and capacity convergence to $u_{\theta _0}$, uniformly over all normalized potentials and hence over all models. The model spaces also collapse for their unscaled metrics, since their diameters are $2A_{\theta _ j}\to 0$. On the other hand, for $A_{\theta _ j}>0$ the rescaled metric $d_{\theta _ j}/A_{\theta _ j}$ makes $\mathcal M(\theta _ j)$ isometric to $\mathcal M(\eta )$, by theorem 13.29.3. Thus the rescaled metric keeps singularity data that disappear in the limiting potentials.
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