Corollary 20.1.27362BC48145F6

Proof. 1 $\implies $ 2. Suppose that 1 holds. Fix a modification $\pi \colon Y\rightarrow X$. Then we need to show that $\mathbb {D}_ Y$ is modified nef. Since modified nefness is a closed condition, after approximating $\mathbb {D}$ by nef Cartier b-divisors, we may assume that $\mathbb {D}$ itself is a nef Cartier b-divisor. We can then find a modification $\pi '\colon Z\rightarrow X$ dominating $\pi $ so that $\mathbb {D}$ is realized by a nef class $\alpha $ on $Z$. Then $\mathbb {D}_ Y$ is nothing but the pushforward of $\alpha $, and hence modified nef thanks to proposition 5.19.17.

2 $\implies $ 1. Suppose that 2 holds. We need to show that $\mathbb {D}$ is nef. Fix a Kähler form $\omega $ on $X$. It suffices to show that $\mathbb {D}+\epsilon \mathbb {D}(\omega )$ is nef for each $\epsilon >0$. After replacing $\mathbb {D}$ by the latter, we may further assume that $\mathbb {D}_ Y$ is big for each modification $\pi \colon Y\rightarrow X$, and $\operatorname{vol}\mathbb {D}_ Y$ has a uniform positive lower bound. In this case, the argument of theorem 20.1.24 shows that $\mathbb {D}=\mathbb {D}(T)$ for some closed positive $(1,1)$-current in $\mathbb {D}_ X$. Therefore, $\mathbb {D}$ is nef, thanks to theorem 20.1.15. $\square$