Proposition 18.17.34A625E059F4C

Proof. Restriction to coordinate sections on which the other potentials are finite determines every active term $a_ iq_ i$ up to a constant, and normalization makes these constants zero. Thus (18.17.2) is well defined, and it identifies $\mathcal S_ a$ with the product of the spaces of curve models in the active coordinates, with their distances multiplied by $a_ i$. Coordinates with $a_ i=0$ do not appear, and for $J(a)=\varnothing $ the empty product is a single point.

The completeness, geodesic and atomic-projection results of corollary 13.28.7, proposition 13.28.11 apply coordinate by coordinate, and the weighted distances add; this proves the metric assertions and the distance formula. The diameter bound is attained by choosing polar measures $A_ i\delta _{x_ i}$ and $A_ i\delta _{y_ i}$ with $x_ i\ne y_ i$ on each active curve. Fiber integration and theorem 13.28.8 give $L^1$ convergence upstairs. The preceding lemma shows that ideal projection does not depend on $f$. Since every model and every pairwise maximum is $\Psi $ plus a pullback from $B$, proposition 18.14.1 gives zero upper volumes, so the upper-volume comparison vanishes.

For analytic density, approximate the atomic part $a_ i\mu _ i^{\mathrm{at}}$ of each scaled polar measure, where $\mu _ i^{\mathrm{at}}=\mu _ i-\mu _ i^{\mathrm{na}}$, by finite atomic measures with rational coefficients and mass at most $a_ iA_ i$. These approximations converge in total variation. Their curve models in the background $a_ i\eta _ i$ have finitely many logarithmic poles with rational coefficients, whose denominators, together with the unit coefficient of $D$, can be cleared in local product monomials. They therefore give analytic models in $\mathcal S_ a$ converging to every ideal model. Conversely, suppose that $\Phi _ a(q)$ has analytic singularities. Restrict it to a section along an active curve on which all other potentials are finite and whose $Z$-coordinate lies outside $D$. By proposition 5.34.1, the restriction, which is $a_ iq_ i$ plus a constant and not identically $-\infty $, has analytic singularities. It is an ordinary model in the positive background $a_ i\eta _ i$, so proposition 18.4.8 shows that it equals its ideal projection. The preceding lemma now shows that $\Phi _ a(q)$ is an ideal model, and closedness gives the analytic-closure assertion. The argument uses rational coefficients for the scaled poles $a_ iq_ i$ themselves. With the rational-coefficient convention of section 5.34, an irrational scaling need not preserve an analytic curve model, since it can give a pole an irrational Lelong number.

For a common active set, the normalized tuple is determined by the potential. The two bounds for $T_{ba}$ follow by comparing each positive weight $b_ i$ with $a_ i$, and atomic projection in each coordinate commutes with this correspondence.

Finally, normalized compactness, theorem 5.25.2, bounds the $L^1(C_ i)$ norms of all the $q_ i$ uniformly. Integration along the compact fibers of $\pi _ i\circ p$ gives constants $K_ i$ bounding the $L^1(M)$ norms of their pullbacks. Off the pole sets, which have smooth volume zero, the difference of the two potentials is the sum of the functions $(a_ i-b_ i)q_ i\circ \pi _ i\circ p$. The triangle inequality proves the uniform estimate, also when some coefficients vanish. Taking $b=0$ gives uniform convergence to $\Psi $, and the diameter formula shows that the diameters tend to zero. $\square$