Lemma 16.13.2691B1A212F1A

Proof. By proposition 10.4.6, the big class contains a Kähler current $\theta +\mathrm{dd}^{\mathrm{c}}\psi $ whose potential $\psi \in \mathrm{PSH}(X,\theta )$ has analytic singularities, and we put $A=\{ \psi =-\infty \} $. Every point of $X$ has a connected coordinate neighborhood $V$ on which $\psi =c\log \sum _ i|g_ i|^2+b$, with $c>0$, $b$ bounded and holomorphic functions $g_ i$ that do not all vanish identically (definition 6.14.5). Thus $A\cap V$ is the common zero set of the $g_ i$, so $A$ is an analytic subset of $X$, different from $X$, and it is closed and pluripolar, being locally the pole set of the plurisubharmonic function $\log \sum _ i|g_ i|^2$. On $V\setminus A$ the continuous function $\sum _ i|g_ i|^2$ is positive, so $\psi $ is locally bounded on $X\setminus A$. If $\varphi $ has minimal singularities, then $\psi \leq \varphi +C'$ for a constant $C'$, so $\varphi $ is locally bounded below on $X\setminus A$; it is bounded above, being upper semicontinuous on the compact space $X$. The last assertion holds because non-pluripolar products give no mass to pluripolar sets. $\square$