Lemma 13.4.25D93FCFBA57F

Proof. Let $E$ be a pluripolar subset of the coordinate domain $U$ on which $T$ lives. By definition 5.47.2 and since $U$ has a countable basis, $E$ is covered by countably many compact sets $K$, each having a compact neighborhood $L$ with $K\subset \operatorname{Int}L$, together with an open neighborhood $W$ of $L$ and a function $V\in \mathrm{PSH}(W)$ such that $E\cap K\subset \{ V=-\infty \} $; shrinking $W$ and subtracting a constant, we may assume $V\leq 0$. The set $\{ V=-\infty \} $ is a pluripolar Borel set, so it suffices to show that $K\cap \{ V=-\infty \} $ has measure zero for the trace measure of $\mathrm{dd}^{\mathrm{c}}v\wedge T$.

Put $\mu =\mathbf1_ LT\wedge \beta ^ r$. This is a finite measure that does not charge $\{ V=-\infty \} $, so $V$ is finite $\mu $-almost everywhere. Hence $\min \{ -V/R,1\} \to 0$ $\mu $-almost everywhere as $R\to \infty $, and by dominated convergence we may choose numbers $R_ j\geq 2^ j$ such that

Put $\widetilde V=\sum _{j\geq 1}\max \{ V/R_ j,-1\} $. The partial sums are plurisubharmonic by proposition 5.9.5, and they decrease, because every term is nonpositive. Their limit satisfies $\widetilde V\geq \sum _ jV/R_ j\geq V$, since $V\leq 0$ and $\sum _ j1/R_ j\leq 1$; hence it is not identically $-\infty $ on any component of $W$, and it is plurisubharmonic by theorem 5.10.1. Moreover $\widetilde V\leq 0$, $\widetilde V=-\infty $ on $\{ V=-\infty \} $, where every term equals $-1$, and by monotone convergence

By lemma 13.4.1, applied on $W$, the function $-\widetilde V$ is integrable on $K$ for the trace measure $\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}$ of $\mathrm{dd}^{\mathrm{c}}v\wedge T$. Hence this measure gives zero mass to $K\cap \{ \widetilde V=-\infty \} \supset K\cap \{ V=-\infty \} $, which proves the first assertion. The second follows by induction on $q$, since $\mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge T=\mathrm{dd}^{\mathrm{c}}v_1\wedge (\mathrm{dd}^{\mathrm{c}}v_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge T)$, where the base current in parentheses is closed and positive of bidimension $(r-q+1,r-q+1)$. $\square$