Corollary 10.4.7856BB9025ACA

Proof. Write the current as $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $, where $\varphi $ has minimal singularities, and normalize $\varphi \leq 0$. Choose $C'\geq 0$ with $-\theta \leq C'\omega $ (lemma 6.14.1). For $0<t\leq \delta /(\delta +C')$ the current

is positive, so $(1-t)\varphi \in \mathrm{PSH}(X,\theta )$. By minimality, $(1-t)\varphi \leq \varphi +C$ for a constant $C$, so $\varphi \geq -C/t$ at every point where $\varphi $ is finite, hence almost everywhere. Since $\varphi $ is quasi-plurisubharmonic, the lower bound holds everywhere, as noted at the beginning of section 10.3. Thus $\varphi $ is bounded, and proposition 10.4.6 shows that the class is Kähler. For the second assertion, suppose that the class contains a Kähler form $\theta +\mathrm{dd}^{\mathrm{c}}u$. Then $u$ is smooth and $u\in \mathrm{PSH}(X,\theta )$, so a potential $\varphi $ with minimal singularities satisfies $u\leq \varphi +C$; hence $\varphi $ is bounded below, and it is bounded above as every quasi-plurisubharmonic function on $X$. $\square$