Theorem 13.30.3C5551396F93B

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

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

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

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 $. $\square$