PomeLemma 5.40.1.294BB4EEA84F Let $\theta $ be a smooth real $(1,1)$-form on $X$, let $K\subseteq X$ be compact and non-pluripolar, and let $w\in C(K,\mathbb R)$. There is a constant $C=C(\theta ,K,w)$ such that every $a\in \mathrm{PSH}(X,\theta )$ satisfying $a\leq w$ quasi-everywhere on $K$ has $\sup _ X a\leq C$.
Lemma 5.40.1294BB4EEA84F
Proof. Otherwise there are such functions $a_ j$ with
\[ M_ j:=\sup _ Xa_ j\geq 2^ j+\max _ Kw. \]
The normalized functions $v_ j=a_ j-M_ j\leq 0$ have uniformly bounded $L^1$ norms by theorem 5.25.2. Choose $A>0$ with $\theta \leq A\omega $. The partial sums
\[ s_ N=\sum _{j=1}^ N2^{-j}v_ j \]
decrease, have uniformly bounded $L^1$ norms, and satisfy $\mathrm{dd}^{\mathrm{c}}s_ N\geq -A\omega $. By corollary 5.25.3, their limit $s$ is a quasi-plurisubharmonic function; in particular, $s$ is not identically $-\infty $.
Let $N_ j\subseteq K$ be a pluripolar set outside which $a_ j\leq w$ on $K$. On $K\setminus \bigcup _ jN_ j$ one has $v_ j\leq -2^ j$ for every $j$, hence $s=-\infty $. The pole set $\{ s=-\infty \} $ of the quasi-plurisubharmonic function $s$ is pluripolar, and so is the countable union of the $N_ j$. Their union would contain all of $K$, contradicting the hypothesis on $K$. $\square$
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