Theorem 20.1.156EB1B4E1D222

Proof. Let $\omega $ be a Kähler form on $X$.

Step 1. Reduce to the case where $T$ is a Kähler current.

Note that $\mathbb {D}(\omega )$ is the Cartier b-divisor realized by $(X,\{ \omega \} )$. The b-divisors $\mathbb {D}(T+\epsilon \omega )=\mathbb {D}(T)+\epsilon \mathbb {D}(\omega )$ decrease to $\mathbb {D}(T)$ as $\epsilon \searrow 0$. Hence, by 20.1.7 and proposition 18.1.27, it suffices to prove our assertion with $T+\epsilon \omega $ in place of $T$.

Step 2. We prove the assertion under the additional assumption that $T$ has analytic singularities.

Let $\pi \colon Y\rightarrow X$ be a modification so that

where $D$ is an effective $\mathbb {Q}$-divisor on $Y$ and $R$ is a closed positive $(1,1)$-current with locally bounded potentials. See theorem 5.18.8 for the existence of $\pi $. Then $\mathbb {D}(T)$ is the nef Cartier b-divisor realized by $(\pi ,\{ R\} )$. Note that (20.1.16) is immediate in this case.

Step 3. We prove the assertion for a general Kähler current $T$. Next, take a closed smooth real $(1,1)$-form $\theta $ in the cohomology class of $T$ and write $T=\theta _{\varphi }$ for some $\varphi \in \mathrm{PSH}(X,\theta )$. Since $T$ is a Kähler current, we choose the quasi-equisingular approximation $(\varphi _ j)_ j$ of $\varphi $ so that, after discarding finitely many terms, each $\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ j$ is a Kähler current. This is possible by Demailly regularization, in the form used in the proof of theorem 18.1.2. We claim that

By definition of this convergence, we need to establish the following: for every modification $\pi \colon Y\rightarrow X$, we have

This follows immediately from theorem 16.9.55 if $\operatorname{Sing}(\pi ^*T)$ has only finitely many components.

Fix a Kähler form $\omega _ Y$ on $Y$. We want to show that for any $\epsilon >0$, we can find $j_0>0$ so that when $j\geq j_0$,

Write the divisorial part of $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}\pi ^*\varphi _ j$ and $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}\pi ^*\varphi $ as

Then $a_ i^ j\leq a_ i$.

We can find an integer $N>0$ large enough so that, in the pseudo-effective order of cohomology classes,

By theorem 16.9.55, we can take $j_0$ large enough so that for $j>j_0$,

Then (20.1.18) follows, and (20.1.17) is established.

As a consequence, $\mathbb {D}(T)$ is nef and the volume can be computed using 20.1.7:

Hence, (20.1.16) follows. $\square$