PomeAC7947F570DBCorollary 20.4.24.AC7947F570DB Let $g:Z\to Y$, $\gamma $ and $\gamma '$ be as in proposition 20.4.22. Let $D_ j$ be prime divisors on $Y$, with integration currents $[D_ j]$, and let $c_ j>0$. If $S=\sum _ j c_ j[D_ j]$ is a positive closed current of finite mass, then
as positive currents of finite mass. The trace masses on $Z$ of the positive tails of this series tend to zero. In particular, $g^*S$ is again purely divisorial.
Proof. Apply the proposition to the increasing finite partial sums $S_ N$ and to their positive closed tails $S-S_ N$. Pullback is additive on positive closed currents, by addition of local potentials. Thus $g^*S-g^*S_ N=g^*(S-S_ N)\geq 0$, and its trace mass is at most $C_ g$ times the trace mass of $S-S_ N$, which tends to zero. The pullback of a finite divisor is the usual pullback of its local equations, which identifies $g^*S_ N$ with the corresponding partial sum of the series, and hence $g^*S$ with its limit. Its terms are integration currents of prime divisors with nonnegative coefficients, and regrouping this countable positive sum of finite mass does not change it. $\square$
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