PomeLemma 16.28.12.7A591F1F87FA For every pluripolar set $E\subseteq \mathbb C^ n$ and every $\delta >0$, there is $b\in \mathcal L_\delta $ with $b=-\infty $ on $E$. Given a non-pluripolar compact set $K$, it can be chosen with $b\leq 0$ on $K$.
Lemma 16.28.127A591F1F87FA
Proof. Put $E_ j=E\cap \overline B(0,j)$. Each $E_ j$ is pluripolar as a subset of $\mathbb P^ n$: it is locally pluripolar near every affine point, and a neighborhood of the hyperplane at infinity does not meet this bounded set. By Josefson's theorem on compact manifolds, [Theorem 1.1.5, SGPT], there is a quasi-plurisubharmonic function on $\mathbb P^ n$, not identically $-\infty $, that equals $-\infty $ on $E_ j$. Rescaling it so that it becomes $\theta _1$-plurisubharmonic and subtracting its supremum gives
\[ p_ j\in \mathrm{PSH}(\mathbb P^ n,\theta _1),\qquad p_ j\leq 0,\qquad p_ j|_{E_ j}=-\infty . \]
The $L^1$ norms of these normalized potentials are uniformly bounded by theorem 5.25.2. The partial sums
\[ p_ N=\sum _{j=1}^ N2^{-j}p_ j \]
decrease, have uniformly bounded $L^1$ norms, and remain $\theta _1$-plurisubharmonic because $\theta _1\geq 0$ and the coefficients sum to at most one. By corollary 5.25.3, their limit $p$ belongs to $\mathrm{PSH}(\mathbb P^ n,\theta _1)$. Every point of $E$ is a pole of one of the summands, hence $p=-\infty $ on $E$. By lemma 16.28.9,
\[ b=\delta (p+\chi /2)\in \mathcal L_\delta \]
has the required properties. Its supremum on $K$ is finite, since $b$ is bounded above on the compact set $K$ and is not identically $-\infty $ on $K$, which is non-pluripolar. Subtracting this supremum gives $b\leq 0$ on $K$. $\square$
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