Proposition 18.15.8.6DEB26AFE67E The set $\mathcal S_\Psi $ consists of normalized ordinary models in background $\Theta $, and $(\mathcal S_\Psi ,\rho _\Psi )$ is a complete geodesic metric space. The upper-volume comparison $d_\Theta $ of definition 13.22.1, which is formed from the upper singularity volumes of $a$, $b$ and $\max (a,b)$, vanishes identically on this set:
\[ d_\Theta (a,b)=0\qquad (a,b\in \mathcal S_\Psi ). \]
The ideal models in $\mathcal S_\Psi $ form a closed geodesically convex subspace, equal to the $\rho _\Psi $-closure of the models with analytic singularities. Ideal projection is a $1$-Lipschitz retraction onto this subspace and selects the unique nearest ideal model. For $a=\Phi (q)$, the distance from $a$ to this subspace is the mass of the nonatomic part of $\mu _ q$. If $\rho _\Psi (a_ j,a)\to 0$, then $a_ j\to a$ in $L^1(M)$ for every smooth positive volume form.
Proof.
The identities for the zero obstacle in proposition 18.15.1 give
\[ P_\Theta [\Phi (q)]=\Phi (q),\qquad P_\Theta [\Phi (q)]_{\mathcal I} =\Phi (P_\eta [q]_{\mathcal I}). \]
Both summands of $\Phi (q)$ have supremum zero on their respective factors, so $\Phi (q)$ is normalized. Restriction to any section outside $D$ shows that $\Phi $ is injective. The classification in lemma 13.28.1 and the distance formula in theorem 13.28.4 therefore make $\Phi $ an isometry from the space of curve models onto $\mathcal S_\Psi $. Completeness and the existence of geodesics follow from corollary 13.28.7.
By proposition 18.14.1, every ordinary representative of $\Phi (q)$ has zero extended mass. Moreover,
\[ \max (\Phi (q_1),\Phi (q_2)) =\Psi +\max (q_1,q_2)\circ p, \]
and the same proposition, applied to the curve potential $\max (q_1,q_2)$, gives zero upper volume for the maximum. Thus all three upper volumes in the definition of $d_\Theta $ vanish.
Under $\Phi $, ideal projection corresponds to taking the atomic part of $\mu _ q$. The assertions on closedness, convexity and nearest points therefore follow from corollary 13.28.7, proposition 13.28.11. To prove analytic closure upstairs, approximate an atomic measure by finite atomic measures with rational coefficients and mass less than $\int _ C\eta $. By lemma 20.3.20, their curve models have the corresponding logarithmic poles up to bounded remainders. Adding $\Psi $ preserves an analytic presentation: in the local product monomial, clear the denominators of the rational pole coefficients together with the unit coefficient of $D$. The resulting models in $\mathcal S_\Psi $ converge for $\rho _\Psi $. Conversely, by proposition 18.4.8, every ordinary model with analytic singularities in this locus is equal to its ideal projection, so the closure of these models lies in the closed ideal subspace.
Finally, if $\rho _\Psi (\Phi (q_ j),\Phi (q))\to 0$, then theorem 13.28.8 gives $q_ j\to q$ in $L^1(C)$. Off the pole sets, which have smooth volume zero, the difference of the two product potentials is $(q_ j-q)\circ p$. For a smooth positive volume form $\mathrm dV$ on $M$, integration along the compact fibers makes $p_*\mathrm dV$ a smooth positive density on $C$, and Fubini's theorem gives
\[ \int _ M|\Phi (q_ j)-\Phi (q)|\, \mathrm dV \leq C_{\mathrm dV}\| q_ j-q\| _{L^1(C,\eta )}\longrightarrow 0, \]
where $C_{\mathrm dV}$ bounds the density of $p_*\mathrm dV$ with respect to $\eta $.
$\square$
Pome
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