PomeLemma 5.19.18.9F062B47A2B4 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Let $T$ be a non-divisorial current on $Y$. Then $\pi _*T$ is non-divisorial.
Lemma 5.19.189F062B47A2B4
Proof. Let $D$ be a prime divisor on $X$. It follows from Zariski's main theorem theorem 9.10.5 that $D$ is not contained in the exceptional locus of $\pi $. Let $D'$ be the strict transform of $D$. Thanks to Siu's semicontinuity theorem theorem 5.16.7, we have
\[ \nu (\pi _*T,D)=\nu (T,D')=0. \]
Hence $\pi _*T$ is non-divisorial. $\square$
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