Proposition 10.4.6106B224C1E35

Proof. Write the given current as $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ with $\varphi \in \mathrm{PSH}(X,\theta )$. If $\varphi $ is bounded, then every $\mathcal I(m\varphi )$ is the unit ideal, so the approximations $G_ m$ are smooth and satisfy $\theta +\mathrm{dd}^{\mathrm{c}}G_ m\geq -(c/m)\omega $; hence the class is nef. If moreover $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $, then $\varphi \in \mathrm{PSH}(X,\theta -\delta \omega )$, and the approximations for the background $\theta -\delta \omega $ satisfy $\theta +\mathrm{dd}^{\mathrm{c}}G_ m\geq (\delta -c/m)\omega $; for large $m$ the smooth form $\theta +\mathrm{dd}^{\mathrm{c}}G_ m$ is a Kähler form in the class. For the last assertion, let $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq \delta \omega $ be a Kähler current in a big class, with $\varphi $ not necessarily bounded. The same inequality shows that $\theta +\mathrm{dd}^{\mathrm{c}}G_ m$ is a Kähler current for large $m$, and $G_ m$ has gentle analytic singularities of type $(m^{-1},\mathcal I(m\varphi ))$ by lemma 6.17.1. $\square$