Proposition 10.5.820DDDD0E7ECB

Proof. Choose $C$ with $\pi ^*\omega \leq C\omega _ Y$. Since $\pi _*$ preserves positivity, lemma 10.5.7 gives $\pi _*\omega _ Y\geq C^{-1}\pi _*\pi ^*\omega =C^{-1}\omega $. Hence a current $S\geq \delta \omega _ Y$ on $Y$ has $\pi _*S\geq \delta C^{-1}\omega $, which proves the assertion for $\pi _*$.

Let $\alpha $ be big on $X$ and $T\in \alpha $ with $T\geq \delta \omega $. Choose a smooth closed representative $\eta $ of the class $\{ \pi _*\omega _ Y\} $, a constant $B>0$ with $\eta \leq B\omega $ (lemma 6.14.1), and $t>0$ with $tB\leq \delta $, so that $T-t\eta \geq 0$. The positive current $T-t\eta +t\pi _*\omega _ Y$ lies in $\alpha $. By lemma 10.5.7, applied to the smooth, hence non-divisorial, current $\omega _ Y$, its pullback is

This is a Kähler current in $\pi ^*\alpha $, so $\pi ^*\alpha $ is big. $\square$