PomeLemma 20.4.17.6370BDE30D1B On a connected compact complex manifold $Z$, let $\beta $ be a smooth real $(1,1)$-form and $v$ a bounded quasi-plurisubharmonic function. If $R=\beta +\mathrm{dd}^{\mathrm{c}}v\geq 0$ is closed, then $\{ R\} _{\mathrm{BC}}$ is nef.
Lemma 20.4.176370BDE30D1B
Proof. Since $R$ is closed, so is $\beta $. Fix a Hermitian metric $\gamma $. By proposition 5.31.2, there are smooth functions $v_ j$ and numbers $\epsilon _ j\downarrow 0$ such that
\[ \beta +\mathrm{dd}^{\mathrm{c}}v_ j\geq -\epsilon _ j\gamma . \]
These forms are closed representatives of $\{ R\} _{\mathrm{BC}}$. Given $\epsilon >0$, choosing $j$ with $\epsilon _ j\leq \epsilon $ gives the estimate required in the definition of nefness. $\square$
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