Proposition 18.11.28B44DB3F02D3

Proof. The smooth function $\theta '/\theta $ has integral one against the probability measure $\theta $, and it is not constant: otherwise $\mathrm{dd}^{\mathrm{c}}F=0$, and the harmonic function $F$ would be constant on $Y$. Choose $c$ with $\min _ Y(\theta '/\theta )<c<1$, so that $\theta '\not\geq c\theta $.

Next choose probability measures $\mu _ j$ without atoms, each carried by a polar set, that converge weakly to $\theta $. To construct them, start with finite atomic probability measures converging weakly to $\theta $, as in theorem 18.6.4, and replace each atom by a nonatomic probability measure carried by a polar Cantor set in a small coordinate disk about that atom, keeping its weight; the construction in example 13.28.14 provides such measures in every disk. If the diameters of the disks tend to zero uniformly, uniform continuity of continuous test functions preserves weak convergence, and the finite union of the polar carriers is polar.

Let $q_ j$ be the normalized Green potentials with $\theta +\mathrm{dd}^{\mathrm{c}}q_ j=\mu _ j$ and $\sup _ Yq_ j=0$. By normalized compactness, the sequence has $L^1$ cluster values, and each of them has curvature $\theta $ and supremum zero, hence vanishes. Thus $q_ j\to 0$ in $L^1$. Put $u_ j=cq_ j$. Then

Since $\mu _ j$ has no atoms, all point Lelong numbers of $u_ j$ vanish. Because modifications of a smooth curve are isomorphisms, theorem 5.37.3 gives trivial multiplier ideals at every weight. Thus $P_\theta [u_ j]_{\mathcal I}=0$, all controlled ideal relations with $[0]$ hold, and every divisorial distance to $[0]$ vanishes. The bounds above give convergence of the ordinary envelopes to zero in $L^1$. Consequently $[u_ j]\to [0]$ in $\tau _{\theta ,0}$.

Set $v_ j=P_{\theta ,F}[u_ j]$. Every competitor $h\preceq u_ j$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}h\geq c\mu _ j$, by corollary 5.27.15. The family of competitors below $F$ contains a translate of $u_ j$ and is stable under finite maxima, so Choquet selection and distributional convergence of increasing maxima show that $v_ j$ satisfies the same inequality. If $[u_ j]\to [0]$ in $\tau _{\theta ,F}$, the ordinary-envelope coordinate would give $v_ j\to P_{\theta ,F}[0]=F$ in $L^1$, because $F$ is a bounded $\theta $-plurisubharmonic competitor. Passing to the limit in the current inequality would then give $\theta '\geq c\theta $, a contradiction.

By proposition 18.11.1, $[u]\mapsto [u-F]$ is a uniform isomorphism from $\tau _{\theta ,F}$ onto $\tau _{\theta ',0}$, so the sequence above shows that $\Gamma _ F$ is discontinuous. The same argument applied to $\theta '$ and $-F$ shows that the inverse is discontinuous. Under this transport, continuity of $\Gamma _ F$ and of its inverse are precisely the two comparisons between $\tau _{\theta ,0}$ and $\tau _{\theta ,F}$, which proves incomparability. $\square$