Lemma 18.1.38`30F0E1037DAE`

Lemma 18.1.38.30F0E1037DAE Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Then for any non-divisorial closed positive $(1,1)$-current $T$ on $Y$, we have

\[ \pi ^* \pi _* T=T+\sum _{i=1}^ N c_ i [E_ i] \]

for finitely many $\pi $-exceptional divisors $E_ i$ and $c_ i>0$.

Proof. Let $E$ be the exceptional locus of $\pi $. Then

\[ T=\mathds {1}_{Y\setminus E}\pi ^* \pi _* T. \]

Therefore,

\[ \pi ^* \pi _* T-T=\mathds {1}_ E\pi ^* \pi _* T, \]

which has the stated form, due to the support theorems, see [Section 8, DemBook]. $\square$

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@misc{pome-tag-30F0E1037DAE,
  author = {{Pome contributors}},
  title = {Lemma 18.1.38},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/30F0E1037DAE/}
}

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Lemma 18.1.3830F0E1037DAE

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Lemma 18.1.38`30F0E1037DAE`