Lemma 5.44.102E102F06172B

Proof. Choose finitely many pairs of concentric coordinate balls $B'_ a\Subset B_ a$ such that the $B'_ a$ cover $X$, and fixed smooth functions $q_ a,r_ a$ on $B_ a$ with $\mathrm{dd}^{\mathrm{c}}q_ a\geq \omega $ and $\mathrm{dd}^{\mathrm{c}}r_ a\geq \theta $, with $r_ a\leq 0$. Then $\psi +q_ a$ is bounded and plurisubharmonic, $w+r_ a$ is negative and plurisubharmonic, and positivity of bounded products gives

The last step is the weighted Chern–Levine–Nirenberg estimate of section 5.27. Summing over the cover and using the normalized $L^1$ bound of theorem 5.25.2 proves the lemma. $\square$