Theorem 6.16.3.E9EBCA5A681D There is a constant $c>0$, depending only on $(X,\omega )$ through the fixed geometry, with the following property. Let $\theta $ be a smooth closed real $(1,1)$-form on $X$, let $\varphi \in \mathrm{PSH}(X,\theta )$, and let $m\geq 1$ be an integer. Then there is a quasi-plurisubharmonic function $\varphi _ m$ on $X$ such that
\[ \theta +\mathrm{dd}^{\mathrm{c}}\varphi _ m\geq -\frac cm\omega , \qquad \varphi _ m\geq \frac1m\log \sum _ i|g_ i|^2+O(1)\quad \text{locally}, \]
where the $g_ i$ are local generators of $\mathcal I(m\varphi )$. The bounded term may depend on $m$, $\theta $, $\varphi $ and the point.
Proof.
By lemma 6.16.2, the functions $v_{j,m}\in \mathrm{PSH}(B_ j,\theta )$ satisfy $v_{j,m}\leq v_{l,m}+C_ B/m$ on $\overline{U'_ j}\cap \overline{U'_ l}$, also at their common zeros. Hence lemma 6.16.1 with $a=C_ B/m$ gives a quasi-plurisubharmonic function $\varphi _ m$ with $\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ m\geq -(C_ BC_\chi /m)\omega $, so the first conclusion holds with $c=C_ BC_\chi $, increased if necessary to make it positive. On $U_ j$ the function $\varphi _ m$ dominates $v_{j,m}=m^{-1}\log K_{j,m}-h_ j$. Since $h_ j$ is smooth, $\mathcal I(m(\varphi +h_ j))=\mathcal I(m\varphi )$, and the comparison of lemma 6.15.3 gives $v_{j,m}\geq m^{-1}\log \sum _ i|g_ i|^2+O(1)$ near each point of $U_ j$. This proves the second conclusion.
$\square$
Pome
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