PomeCorollary 18.1.41.1F0FA3B61EBC Let $\pi \colon X\rightarrow Z$ be a proper bimeromorphic morphism from $X$ to a Kähler manifold $Z$. Assume that $T$ is an $\mathcal{I}$-good closed positive $(1,1)$-current on $X$. Then so is $\pi _*T$.
Corollary 18.1.411F0FA3B61EBC
Proof. We may assume that $\int _ X T^ n>0$. Then by corollary 18.1.39,
\[ \operatorname{vol}\pi _*T=\operatorname{vol}T>0 \]
as well. Since $T$ is $\mathcal{I}$-good, we have
\[ \operatorname{vol}T=\int _ X T^ n. \]
But $\int _ X T^ n=\int _ Z (\pi _*T)^ n$, so
\[ \operatorname{vol}\pi _*T=\int _ Z (\pi _*T)^ n>0. \]
It follows that $\pi _*T$ is $\mathcal{I}$-good. $\square$
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