Lemma 13.8.1663159428707

Proof. We replace the factors $T_ j$ by $T'_ j$ one at a time; the intermediate tuples satisfy the same hypotheses, so it suffices to treat one step, in which a single factor changes. Write it as $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ before and $\theta +\mathrm{dd}^{\mathrm{c}}\psi $ after the change, and let $\Theta $ be the non-pluripolar product of the remaining factors (some primed, some not) wedged with $\omega ^{n-p}$. By theorem 13.5.4, theorem 13.3.1, $\Theta $ is a closed positive current of bidimension $(1,1)$, and it gives zero mass to the set $A$, which is pluripolar and Borel, being closed and locally contained in the pole sets of plurisubharmonic functions.

On the open set $X\setminus A$ all potentials are locally bounded, so there the non-pluripolar products of the remaining factors are Bedford–Taylor products, and by proposition 13.4.3 the products of the two tuples, wedged with $\omega ^{n-p}$, are the Bedford–Taylor products $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge \Theta $ and $(\theta +\mathrm{dd}^{\mathrm{c}}\psi )\wedge \Theta $ with base $\Theta $; here $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge \Theta =\mathrm{dd}^{\mathrm{c}}(h+\varphi )\wedge \Theta $ for smooth local potentials $h$ of $\theta $. More generally, if $\Omega '$ is a smooth closed real $(1,1)$-form and $\varrho \in \mathrm{PSH}(X,\Omega ')$ is locally bounded on $X\setminus A$, then the measure $(\Omega '+\mathrm{dd}^{\mathrm{c}}\varrho )\wedge \Theta $ on $X\setminus A$ is the restriction of a global non-pluripolar product wedged with $\omega ^{n-p}$, so it has finite mass by theorem 13.5.4. Since none of these products charges $A$, it suffices to prove

We prove it by applying Stokes' formula on $X\setminus A$ to functions that equal $\varphi -k$ near $A$. After adding constants we may assume $\varphi \geq \psi $ and $\tau \leq 0$. Since $\tau $ is quasi-plurisubharmonic and $X$ is compact, there is $C_\tau \geq 0$ with $C_\tau \omega +\mathrm{dd}^{\mathrm{c}}\tau \geq 0$ (lemma 6.14.4(4)). Put $\Omega =(C_\tau +1)\omega $. Fix $0<\varepsilon \leq 1$ and put $f_ k=\max \{ \psi +\varepsilon \tau ,\varphi -k\} $. Since $\theta +\mathrm{dd}^{\mathrm{c}}\psi \geq 0$ and $\varepsilon (C_\tau \omega +\mathrm{dd}^{\mathrm{c}}\tau )\geq 0$, both $\psi +\varepsilon \tau $ and $\varphi -k$ are $(\theta +\Omega )$-plurisubharmonic, hence so is $f_ k$, and $f_ k$ is locally bounded on $X\setminus A$. Since $\psi +\varepsilon \tau -\varphi \leq \varepsilon \tau $, we have $f_ k=\varphi -k$ on the open neighborhood $\{ \tau <-k/\varepsilon \} $ of $A$. Hence on $X\setminus A$ the function $g_ k=f_ k-\varphi +k=\max \{ \psi +\varepsilon \tau -\varphi +k,0\} $ takes values in $[0,k]$, and it vanishes outside the compact set $L_ k=\{ \tau \geq -k/\varepsilon \} \subset X\setminus A$.

Stokes' formula. With a smooth local potential $h$ of $\theta +\Omega $, the Bedford–Taylor recursion gives, on $X\setminus A$,

because $\mathrm{dd}^{\mathrm{c}}(k\Theta )=0$. The signed measure $\lambda _ k=\mathrm{dd}^{\mathrm{c}}(g_ k\Theta )$ vanishes outside $L_ k$. Let $\kappa $ be a smooth function with compact support in $X\setminus A$ that equals one on a neighborhood of $L_ k$. Then $\lambda _ k(X\setminus A)=\int \kappa \, \mathrm d\lambda _ k=(g_ k\Theta )(\mathrm{dd}^{\mathrm{c}}\kappa )=0$, because $\mathrm{dd}^{\mathrm{c}}\kappa $ vanishes near the support of $g_ k\Theta $. Since the measures on the left have finite mass, we obtain

Passage to the limit. On $X\setminus A$ the functions $f_ k$ decrease to the locally bounded function $\psi +\varepsilon \tau $ as $k\to \infty $. By decreasing continuity with the fixed base $\Theta $ (theorem 7.3.7), applied locally, the positive measures $(\theta +\Omega +\mathrm{dd}^{\mathrm{c}}f_ k)\wedge \Theta $ converge weakly on $X\setminus A$ to $(\theta +\Omega +\mathrm{dd}^{\mathrm{c}}(\psi +\varepsilon \tau ))\wedge \Theta $. The mass of the limit on the open set $X\setminus A$ is at most the lower limit of the masses, because it is the supremum of the integrals of test functions $0\leq \kappa \leq 1$ supported in $X\setminus A$, and each such integral is the limit of the corresponding integrals for the approximating measures. Therefore

By additivity of Bedford–Taylor products, the measure on the left is the sum of the positive measures $(\theta +\mathrm{dd}^{\mathrm{c}}\psi )\wedge \Theta $, $\varepsilon (C_\tau \omega +\mathrm{dd}^{\mathrm{c}}\tau )\wedge \Theta $ and $(\Omega -\varepsilon C_\tau \omega )\wedge \Theta $, and the measure on the right is the sum of $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge \Theta $ and $\Omega \wedge \Theta $; all these measures have finite mass on $X\setminus A$. Dropping the second term on the left, we get

and letting $\varepsilon \downarrow 0$ proves the desired inequality. If the differences are bounded, the same argument with the roles of $\varphi $ and $\psi $ exchanged gives equality. $\square$