PomeLemma 8.9.8.A1E60FC7AA88 Let $U$ be a coordinate ball compactly contained in a holomorphic chart. There is a smooth strictly plurisubharmonic $\rho $ on a neighborhood of $\overline U$ such that $\mathrm{dd}^{\mathrm{c}}\rho \geq 2\omega $ and, for every bounded admissible $u$ on $X$, $u+\rho $ is $m$-subharmonic relative to $\omega $ on $U$. Adding $\rho $ to the approximations in 8.6.6 gives uniformly bounded smooth decreasing approximations in this local class.
Lemma 8.9.8A1E60FC7AA88
Proof. Take $\rho =C|z|^2$ in the chart. On the fixed compact set $\overline U$, a sufficiently large $C$ gives $\mathrm{dd}^{\mathrm{c}}\rho \geq 2\omega $. An upper test $q$ for $u+\rho $ at $x$ gives the upper test $q-\rho $ for $u$ there. Hence
\[ \mathrm{dd}^{\mathrm{c}}q=(\omega +\mathrm{dd}^{\mathrm{c}}(q-\rho ))+(\mathrm{dd}^{\mathrm{c}}\rho -\omega ) \in \overline{\Gamma _ m(\omega )}, \]
since the second summand is positive and the cone is preserved by adding nonnegative forms. Smooth addition preserves the canonical representative. The same identity applies pointwise to every smooth admissible approximation. Adding a fixed smooth function preserves pointwise decrease and convergence, and its bounded norm on $\overline U$ preserves the common local bound. $\square$
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