PomeProposition 18.16.5.B61109A06541 The set $\mathcal S_\Psi ^{(m)}$ consists of normalized ordinary models and is a complete geodesic metric space for $\rho _\Psi ^{(m)}$. Its ideal models form a closed geodesically convex subspace, equal to the metric closure of its models with analytic singularities. Ideal projection is the unique nearest-point retraction onto this subspace and is $1$-Lipschitz, and the distance to the subspace is the sum of the nonatomic polar masses. Metric convergence implies $L^1(M)$ convergence for every smooth positive volume form. The upper-volume comparison $d_\Theta $ vanishes on every pair of points of this locus.
Proposition 18.16.5B61109A06541
Proof. The preceding lemma and proposition 18.14.1 give
\[ \begin{gathered} P_\Theta [\Phi (q_1,\ldots ,q_ m)]=\Phi (q_1,\ldots ,q_ m),\\ P_\Theta [\Phi (q_1,\ldots ,q_ m)]_{\mathcal I} =\Phi (q_{1,\mathcal I},\ldots ,q_{m,\mathcal I}), \end{gathered} \]
since the induced zero obstacle is zero; independence of the factors gives normalization. Restricting to sections on which the other potentials are finite shows that if two product potentials are equal, then each coordinate difference is constant, and normalization makes every such constant zero. Thus $\Phi $ identifies this locus with the finite product of the spaces of curve models, equipped with the sum of their metrics.
Completeness follows coordinate by coordinate from corollary 13.28.7. Interpolating all polar measures linearly with the same parameter gives a constant-speed geodesic for the sum metric. Ideal projection takes the atomic part in each coordinate, so the assertions on closedness, convexity, unique nearest points and the Lipschitz bound follow coordinate by coordinate from corollary 13.28.7, proposition 13.28.11, and the distances add. Finite atomic approximations with rational coefficients in each coordinate have local logarithmic coefficients whose denominators, together with the unit coefficient of $D$, can be cleared in a product monomial. They give analytic models that are dense in the ideal subspace. Conversely, every ordinary model with analytic singularities is its own ideal projection by proposition 18.4.8, so the closure of these models is exactly the ideal subspace.
By theorem 13.28.8, metric convergence gives $L^1$ convergence on each curve. Off the pole sets, the difference of two product potentials is the sum of the pulled-back coordinate differences. Integrating each term along the compact fibers of $h_ i$ bounds its $L^1$ norm on $M$ by a constant times its $L^1$ norm on the curve, as in proposition 18.15.8, and the finite sum tends to zero.
Finally, since $\dim Y=m<\dim M$, proposition 18.14.1 shows that every model in this locus has zero upper singularity volume. A pairwise maximum is still $\Psi $ plus the pullback of a $\beta $-plurisubharmonic function on $Y$, so the same proposition gives zero upper volume for it. All three terms of $d_\Theta $ therefore vanish. $\square$
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