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The polar description also compares models in different backgrounds. Using the background as an additional coordinate gives a complete space that includes the zero-area boundary. The singularity coordinate still records the diffuse polar curvature, and taking its atomic part still gives the nearest ideal model.
We keep the compact connected curve $Y$ and the smooth positive form $\eta $ of area one. Let
\[ \mathcal B(Y)=\{ \sigma \in C^\infty (Y,\Lambda ^{1,1}_{\mathbb R}): A_\sigma =\textstyle \int _ Y\sigma \geq 0\} \]
be the set of admissible backgrounds. Choose $C^ k$ norms on smooth forms, defined by a fixed smooth metric on $Y$, and put
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13.30.1
\begin{equation} b(\sigma ,\tau )=\sum _{k=0}^\infty 2^{-k-1} \min \{ 1,\| \sigma -\tau \| _{C^ k}\} . \end{equation}
This is a complete metric for the $C^\infty $ topology on $\mathcal B(Y)$. Indeed, a $b$-Cauchy sequence is Cauchy in every $C^ k$ norm; the limits agree and define a smooth form, whose area is nonnegative by $C^0$ convergence. Conversely, convergence in every $C^ k$ norm gives convergence for $b$: a finite part of the sum becomes small, and the tail is uniformly small.
Let $\mathcal M(\sigma )$ be the space of ordinary models of proposition 13.29.1, and define
\[ \mathscr M(Y)=\{ (\sigma ,p):\sigma \in \mathcal B(Y), p\in \mathcal M(\sigma )\} . \]
For such a pair write $\mu _ p^\sigma =\mathbf1_{\{ p=-\infty \} }(\sigma +\mathrm{dd}^{\mathrm{c}}p)$, and set
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13.30.2
\begin{equation} D((\sigma ,p),(\tau ,q)) =b(\sigma ,\tau )+\| \mu _ p^\sigma -\mu _ q^\tau \| _{\mathrm{TV}}. \end{equation}
Let $\mathscr I(Y)\subset \mathscr M(Y)$ consist of the pairs $(\sigma ,p)$ with $P_\sigma [p]_{\mathcal I}=p$, and let $\mathscr A_+(Y)$ consist of the pairs $(\sigma ,p)$ for which $p$ has analytic singularities and positive non-pluripolar mass in background $\sigma $. For a finite measure $\mu $ on $Y$, write $\mu =\mu ^{\mathrm{at}}+\mu ^{\mathrm{na}}$ for its decomposition into an atomic and a nonatomic part.
Theorem 13.30.3.C5551396F93B The metric space $(\mathscr M(Y),D)$ is complete. Its restriction to each fixed-background fiber is the polar metric $d_\sigma $. The map
\[ R(\sigma ,p)=(\sigma ,P_\sigma [p]_{\mathcal I}) \]
is a $1$-Lipschitz retraction onto the closed subspace $\mathscr I(Y)$, and $R(\sigma ,p)$ is the unique ideal pair nearest to $(\sigma ,p)$. Moreover,
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13.30.4
\begin{equation} \overline{\mathscr A_+(Y)}^{\, D}=\mathscr I(Y),\qquad \operatorname {dist}_ D((\sigma ,p),\mathscr I(Y)) =(\mu _ p^\sigma )^{\mathrm{na}}(Y). \end{equation}
If $(\sigma _ j,p_ j)\to (\sigma ,p)$ for $D$, then $p_ j\to p$ in $L^1(Y,\eta )$ and in capacity for $\eta $, including when $A_\sigma =0$.
Proof.
By proposition 13.29.1, polar curvature identifies $\mathscr M(Y)$ with the set of pairs $(\sigma ,\mu )$ in which $\sigma \in \mathcal B(Y)$ and $\mu $ is a polar measure of mass at most $A_\sigma $. In these coordinates, (13.30.2) is the restriction of the product metric $b+\| \cdot \| _{\mathrm{TV}}$, so it separates pairs and restricts to $d_\sigma $ on every fiber.
Suppose that $(\sigma _ j,\mu _ j)$ is a Cauchy sequence. Completeness of $b$ gives $\sigma _ j\to \sigma $ in the $C^\infty $ topology, and completeness of the space of finite signed measures gives $\mu _ j\to \mu $ in total variation. The limit $\mu $ is positive and satisfies $\mu (Y)\leq A_\sigma $. Choose polar sets $E_ j$ carrying the $\mu _ j$. Their union $E$ is polar, and
\[ \mu (Y\setminus E)\leq \| \mu -\mu _ j\| _{\mathrm{TV}}\longrightarrow 0. \]
Thus $\mu $ is a polar measure and defines a model in background $\sigma $. This proves completeness.
Ideal projection takes the atomic part, by proposition 13.29.1. Taking the atomic part does not increase total variation, so $R$ is $1$-Lipschitz. The positive atomic measures are closed in total variation, since a limit is carried by the countable union of the sets of atoms of the sequence. Therefore $\mathscr I(Y)$ is closed. Let $\mu $ be the polar measure of $p$. For every ideal pair $(\tau ,q)$, whose polar measure $\nu $ is atomic, mutual singularity gives
\[ D((\sigma ,p),(\tau ,q)) =b(\sigma ,\tau )+\mu ^{\mathrm{na}}(Y) +\| \mu ^{\mathrm{at}}-\nu \| _{\mathrm{TV}}. \]
Its unique minimum is attained at $\tau =\sigma $ and $\nu =\mu ^{\mathrm{at}}$, that is, at $R(\sigma ,p)$. This proves the retraction and distance assertions.
