Proposition 18.18.3347997EB7F21

Proof. Restriction to coordinate sections on which the other potentials are finite determines each $q_ i$ up to a constant, and normalization fixes that constant. The preceding lemma and proposition 13.29.1 therefore identify the locus with the finite product of the spaces of curve models, with the sum of their polar metrics; factors of zero area are single points. Completeness, geodesics and the diameter formula follow coordinatewise. Under this identification, corollary 13.28.7, proposition 13.28.11 give closedness, geodesic convexity and the unique nearest-point retraction onto the ideal subspace, and the distances add. Fiber integration transfers $L^1$ convergence on the curves to $M$.

For analytic density, approximate each atomic polar measure by finite atomic measures with rational coefficients and mass at most $B_ i$. By the atomic Green construction and transport under smooth gauge changes, their models in background $\sigma _ i$ have the corresponding finitely many poles with rational coefficients, up to bounded remainders. Coordinates of zero area are smooth. Together with the unit coefficient of $D$, the poles in the coordinates of positive area allow the denominators to be cleared in local product monomials, which gives analytic models in the locus. They converge for the metric (18.18.2). Conversely, by proposition 5.34.1, a product model with analytic singularities restricts, on every coordinate curve section on which the other potentials are finite and whose $Z$-coordinate lies outside $D$, to a function with analytic singularities that is not identically $-\infty $. Each $q_ i$ therefore has polar curvature with finitely many atoms, and it is an ideal model by the classification of curve models. Closedness proves the analytic-closure assertion.

By proposition 18.14.1, applied to the divisor $B\times D$ with weight one, every model and every pairwise maximum in this locus has zero upper volume; for a maximum, use $\Psi $ plus the pullback of the maximum of the two potentials on $B$. Thus all three terms of the upper-volume comparison vanish, even in the backgrounds that are not big by the lemma.

To compare different backgrounds, put $a_ i=B_ i/A_ i$, and for a positive polar measure $\mu $ on $C_ i$ of mass at most $B_ i$ write $p_{\sigma _ i}(\mu )$ for the ordinary model in background $\sigma _ i$ with polar curvature $\mu $, as in proposition 13.29.1. Sending a curve model for $\eta _ i$ with polar measure $\mu _ i$, of mass at most $A_ i$, to $p_{\sigma _ i}(a_ i\mu _ i)$, which for $a_ i=0$ is the unique normalized potential, defines an isometry from the weighted locus $\mathcal S_ a$ of proposition 18.17.3 onto $\mathcal S_\sigma $ that maps the ideal subspaces onto each other. Applying proposition 18.17.5 with $a_ i=B_ i/A_ i$ and $b_ i=B'_ i/A_ i$ gives (18.18.4), the exact value for ordered areas, the contractions and the diameters of their fibers. In curve coordinates the contraction is

In the second case $B'_ i=0$, because $B'_ i\leq B_ i$. Atomic projection commutes with both operations. As in theorem 13.29.3, irrational area ratios need not preserve individual analytic models, but the ideal subspaces are preserved for every ratio.

If $B_ i>0$, the models with polar measure $B_ i\delta _ x$ for infinitely many distinct points $x\in C_ i$, with all other coordinates fixed at their models with zero polar curvature, form an infinite set of ideal product models with mutual distances $2B_ i$. This excludes total boundedness of both loci. If all areas are zero, every curve coordinate is fixed at its unique normalized smooth potential, which proves the last assertion. $\square$