Pome2A843EB4B4AEProposition 5.28.6.2A843EB4B4AE Let $\theta $ be a smooth real $(1,1)$-form, let $h$ be a Hermitian form on $X$, and let $u\in \mathrm{PSH}(X,\theta )\cap C^{1,1}(X)$. If
for a constant $a>0$, then $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \delta h$ as a current for some $\delta >0$.
Proof. Put $T=\theta +\mathrm{dd}^{\mathrm{c}}u$. Its measurable coefficients are bounded on the compact manifold $X$, so there is $K>0$ such that, almost everywhere, the eigenvalues of $T$ relative to $h$ lie in $[0,K]$. By proposition 5.28.5, their product is the density of $T^ n$ with respect to $h^ n$, and hence it is at least $a$. Each eigenvalue is therefore at least $a/K^{n-1}$ almost everywhere. This inequality of matrices almost everywhere gives the asserted inequality of currents with $\delta =a/K^{n-1}$. $\square$
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