Lemma 18.16.1D9950FF0BE1C

Proof. The function $w$ is $\beta $-plurisubharmonic and has supremum zero. We prove $P_\beta [w]=w$ by induction on the number of factors, the case of one factor being the identity $P_{\eta _1}[q_1]=q_1$. Let $v\leq 0$ be an ordinary competitor, and replace it by $\max (v,w)$. This preserves $v\preceq w$ and ensures that the restrictions of $v$ to slices are not identically $-\infty $ wherever the corresponding curve potentials are finite.

Write $Y=Y'\times C_ m$, and take the regularized partial supremum

The restrictions of $v$ to the slices $\{ y'\} \times C_ m$ are $\eta _ m$-plurisubharmonic or identically $-\infty $. They are bounded above by zero, and at least one of them is not identically $-\infty $, since $v\geq w$. The local upper-envelope theorem from section 5.21 therefore makes $a$ an $\eta _ m$-plurisubharmonic function. The global inequality $v\leq w+C$ gives $a\leq q_ m+C$, because the other normalized potentials have supremum zero. Thus $a$ is a competitor for $P_{\eta _ m}[q_ m]$, and $a\leq P_{\eta _ m}[q_ m]=q_ m$.

For $t$ with $q_ m(t)$ finite, the function $v(\cdot ,t)-q_ m(t)$ is a potential in the product background on $Y'$, not identically $-\infty $; it is nonpositive and at least as singular as $\sum _{i<m}q_ i\circ \pi _ i$. The induction hypothesis bounds it by that sum. Hence $v\leq w$ off the poles of $q_ m$. At those poles the original bound $v\leq w+C$ gives the same inequality. Taking the envelope proves the ordinary identity.

For the ideal identity, let $E_ i$ be the atomic part of the polar curvature of $q_ i$, and take the Green potentials $g_ i$ of lemma 20.3.20. They satisfy

By lemma 5.37.7, lemma 5.37.4, $g=\sum _ i g_ i\circ \pi _ i$ is multiplier-ideal equivalent to $w$. Its complex Hessian is a closed countable sum of the divisors $\pi _ i^{-1}(x)$ minus the smooth form $\sum _ i E_ i(C_ i)\pi _ i^*\eta _ i/A_ i$. Each coordinate fiber has constant mass for the product metric, so the divisor part has finite mass. Moreover $g$ is $\beta $-plurisubharmonic since $E_ i(C_ i)\leq A_ i$. Therefore corollary 18.4.9 gives $P_\beta [g]_{\mathcal I}=P_\beta [g]$. Since $g$ is ordinarily equivalent to $\sum _ iq_{i,\mathcal I}\circ \pi _ i$, the ordinary identity, applied to the models $q_{i,\mathcal I}$, shows that this envelope is $\sum _ iq_{i,\mathcal I}\circ \pi _ i$. As $w\sim _{\mathcal I}g$, the ideal envelope of $w$ is the same, which proves the assertion. $\square$