PomeLemma 23.1.46.712BD3EA719C Every bounded invariant $\omega $-plurisubharmonic function $u$, with $-L\leq u\leq 0$, admits smooth invariant $\omega $-plurisubharmonic approximants with the same bounds converging to $u$ in capacity.
Lemma 23.1.46712BD3EA719C
Proof. The compact group $G=U(3)\times (\mathbb R/\mathbb Z)$ acts transitively and holomorphically on $X$ by unitary transformations and positive scalar multiplication. It preserves $\omega $. Choose a nonnegative smooth central approximate identity $\rho _\epsilon $ on $G$, supported near its identity, and put
\[ u_\epsilon (p)=\int _ Gu(gp)\rho _\epsilon (g)\, \mathrm dg. \]
Convolution on a compact homogeneous space is smooth. Averaging preserves the current inequality and the bounds, and centrality makes convolution commute with the $U(2)$ action. Transporting the bounded tests defining capacity shows that $\operatorname {Cap}_\omega $ is $G$-invariant.
Suppose $L>0$. By quasi-continuity, choose an open set $O$ of arbitrarily small capacity such that $u$ is continuous on $X\setminus O$. When $g$ is sufficiently close to the identity, $u(gp)$ and $u(p)$ are uniformly close whenever both points lie outside $O$. Thus, outside $O$ and for small kernel support,
\[ \{ |u_\epsilon -u|>\delta \} \subset \{ P_\epsilon \mathbf1_ O>\delta /(4L)\} ,\qquad P_\epsilon \mathbf1_ O(p)=\int _ G\mathbf1_ O(gp)\rho _\epsilon (g)\, \mathrm dg. \]
Markov's inequality, followed by capacity invariance, gives
\[ \begin{split} \operatorname {Cap}_\omega \{ P_\epsilon \mathbf1_ O>\delta /(4L)\} & \leq \frac{4L}{\delta }\sup _{0\leq v\leq 1} \int _ XP_\epsilon \mathbf1_ O\, (\omega +\mathrm{dd}^{\mathrm{c}}v)^3\\ & \leq \frac{4L}{\delta }\operatorname {Cap}_\omega (O), \end{split} \]
where the supremum runs over $v\in \mathrm{PSH}(X,\omega )$. Letting the capacity of $O$ tend to zero proves the claim. If $L=0$, $u=0$ and no approximation is needed. $\square$
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