For the analytic closure, let $(\sigma ,q)$ be an ideal pair. If $A_\sigma >0$, finite atomic measures with rational coefficients and total mass strictly less than $A_\sigma $ approximate every positive atomic measure of mass at most $A_\sigma $ in total variation. Their models in background $\sigma $ are analytic, by the classification on curves and the gauge transport of proposition 13.29.1, and they have positive non-pluripolar mass. If $A_\sigma =0$, the polar measure of $q$ is zero. The pairs with backgrounds $\sigma +j^{-1}\eta $ and zero polar measure converge to $(\sigma ,q)$ for $D$. Their models are bounded, so they have analytic singularities with trivial ideal, and their non-pluripolar masses are $j^{-1}>0$. Hence every ideal pair lies in the closure of $\mathscr A_+(Y)$. Conversely, an analytic model is an ideal model by proposition 18.4.8, and $\mathscr I(Y)$ is closed. This proves (13.30.4).
Finally, let $(\sigma _ j,p_ j)\to (\sigma ,p)$, and write $\mu _ j=\mu _{p_ j}^{\sigma _ j}$ and $\mu =\mu _ p^\sigma $. If $A_\sigma >0$, put $\lambda _ j=A_{\sigma _ j}/A_\sigma $. For all sufficiently large $j$ these numbers are positive and tend to one, and
\[ \| \lambda _ j^{-1}\mu _ j-\mu \| _{\mathrm{TV}} \leq \lambda _ j^{-1}\| \mu _ j-\mu \| _{\mathrm{TV}} +|\lambda _ j^{-1}-1|\, \mu (Y)\longrightarrow 0. \]
Now apply corollary 13.29.7. If $A_\sigma =0$, the backgrounds converge uniformly and all areas are nonnegative, and the uniform collapse of theorem 5.29.23 gives $L^1$ and capacity convergence to the unique normalized potential $u_\sigma =p$ in background $\sigma $.
$\square$
The $C^\infty $ topology on backgrounds is needed for completeness: a sequence of smooth forms that is Cauchy for the uniform norm can have a limit that is not smooth. On a fixed background of positive area, $D$ induces on models of positive mass the topology of theorem 13.27.4, but it allows sequences to converge to models of zero mass. In particular, the Cauchy sequences for $D$ need not satisfy the condition on reciprocal volumes in that theorem.
Smooth gauge changes and positive scaling act compatibly on all fibers of the joint space.
Corollary 13.30.5.8918C681057E For $f\in C^\infty (Y,\mathbb R)$, the map
\[ \mathcal G_ f(\sigma ,p) =(\sigma +\mathrm{dd}^{\mathrm{c}}f,P_{\sigma +\mathrm{dd}^{\mathrm{c}}f}[p-f]) \]
is an isometry of $\mathscr M(Y)$, with inverse $\mathcal G_{-f}$ and $\mathcal G_ g\mathcal G_ f=\mathcal G_{f+g}$. It preserves analytic models and commutes with $R$.
For $\lambda >0$, the bijection $\mathcal H_\lambda (\sigma ,p)=(\lambda \sigma ,\lambda p)$ commutes with $R$ and satisfies
\[ \min \{ 1,\lambda \} D(x,y) \leq D(\mathcal H_\lambda x,\mathcal H_\lambda y) \leq \max \{ 1,\lambda \} D(x,y). \]
Proof.
The gauge transport in proposition 13.29.1 preserves the polar measure and, after subtraction of $f$, the ordinary type. In the coordinates $(\sigma ,\mu )$ of the proof of theorem 13.30.3, $\mathcal G_ f$ is therefore the map $(\sigma ,\mu )\mapsto (\sigma +\mathrm{dd}^{\mathrm{c}}f,\mu )$. It does not change the area, and $b(\sigma +\mathrm{dd}^{\mathrm{c}}f,\tau +\mathrm{dd}^{\mathrm{c}}f)=b(\sigma ,\tau )$. This proves that $\mathcal G_ f$ is an isometry and the laws for inverses and compositions. Taking the atomic part commutes with this map, and subtraction of a smooth function and bounded changes preserve analytic singularities.
Scaling the competitors of the ordinary envelope shows that $\lambda p$ is an ordinary model for $\lambda \sigma $. In the coordinates $(\sigma ,\mu )$, $\mathcal H_\lambda $ is the map $(\sigma ,\mu )\mapsto (\lambda \sigma ,\lambda \mu )$, with inverse $\mathcal H_{\lambda ^{-1}}$, and taking the atomic part commutes with scaling. For $r\geq 0$,
\[ \min \{ 1,\lambda \} \min \{ 1,r\} \leq \min \{ 1,\lambda r\} \leq \max \{ 1,\lambda \} \min \{ 1,r\} . \]
Apply this to every $C^ k$ norm in (13.30.1); the total-variation term is multiplied by $\lambda $. The two bounds follow.
$\square$
Pome
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