Theorem 13.5.43783DE0B4D00

Proof. The decomposition $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j$ exists by theorem 10.2.15. Since $\tau _ j\geq \theta _ j$ and $\tau _ j\geq 0$, $\varphi _ j$ is $\tau _ j$-plurisubharmonic, and so is every constant; hence $\max \{ \varphi _ j,-k\} $ is a bounded $\tau _ j$-plurisubharmonic function, and

is a closed positive current. By proposition 13.5.1, the total mass $\int _ XR_{1,k}\wedge \cdots \wedge R_{p,k}\wedge \omega ^{n-p}$ equals the right side of (13.5.5) for every $k$.

Local finiteness. Let $W$ be a relatively compact coordinate chart, and choose smooth functions $h_ j$ and $h$ near $\overline W$ with $\mathrm{dd}^{\mathrm{c}}h_ j=\theta _ j$ and $\mathrm{dd}^{\mathrm{c}}h=\omega $. Then $u_ j=\varphi _ j+h_ j$ is a plurisubharmonic potential of $T_ j$ on $W$. Let $B$ be an integer with $h_ j\leq B$ on $W$ for all $j$, fix an integer $l\geq 1$, and put $O_ l(u)=\bigcap _ j\{ u_ j>-l\} $. On $W\cap O_ l(u)$ we have $\varphi _ j>-l-B$, so for every integer $k\geq l+B$ the bounded plurisubharmonic functions $\max \{ u_ j,-l\} +Ch$ and $\max \{ \varphi _ j,-k\} +h_ j+Ch$ both equal $u_ j+Ch$ there; the second is a local potential of $R_{j,k}$. By plurifine locality,

Expanding the left side into positive terms gives

Let $K\subset W$ be compact. On a neighborhood of $K$ in $W$ we have $\beta \leq c\, \omega $ for some $c>0$ (lemma 6.14.1), so by the last display and (13.2.8), the trace masses on $K$ of the truncated local products are at most $c^{n-p}$ times the right side of (13.5.5), uniformly in $l$. Hence the non-pluripolar product is locally finite everywhere, and $\langle T_1\wedge \cdots \wedge T_ p\rangle $ is a closed positive current by theorem 13.3.1.

Mass bound. By the same argument, the currents $T_ j+C\omega =\tau _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j$, with local potentials $u_ j+Ch$, have a locally finite non-pluripolar product. The sets $G_ k=\bigcap _ j\{ \varphi _ j>-k\} $ increase and exhaust the complement of the union of the pole sets of the $\varphi _ j$. On $G_ k$ the global truncations $\max \{ \varphi _ j,-k\} $ coincide with $\varphi _ j$. If $B'$ is an integer with $h_ j+Ch\geq -B'$ on $W$ for all $j$, then on $G_ k\cap W$ all local potentials satisfy $u_ j+Ch>-k-B'$, so they coincide there with their truncations at level $k+B'$, and also with the local potentials $\max \{ \varphi _ j,-k\} +h_ j+Ch$ of $R_{j,k}$. By plurifine locality and lemma 13.2.2, $\mathbf1_{G_ k}\bigwedge _ jR_{j,k}=\mathbf1_{G_ k}\langle \bigwedge _ j(T_ j+C\omega )\rangle $, and these restrictions increase to $\langle \bigwedge _ j(T_ j+C\omega )\rangle $. Their total masses against $\omega ^{n-p}$ are at most the right side of (13.5.5), hence so is that of the limit. Finally, expanding $\langle \bigwedge _ j(T_ j+C\omega )\rangle $ by multilinearity (proposition 13.2.6) into a sum of positive currents, one of which is $\langle \bigwedge _ jT_ j\rangle $, we obtain $\langle \bigwedge _ j T_ j\rangle \leq \langle \bigwedge _ j(T_ j+C\omega )\rangle $, which proves (13.5.5). $\square$