Lemma 23.2.62.D4DABE9DA177 Let $\pi :Y\to X$ be a modification between connected compact complex manifolds, and let $\theta $ be a smooth closed real $(1,1)$-form on $X$. Then
\[ \{ \theta \} \text{ is pseudoeffective} \quad \Longleftrightarrow \quad \{ \pi ^*\theta \} \text{ is pseudoeffective}. \]
The pullback on real Bott–Chern cohomology in bidegree $(1,1)$ is injective.
Proof.
By proposition 10.11.2, both pullback and pushforward preserve pseudoeffectivity; the pullback of potentials is the one used in lemma 5.43.19. Since $\pi $ has degree one,
\[ \pi _*\pi ^*\{ \theta \} =\{ \theta \} . \]
Indeed, the identity of currents $\pi _*\pi ^*\theta =\theta $ follows by integration over the locus where $\pi $ is an isomorphism, whose complement has volume zero. This left inverse proves injectivity and the reverse implication for pseudoeffectivity.
$\square$
Pome
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