Pome191D96316EC3Proposition 5.19.17.191D96316EC3 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$.
4F00658A644BFor any nef class $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$, $\pi ^*\alpha $ is nef.
98B07470D933For any modified nef class $\beta \in \mathrm{H}^{1,1}(Y,\mathbb {R})$, $\pi _*\beta $ is modified nef.
Proof. Only 2 requires a proof. Fix a Kähler class $\gamma $ on $Y$. Since the modified nef cone is closed, replacing $\beta $ by $\beta +\epsilon \gamma $ and then letting $\epsilon \to 0+$ reduces the proof to the case where $\beta $ is big as well. Let $T$ (resp. $S$) be a current with minimal singularities in $\pi _*\beta $ (resp. in $\beta $), and let $D$ be a prime divisor on $X$. It suffices to show that
Since $\beta $ is modified nef, $S$ is non-divisorial. Hence by 5.19.18 below, $\nu (\pi _*S,D)=0$. Since $T$ has minimal singularities in $\pi _*\beta $, we have $T\succeq \pi _*S$, and the desired equality follows. $\square$
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