PomeLemma 18.27.3.E5C4A193363D Let $B$ be a smooth real $(1,1)$-form. If $q'$ is quasi-plurisubharmonic, all its point Lelong numbers vanish, and $B+\mathrm{dd}^{\mathrm{c}}q'\geq \epsilon h$ with $\epsilon >0$, then there is a smooth real $f$ such that $B+\mathrm{dd}^{\mathrm{c}}f\geq \epsilon h/2$.
Lemma 18.27.3E5C4A193363D
Proof. Apply the classical closed-current regularization of 5.36.3 to $T=\mathrm{dd}^{\mathrm{c}}q'$ with smooth representative zero and lower bound $\gamma =\epsilon h-B$. The lower bound is allowed to be nonclosed. More explicitly, fix any positive attenuation level. Its Lelong level set is empty, so the regularization gives global smooth $f_ k$ and
\[ \mathrm{dd}^{\mathrm{c}}f_ k\geq \gamma -C\lambda _ kh-\epsilon _ kh, \]
where the continuous nonnegative $\lambda _ k$ decrease pointwise to zero and $\epsilon _ k\downarrow 0$. A fixed smooth metric on $TX$ has bounded curvature, so a sufficiently large multiple $Ch$ supplies the curvature bound in that theorem. Dini's theorem gives uniform convergence of $\lambda _ k$ to zero. Choose $k$ with $C\| \lambda _ k\| _\infty +\epsilon _ k\leq \epsilon /2$ and set $f=f_ k$. The smooth approximants need not remain in the ordinary type of $q'$. $\square$
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