Lemma 16.27.149691FF58A24

Proof. Apply theorem 6.17.6 with $\kappa $ as the Kähler form of reference. It gives potentials $\Phi _ j$ decreasing to $p$ with $\kappa +\mathrm{dd}^{\mathrm{c}}\Phi _ j\geq -\varepsilon _ j\kappa $, where $\varepsilon _ j\downarrow 0$, and with gentle analytic singularities of type $(2^{-j},\mathcal I(2^ jp))$. Since $p$ is bounded, $\mathrm e^{-2^ jp}$ is bounded, so $\mathcal I(2^ jp)$ is the unit ideal. In the local presentations $\Phi _ j=2^{-j}\log \sum _ i|g_ i|^2+b$ of definition 6.14.5, the functions $g_ i$ therefore generate the unit ideal and have no common zero, so $\Phi _ j$ is bounded. Hence the smooth function $\exp (2^ j\Phi _ j)$ has no zeros, and $\Phi _ j=2^{-j}\log \exp (2^ j\Phi _ j)$ is smooth. Put $p_ j=\Phi _ j/(1+\varepsilon _ j)$. Then

The bounds $p\leq \Phi _ j\leq \Phi _1$ give $|p_ j|\leq |\Phi _ j|\leq \max \{ \sup _ X|p|,\sup _ X|\Phi _1|\} $, and $p_ j\to p$ pointwise because $\Phi _ j\to p$ and $\varepsilon _ j\to 0$ (unlike the $\Phi _ j$, the $p_ j$ need not decrease). $\square$