Lemma 10.3.85755B2D3C604

Proof. The family whose upper envelope defines $V_\theta $ in definition 10.3.7 is nonempty, because $\varphi -\sup _ X\varphi $ belongs to it for every $\varphi \in \mathrm{PSH}(X,\theta )$, and it is bounded above by $0$. Theorem 5.10.2, applied on coordinate balls after adding smooth local potentials of $\theta $, therefore gives $V_\theta \in \mathrm{PSH}(X,\theta )$ and $V_\theta \leq 0$; and $\sup _ XV_\theta =0$, because $V_\theta \geq \varphi -\sup _ X\varphi $ for every $\varphi $. The same inequality reads $\varphi \leq V_\theta +\sup _ X\varphi $, so $V_\theta $ has minimal singularities. Any two potentials with minimal singularities are each more singular than the other, so their difference is bounded on the set where both are finite. Replacing $\theta $ by $\theta +\mathrm{dd}^{\mathrm{c}}h$ with $h$ smooth replaces $\mathrm{PSH}(X,\theta )$ by $\mathrm{PSH}(X,\theta )-h$; since $h$ is bounded, the notion of minimal singularities depends only on the class $\{ \theta \} $. $\square$