Proposition 18.16.29E5A9E7C5770

Proof. Write $e_ i=P_{\sigma _ i,F_ i}[u_ i]$. Sums of ordinary competitors in the separate coordinates are ordinary competitors on the product, and their regularized supremum is $\sum _ i e_ i\circ \pi _ i$. This gives the lower bound in the first identity.

For the upper bound we use induction on the number of factors, the case of one factor being trivial. Let $v\leq F$ be an ordinary competitor, so that $v\leq u+C$ for some constant $C$. Choose a constant $K$ with $u-K\leq F$ and replace $v$ by $\max (v,u-K)$. This preserves both bounds and ensures that the restrictions of $v$ to the slices used below are not identically $-\infty $.

Write $Y=Y'\times C_ m$, and on $Y'$ put $u'=\sum _{i<m}u_ i\circ \pi _ i$ and $F'=\sum _{i<m}F_ i\circ \pi _ i$. The regularized partial supremum

is $\sigma _ m$-plurisubharmonic by the local upper-envelope theorem of section 5.21: the functions $t\mapsto v(y',t)-F'(y')$ have the common upper bound $F_ m$, and at least one of them is not identically $-\infty $, by the lower bound $v\geq u-K$. The bound $v\leq u+C$ gives

Thus $a$ is a competitor for $e_ m$, and $a\leq e_ m$. If $u_ m(t)$ is finite, so is $e_ m(t)$, since a sufficiently low translate of $u_ m$ is a competitor for $e_ m$. The function $v(\cdot ,t)-e_ m(t)$ is then a potential in the product background on $Y'$, not identically $-\infty $, lying below $F'$ and at least as singular as $u'$. By induction, $v(\cdot ,t)\leq \sum _{i<m}e_ i\circ \pi _ i+e_ m(t)$. At a pole of $u_ m$, the bound $v\leq u+C$ already gives $v=-\infty $. Every competitor is therefore bounded by $\sum _ i e_ i\circ \pi _ i$, and taking the regularized supremum proves the ordinary identity.

For the ideal identity, put $B_ i=\int _{C_ i}\sigma _ i\geq 0$ and $\widehat\eta _ i=\eta _ i/A_ i$. By lemma 5.29.22, choose smooth functions $\chi _ i$ with

Let $E_ i$ be the atomic part of $\sigma _ i+\mathrm{dd}^{\mathrm{c}}u_ i$. If $B_ i>0$, apply lemma 20.3.20 to $u_ i+\chi _ i$ in the positive background $B_ i\widehat\eta _ i$. Subtracting $\chi _ i$ from the resulting atomic Green potential gives a $\sigma _ i$-plurisubharmonic function $g_ i$ such that

the second relation uses the smooth gauge bounds of proposition 18.5.1. If $B_ i=0$, lemma 5.29.22 gives $u_ i=-\chi _ i+c_ i$ for a constant $c_ i$, and $E_ i=0$; taking $g_ i=-\chi _ i$ gives the same conclusions. In either case,

Put $g=\sum _ i g_ i\circ \pi _ i$. Pullback and addition, lemma 5.37.7, lemma 5.37.4, give $g\sim _{\mathcal I}u$. The complex Hessian of $g$ is a countable sum of divisor currents supported on coordinate fibers, minus a smooth form. This divisor current has finite mass, because each coordinate fiber has constant mass for a fixed product metric and each $E_ i$ has finite mass. Thus lemma 5.38.5 applies, even when some $B_ i$ vanish. Every ideal competitor $v\leq F$ for $u$ therefore satisfies $v\preceq g$, and is bounded by $P_{\gamma ,F}[g]$.

Set $r_ i=P_{\sigma _ i,F_ i}[u_ i]_{\mathcal I}$. Invariance of the ordinary envelope under ordinary equivalence and proposition 18.12.2 give

By the ordinary identity already proved, applied to $g$, the ideal envelope of $u$ is therefore at most $\sum _ i r_ i\circ \pi _ i$. Conversely, each $r_ i$ lies below $F_ i$ and is multiplier-ideal equivalent to $u_ i$, again by proposition 18.12.2. Their sum is a $\gamma $-plurisubharmonic ideal competitor for $u$ below $F$, which proves the reverse inequality. No strict positivity of the backgrounds is needed in this comparison. $\square$