Proposition 18.26.26FC2CCCE62C86

Proof. Place the polydisk $D$ of lemma 18.26.16 in a coordinate chart, and use one cutoff function $0\leq \chi \leq 1$, compactly supported there and equal to one near $S=K^ n$. For $a\in B$, extend $H_ a=\chi h_ a$ by zero and put $u_ a=tH_ a$. We first prove a curvature bound that is uniform in $a$.

Choose a compact neighborhood $\Sigma $ of the supports of the derivatives of $\chi $, disjoint from $S$. The isotropic maximum $\max _ j g(z_ j)$ is bounded below on $\Sigma $ by $-B_0$, for some $B_0>0$. Since $g\leq 0$ and $a_ j\leq \beta $, it follows that $-\beta B_0\leq h_ a\leq 0$ there. If the $j$th branch $a_ jg(z_ j)$ attains the maximum, then $a_ jg(z_ j)\geq -\beta B_0$, so $g(z_ j)\geq -\beta B_0/\alpha $. Within the fixed compact range of the coordinates, this superlevel set stays away from $K$, and $g$ and its gradient are bounded on a neighborhood of it. Thus all maximizing branches have uniformly bounded gradients, and the maxima $h_ a$ satisfy one local Lipschitz bound on $\Sigma $, independent of $a\in B$.

In the distributional product rule for $\mathrm{dd}^{\mathrm{c}}(\chi h_ a)$, the term $\chi \mathrm{dd}^{\mathrm{c}}h_ a$ is positive. The remaining terms contain derivatives of $\chi $ together with $h_ a$ or its first derivatives, so the preceding bounds control their coefficients uniformly. The cutoff argument of proposition 18.26.19 therefore gives one $C>0$ with $\mathrm{dd}^{\mathrm{c}}H_ a\geq -C\omega $ for every $a\in B$. Choose $t_0<1/C$ with $t_0^ n\beta ^ n<A-\delta $. Then all $u_ a$, $0<t\leq t_0$, are normalized and satisfy $\omega +\mathrm{dd}^{\mathrm{c}}u_ a\geq (1-Ct)\omega $.

By theorem 11.1.7, all bounded $\omega $-plurisubharmonic functions have mass $A$, so $v_-(\omega )=v_+(\omega )=A$. For any $v\sim u_ a$, the global limit of the measures of the truncations has total mass $A$, as in the proof of corollary 18.26.20. Near $S$, the difference between $v$ plus a smooth local correction and $th_ a$ is bounded. By lemma 5.27.19, lemma 18.26.16, the residual measure of $v$ is

extended by zero from the chart. Every corrected closed power of lower degree gives $S$ zero coefficient mass, so this is also the restriction to $S$ of the global limit of the truncations. Off $S$ that limit equals the non-pluripolar measure, which proves that every such representative has mass

Hence, by theorem 13.19.5, ordinary projection preserves both singularity volumes, and by theorem 13.19.6 it preserves the residual measure. Setting $p_ a=P_\omega [u_ a]$ proves the formulas for the residual measure and the volumes simultaneously for all $a\in B$.

For $a,b\in B$ and $c_ j=\min (a_ j,b_ j)$, positivity of the weights and $g\leq 0$ give, pointwise and also at the poles,

The invariance of the comparison under ordinary projection in proposition 13.22.4 and the definition of the comparison give

For the boxes $R_ a=\prod _ j[0,a_ j]\subset \mathbb R^ n$ and Lebesgue measure $\lambda $, the last expression is $t^ n\lambda (R_ a\mathbin {\triangle }R_ b)$, since $R_ a\cap R_ b=R_ c$; equivalently, it is $t^ n\| \mathbf1_{R_ a}-\mathbf1_{R_ b}\| _{L^1(\lambda )}$. This proves the triangle inequality within the family. A slab in the coordinate where $|a_ j-b_ j|$ is largest, with every other coordinate in $[0,\alpha ]$, lies in the symmetric difference up to its boundary, and its volume gives the lower bound. The symmetric difference is covered by the $n$ coordinate slabs between $a_ j$ and $b_ j$, with every other coordinate in $[0,\beta ]$, and their total volume gives the upper bound. In particular, the distance separates the parameters, and the two estimates show that it induces the Euclidean topology of $B$. A sequence that is Cauchy for the distance has Cauchy weights in the Euclidean sense, whose limit lies in $B$, and the upper bound gives convergence for the distance. Compactness follows from the same bounds.

The support and Lelong-number argument in the proof of proposition 18.26.17 applies for every positive weight vector. Every exceptional divisor mapping into $S$ maps to a point, and comparison with $ta_1g(z_1\circ \pi )$ shows that the generic Lelong number along it is zero. Thus every first polar current is zero. The same comparison gives zero Lelong numbers at all points and trivial multiplier ideals at all weights. The strict volume deficit $t^ n\prod _ j a_ j$ shows that $p_ a\ne 0$, whereas the ideal envelope of $p_ a$ is zero, so $p_ a$ is not an ideal model.

Finally, the constraint $\prod _ ja_ j=P$ defines a closed subset of $B$, so the subfamily is complete. Taking logarithms coordinatewise identifies it with

For $\alpha ^ n<P<\beta ^ n$, this convex polytope meets the interior of the cube at the point with $x_ j=(\log P)/n$ for all $j$, and it has dimension $n-1$. The formulas for the volume and the residual measure are constant on it, while the distance separates its members. For example, taking $\alpha =1/2$, $\beta =2$, $P=1$, $a=(2,1/2,1,\ldots ,1)$ and $b=(1,1,1,\ldots ,1)$ gives two models with common residual measure $t^ n\rho $, both volumes equal to $A-t^ n$, and distance $t^ n$. $\square$