Proposition 18.14.4A399EFC10B9E

Proof. The ordinary order $u\preceq v$ is equivalent to $w\preceq \varphi $. One implication follows by addition and pullback. Conversely, cancel $\Psi $ outside $H$, which has smooth volume zero; Fubini's theorem in product charts gives the base inequality almost everywhere, and lemma 5.24.3 makes it hold everywhere. Membership in the relative envelope class requires, for every $a\in (0,1)$, a stretching competitor $h\in \mathrm{PSH}(M,\Theta )$ with

A base competitor $c\in \mathrm{PSH}(Y,\beta )$ with $a\varphi +(1-a)c\leq w$ gives such an $h$, namely $h=\Psi +c\circ p$.

Conversely, put $\rho _ j=\nu (w\circ p,H_ j)$ and $\sigma _ j=\nu (\varphi \circ p,H_ j)$. The ordinary order $w\preceq \varphi $ gives $\rho _ j\geq \sigma _ j\geq 0$. Comparing generic Lelong numbers along $H_ j$ in the inequality above gives

Consequently $\nu (h,H_ j)\geq \lambda _ j+(\rho _ j-a\sigma _ j)/(1-a)\geq \lambda _ j$. Divisorial removal and descent along the fibers give $h=\Psi +c\circ p$ with $c\in \mathrm{PSH}(Y,\beta )$. Canceling $\Psi $ outside $H$ and the pole sets, and applying Fubini's theorem, we obtain $a\varphi +(1-a)c\leq w$ almost everywhere on the base, and hence on all of $Y$ by lemma 5.24.3. This proves the first equivalence.

For the ideal reference, theorem 18.4.2 gives $q_ M\sim _{\mathcal I}u$. The generic divisorial coefficients of $q_ M$ are therefore at least $\lambda _ j$, and removal and descent give

Cancellation in lemma 5.37.4 gives $b\circ p\sim _{\mathcal I}w\circ p$. By lemma 5.37.7, these ideals are extensions of the base ideals, so restriction along any local section gives $b\sim _{\mathcal I}w$, even over a fiber contained in $H$.

Put $C=\max \{ 0,\sup _ M\Psi \} $ and $K=\max \{ 0,\sup _ Yb\} $. The potential $\Psi +q_ Y\circ p-C$ is nonpositive and multiplier-ideal equivalent to $u$, so it is a competitor for $q_ M$. Likewise, $b-K$ is a competitor for $q_ Y$. Canceling $\Psi $ in the first inequality almost everywhere and extending as above gives

Thus $b\sim q_ Y$. The first equivalence with reference $b$, together with the invariance of the class under ordinarily equivalent references in proposition 16.28.62, proves the second equivalence. This argument works with the ideal model itself and does not require the induced fiber obstacle to be finite. If $w$ belongs to the base relative class, proposition 18.12.2 gives equality of the two envelopes upstairs below every smooth obstacle.

For any ordinary representative of $u$ or $q_ M$, removal and descent give the form $\Psi +c\circ p$ with $c\in \mathrm{PSH}(Y,\beta )$. Outside $H$, replace $c$ locally by $\max (c,-k)$ and keep $\Psi $. The resulting bounded quasi-plurisubharmonic potentials have currents pulled back from the $d$-dimensional base, so their $n$th products vanish by lemma 5.28.4, since $n>d$. By plurifine locality, this product agrees with the original extended measure on $\{ c\circ p>-k\} $. These sets cover the locus where $c\circ p$ is finite, and the pole sets and $H$ receive no mass. This proves the mass and volume assertions. $\square$