Proposition 10.5.6C811386D768F

Proof. By lemma 6.14.1 on the compact manifold $Y$, $\pi ^*\omega \leq C\omega _ Y$ for a constant $C$. If $\theta _\varepsilon \geq -\varepsilon \omega $ are nef representatives of a class on $X$, then $\pi ^*\theta _\varepsilon \geq -C\varepsilon \omega _ Y$, so the pullback class is nef. Let $S$ be a non-divisorial closed positive current on $Y$ and $D\subset X$ a prime divisor, with strict transform $D'$. Over $X\setminus A$ the map $\pi $ is a biholomorphism, which identifies $\pi _*S$ with $S$ and $D\setminus A$ with $D'\setminus \pi ^{-1}(A)$, and Lelong numbers are invariant under biholomorphisms (proposition 6.7.1). The generic value of the Lelong numbers of $S$ along $D'$ is attained on a dense subset of $D'$ (proposition 6.9.7), hence at some point $y$ of the nonempty open subset $D'\setminus \pi ^{-1}(A)$. Therefore $\nu (\pi _*S,D)\leq \nu (\pi _*S,\pi (y))=\nu (S,y)=\nu (S,D')=0$. Hence $\pi _*S$ is non-divisorial.

Let $\beta $ be a modified nef class on $Y$ and $t>0$. The class $\beta +t\{ \omega _ Y\} $ is big, and it is modified nef because $\mathcal{MN}$ is a convex cone containing the Kähler classes. By proposition 10.5.2, a positive current with minimal singularities in this class is non-divisorial. Its pushforward is a non-divisorial positive current in $\pi _*\beta +t\pi _*\{ \omega _ Y\} $, so this class is modified nef: the definition is satisfied with $T_\varepsilon $ equal to this current for every $\varepsilon $. These classes tend to $\pi _*\beta $ as $t\downarrow 0$, and $\mathcal{MN}$ is closed, so $\pi _*\beta $ is modified nef. $\square$