Corollary 18.14.50B566FB6A5E6

Proof. The forward implication is part of proposition 18.14.1. For the converse, we approximate a smooth obstacle on the base by the induced obstacles of continuous obstacles upstairs. Suppose that the envelopes upstairs agree below every smooth obstacle. Uniform obstacle comparison extends this equality to continuous real obstacles. Fix a smooth real function $B$ on $Y$ and put $G=\widehat0$. The continuous obstacles

have induced base obstacles

with value $+\infty $ on $Z$. By the reduction and the assumed equality,

Write $q_ B=P_{\beta ,B}[w]_{\mathcal I}$. Then $q_ B\leq q_ j\leq q_1$, and the $q_ j$ decrease to a $\beta $-plurisubharmonic potential $r$, not identically $-\infty $, with convergence in $L^1$; indeed, their negative parts are bounded by that of $q_ B$, and corollary 5.25.3 applies. The set $Z$ has smooth volume zero: in each product chart of the submersion, the preimage of $Z$ lies in $H$, which has smooth volume zero, and Fubini's theorem gives the assertion on the base. Off $Z$, the bounds $q_ j\leq G_ j\downarrow B$ give $r\leq B$, and almost-everywhere order in lemma 5.24.3 extends the inequality to all of $Y$.

Each $q_ j$ is multiplier-ideal equivalent to $w$. To see this without assuming $G_ j$ finite, put $Q_0=P_\Theta [u]_{\mathcal I}$ and $Q_ j=P_{\Theta ,F_ j}[u]_{\mathcal I}$. Translation of the competitor families gives

Thus $Q_ j\sim Q_0\sim _{\mathcal I}u$ by theorem 18.4.2. Since $Q_ j=\Psi +q_ j\circ p$, cancellation and reflection of multiplier ideals under submersions, lemma 5.37.4, lemma 5.37.7, give $q_ j\sim _{\mathcal I}w$. The same bounded comparison with $P_\beta [w]_{\mathcal I}$ gives $q_ B\sim _{\mathcal I}w$.

Let $\pi $ be any smooth local modification of a coordinate neighborhood in the base. The bounds $q_ B\circ \pi \leq q_ j\circ \pi \leq q_1\circ \pi $ give decreasing local $L^1$ convergence to $r\circ \pi $. At every point $x$ of this modification, lemma 5.38.7 and the ordinary order give

All these Lelong numbers are therefore equal, and theorem 5.37.3 gives $r\sim _{\mathcal I}w$. Together with $r\leq B$, this makes $r$ a competitor for $q_ B$. Hence $r=q_ B$.

Put $\epsilon _ j=\max \{ 0,\sup _ Y(q_ j-B)\} $. The moving-point estimate in lemma 5.25.1 gives $\epsilon _ j\to 0$: otherwise a subsequence of maximizers of $q_ j-B$ converges to some $y$, where that estimate contradicts $q_ B(y)\leq B(y)$. Every ordinary competitor $b$ below $G_ j$ satisfies $b\leq q_ j$, so $b-\epsilon _ j\leq B$ and $b-\epsilon _ j\preceq w$. Taking regularized suprema yields

Since $q_ B\leq q_ j$, letting $j\to \infty $ proves $q_ B\leq P_{\beta ,B}[w]$. The reverse inequality always holds, so the two envelopes agree below $B$. Since $B$ was arbitrary, this proves the converse.

Under the relative-class hypothesis, proposition 18.12.2 gives equality of the two base envelopes below every smooth obstacle, and the equivalence just proved gives the stated equality upstairs.

In the curve case, put $G=\widehat0$ and $L=-\sup _ M\Psi $. The uniform bound in Step 2 of the proof of proposition 18.14.1 gives a finite $K$ with $\sup _ Yb\leq K$ for every $b\in \mathrm{PSH}(Y,\beta )$ below $G$. Since $G\geq L$, translation of the two competitor families gives

By proposition 13.29.1, corollary 5.27.15, the polar curvatures of these two base envelopes are $\mu _ w$ and $D_ w$, respectively. Thus equality of the envelopes upstairs below zero forces $\mu _ w=D_ w$. Conversely, under this equality of measures, corollary 13.29.2 gives equality below every smooth obstacle on the base. As in Step 5 of the proof of proposition 18.14.1, this extends to every induced obstacle $\widehat F$, and the reduction gives equality upstairs below every smooth obstacle. Taking $F=0$ proves the remaining implication. The same curve criterion identifies the equality of measures with membership of $w$ in the base relative envelope class of $q$, and the equivalence for ideal references in proposition 18.14.4 gives the last assertion upstairs. $\square$