18.17 Varying the curve coefficientsE1C8539A70B7

Proof. For $a_ i>0$, scaling the ordinary competitors shows that $a_ iq_ i$ is an ordinary model for $a_ i\eta _ i$. Its polar curvature is $a_ i\mu _ i$, where $\mu _ i$ is the polar curvature of $q_ i$. Taking atomic parts and applying theorem 13.29.3, we see that its ideal model is $a_ iq_{i,\mathcal I}$.

Put $w_ a=\sum _{i\in J(a)}a_ iq_ i\circ \pi _ i$. If there are at least two active factors, lemma 18.16.1 makes the corresponding sum an ordinary model on the product of the active curves, with its positive product background. For one active factor, this is the identity on the curve. The identity is preserved under pullback to $Y$, by proposition 5.43.5: every competitor in the pulled-back background descends, and so do its obstacle bound and its ordinary order, since the projection is surjective. If there are no active factors, the normalized model is zero by the maximum principle. Hence $P_{\beta _ a}[w_ a]=w_ a$ in all cases.

On $B=Y\times C_0$, use the background $\beta _{a,f}=\pi ^*\beta _ a+(f\circ \pi )s_0^*\tau $. As in corollary 18.16.6, replace an ordinary competitor by its maximum with $w_ a\circ \pi $ and restrict it to each section $Y\times \{ z\} $. This gives $P_{\beta _{a,f}}[w_ a\circ \pi ]=w_ a\circ \pi $.

For the ideal identity, take $g_ a=\sum _{i\in J(a)}a_ i g_ i\circ \pi _ i\circ \pi $, with the atomic Green potentials $g_ i$ of lemma 20.3.20. Scaling, addition and pullback preserve multiplier-ideal equivalence by lemma 5.37.4, lemma 5.37.7. The complex Hessian of $g_ a$ is a closed countable divisor current of finite mass minus a smooth form, and $g_ a$ is $\beta _{a,f}$-plurisubharmonic. It is ordinarily equivalent to $w_{a,\mathcal I}\circ \pi $, where $w_{a,\mathcal I}=\sum _{i\in J(a)}a_ iq_{i,\mathcal I}\circ \pi _ i$. By lemma 5.38.5, every ideal competitor is an ordinary competitor for $g_ a$, so by the ordinary identity applied to the models $a_ iq_{i,\mathcal I}$ it is bounded by $w_{a,\mathcal I}\circ \pi $. The latter is itself an ideal competitor, so the ideal envelope equals it. When $J(a)$ is empty, both envelopes on $B$ are zero directly. Finally, proposition 18.14.1, applied to $M\to B$ and the divisor $B\times D$ with weight one, gives the identities upstairs, and independence of the factors gives normalization. $\square$

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. $\square$

Proof. Use the correspondence

Every potential in either locus arises from such a tuple, with arbitrary normalized models in its coordinates of zero coefficient, so both projections of $R_{ab}$ are surjective. For any two tuples, put $d_ i=\| \mu _ i-\nu _ i\| _{\mathrm{TV}}$. By (18.17.2), the difference of their two distances is $\sum _ i(a_ i-b_ i)d_ i$, with $0\leq d_ i\leq 2A_ i$. Thus the distortion of this correspondence satisfies $\operatorname {dis}R_{ab}\leq 2\sum _ i|a_ i-b_ i|A_ i$, and lemma 15.19.20 gives the upper bound in (18.17.6). The diameter formula in proposition 18.17.3 and the same lemma give the lower bound.

Restricting the correspondence to tuples of ideal curve models, we obtain a correspondence between the two ideal subspaces with the same distortion bound. The atomic measures $A_ i\delta _{x_ i}$ and $A_ i\delta _{y_ i}$ show that the ideal subspaces have the same diameters as the full spaces, so the lower bound also holds for them. When $b\leq a$, the upper and lower bounds coincide, including when $b=0$.

For the transition map, every active coordinate of $b$ is active for $a$, and uniqueness of the normalized active tuple makes the map well defined. Every $b$-tuple extends to an $a$-tuple, so the map is surjective, and $\sum _ i b_ i d_ i\leq \sum _ i a_ i d_ i$ proves the Lipschitz bound. Atomic projection in each coordinate commutes with discarding and rescaling coordinates. In a fiber, every coordinate with $b_ i>0$ is fixed, while the coordinates with $a_ i>0$ and $b_ i=0$ vary freely. The diameter of the fiber is therefore the sum above, by the calculation of the diameter on each curve.

Finally, fix an active coordinate $i$ and all other coordinates at their zero models. Let $q_ i$ have polar measure $A_ i\delta _ x$, with $x$ ranging over infinitely many distinct points of $C_ i$. The resulting potentials are ideal models, and any two of them are at distance $2a_ iA_ i$. This infinite separated set excludes total boundedness of both spaces. In particular, the comparison above is between complete bounded spaces that are not compact. $\square$