Corollary 13.6.1BEC54934FC2C

Proof. In product charts choose smooth local potentials $a(x)$ of $\theta $ and $b(z)$ of $\gamma $. Then $W+a+b$ is plurisubharmonic, so by proposition 5.9.3 its restriction $W_ z+a+b(z)$ to a fiber is plurisubharmonic in $x$ or identically $-\infty $; the second alternative is excluded, because $W_ z\geq \varphi -C$ and $\varphi $ is not identically $-\infty $ on any open set. Hence $W_ z$ is quasi-plurisubharmonic with $\theta +\mathrm{dd}^{\mathrm{c}}W_ z\geq 0$, that is, $W_ z\in \mathrm{PSH}(X,\theta )$. All the products in (13.6.2) are finite by theorem 13.5.4, and (13.5.5) bounds the fiber masses uniformly in $z$, because the shifted background can be chosen from the fixed form $\theta $, independently of $W_ z$.

Put $w=W+a+b$, $v=\Phi +a+b$ and $O_ k=\{ w>-k\} \cap \{ v>-k\} $ in a product chart. Let $\rho \geq 0$ be a smooth function with compact support in the chart, let $G\geq 0$ be bounded and Borel, and let $\eta \geq 0$. Apply (13.4.6) to the bounded functions $\max \{ w,-k\} $ and $\max \{ v,-k\} $, with multiplicities $j$ and $r-j$, and to the weight $\rho G\mathbf1_{O_ k}$. The set $O_ k$ is contained in the level-$k$ set of the tuple formed by $j$ copies of $w$ and $r-j$ copies of $v$ (if $j=0$ or $j=r$, one of the two conditions defining $O_ k$ concerns a potential that does not occur in the tuple). Hence, by lemma 13.2.2, the restriction to $O_ k$ of the bounded mixed product equals the restriction to $O_ k$ of the non-pluripolar product, on the total space and, with $(O_ k)_ z$ in place of $O_ k$, on every fiber. These restrictions increase with $k$, and they increase to the non-pluripolar products, because the complements of $\bigcup _ kO_ k$ and of $\bigcup _ k(O_ k)_ z$ are contained in the pole sets of $w,v$ and of $w_ z,v_ z$, which are pluripolar and carry no mass. Monotone convergence, on the total space and in $z$, therefore gives the fiber formula for the non-pluripolar products with weight $\rho G$.

Taking $G=1$ gives the unweighted identity. Since $\Phi -C\leq W\leq \Phi +C$, the potentials $W$ and $\Phi $ have the same pole set, and so do $W_ z$ and $\varphi $; take $G=W-\Phi +C\in [0,2C]$ off this set and $G=C$ on it, and subtract $C$ times the unweighted identity. Summing over a finite partition of unity by such functions $\rho $ (proposition 5.2.6), and writing a signed $\eta $ as a difference of nonnegative forms, gives (13.6.2). All integrals are finite by the uniform mass bound and $|W-\Phi |\leq C$, and the values assigned on the common pole set do not matter, because this set is pluripolar and carries no mass for any of the products. $\square$