Lemma 5.29.2031C5645526CA

Proof. For $n=1$, Stokes' theorem gives $\int _ X\theta _ v=\int _ X\theta $, so assume $n\geq 2$. Choose a fixed Hermitian form $\Omega \geq \theta $. All currents on the left-hand side are positive, and $\Omega _\psi \geq \theta _\psi $. Hence Stokes' theorem for bounded potentials gives

The first term on the right is bounded independently of $v$. To estimate the second, differentiate:

where the second term is absent for $n=2$. Since $\mathrm{dd}^{\mathrm{c}}\Omega $ and $\mathrm d\Omega \wedge \mathrm d^{\mathrm c}\Omega $ are smooth forms, the absolute values of these terms are bounded by fixed multiples of the positive measures $\Omega ^2\wedge \Omega _\psi ^{n-2}$ and $\Omega ^3\wedge \Omega _\psi ^{n-3}$. The weighted local Chern–Levine–Nirenberg inequality of section 5.27 bounds the integrals of $|v|$ against these measures. Indeed, on each larger coordinate ball, add fixed smooth corrections to $v$ and $\psi $, shift the corrected $v$ so that it becomes negative, expand in the corrected positive Hessian of $\psi $, and apply that inequality on a smaller ball. The resulting bound is

By theorem 5.25.2, normalized $\theta $-plurisubharmonic functions have uniformly bounded $L^1$ norms, and summing the local bounds proves the assertion. $\square$