Corollary 10.4.12B72607AB8B41

Proof. Near a point, let $h$ be smooth with $\mathrm{dd}^{\mathrm{c}}h=\theta $. Off the support of $D$ the function $v+h=(\varphi +h)-\sum _ ia_ i\log |s_ i|^2$ is plurisubharmonic, since $\log |s_ i|^2$ is pluriharmonic there, and it is bounded above. On a connected coordinate neighborhood of the point, the support of $D$ lies in the zero set of the holomorphic function $\prod _ is_ i\not\equiv 0$, so theorem 5.11.1 extends $v+h$ to a plurisubharmonic function $\widetilde v$. The values of $\widetilde v$ on the support of $D$ are upper limits of values of $v+h$, so $\widetilde v$ is bounded, and $\widetilde v=v+h$ outside the null set $\operatorname{Supp}D$. Since $\mathrm{dd}^{\mathrm{c}}\log |s_ i|^2=[D_ i]$ (lemma 9.3.19), the current $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ satisfies $T-[D]=\mathrm{dd}^{\mathrm{c}}\widetilde v\geq 0$ near the point, where $[D]=\sum _ ia_ i[D_ i]$. Thus $T-[D]$ is a positive current in $\{ \theta \} -\{ D\} $ with locally bounded potentials. If $T\geq \delta \omega $, apply the same argument to $T-\delta \omega $, whose potential has the same logarithmic singularities; then $T-[D]\geq \delta \omega $. In both cases proposition 10.4.6 gives the assertion. $\square$