Lemma 20.1.7.F5B2C1F1E318 Let $(\mathbb {D}_ i)_{i\in I}$ be a net of pseudo-effective b-divisors converging to a Weil b-divisor $\mathbb {D}$. Then
3BFB2E87E034
20.1.8
\begin{equation} \varlimsup _{i\in I}\operatorname{vol}\mathbb {D}_ i\leq \operatorname{vol}\mathbb {D}. \end{equation}
If the net is decreasing, then
\[ \lim _{i\in I}\operatorname{vol}\mathbb {D}_ i=\operatorname{vol}\mathbb {D}. \]
Proof.
First observe that $\mathbb {D}$ is pseudo-effective, since $\mathbb {D}_ i\geq 0$ for all $i$.
Let $\pi \colon Y\rightarrow X$ be a modification. Then
\[ \operatorname{vol}\mathbb {D}_ Y=\lim _{i\in I} \operatorname{vol}\mathbb {D}_{i,Y}\geq \varlimsup _{i\in I} \operatorname{vol}\mathbb {D}_ i. \]
The inequality (20.1.8) follows. As for the decreasing case, it suffices to observe that both sides of (20.1.8) can be written as
\[ \inf _ i \inf _{\pi \colon Y\rightarrow X} \operatorname{vol}\mathbb {D}_{i,Y}. \]
This completes the proof.
$\square$
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