Theorem 21.13.398F3CCCDB83EC

Proof. We first reduce to the case where $D|_{U_{\sigma }}=0$. In fact, changing $D$ to $D+\operatorname{div}\chi ^{m_{\sigma }}$ would result in changing $F_{\theta }$ to $F_{\theta }-m_{\sigma }$. Hence, $F_{\varphi }$ changes to $F_{\varphi }-m_{\sigma }$. Therefore, $\Delta (\theta ,\varphi )$ becomes $\Delta (\theta ,\varphi )-m_{\sigma }$. Taking (21.13.6) into consideration, we may assume that $m_{\sigma }=0$.

Step 1. We first reduce to the case where $\theta _{\varphi }$ is a Kähler current.

By 13.13.47, we can find $\psi \in \mathrm{PSH}(X,\theta )$ such that $\psi \leq \varphi $ and $\theta _{\psi }$ is a Kähler current. Taking the average along $T_ c$, we may assume that $\psi $ is $T_ c$-invariant.

For each $t\in (0,1)$, we let

Suppose that Kähler current case is known. Then we get

for any $t\in (0,1)$. It follows from theorem 21.11.56 that

for any $t\in (0,1)$. Thanks to theorem 19.4.30, we have

Comparing the volumes of both sides using proposition 21.13.28, (19.4.20), and the $\mathcal{I}$-goodness of toric singularities example 18.1.56, we find that

In particular, we conclude (21.13.40).

Step 2. We handle the case where $\theta _{\varphi }$ is a Kähler current.

Let $(\varphi _ j)_ j$ be the quasi-equisingular approximation given by 21.13.38.

Now assume that the result is known for each $\varphi _ j$. Then

In particular, thanks to proposition 19.4.28,

for each $j\geq 1$. It follows from theorem 19.4.30 that

Comparing the volumes of both sides using proposition 21.13.28, (19.4.20), and example 18.1.56, we conclude (21.13.40).

Step 3. It remains to prove the assertion for the analytic approximants appearing in Step 2.

By 21.13.38, these approximants may be chosen in the form

where $q\in \mathbb Z_{>0}$ and $s_1,\ldots ,s_ a\in \mathrm{H}^0(X,L^ q)$ are toric eigen-sections.

Let $m_1,\ldots ,m_ a\in qP_ D\cap M$ be the lattice points corresponding to $s_1,\ldots ,s_ a$.

Observe that

where we have applied (21.13.23) on the second line and 21.11.62 on the third line. Since $m_1,\ldots ,m_ a\in qP_ D$, the preceding formula also shows that

Thus, if $m\in k\Delta (\theta ,\varphi )\cap M$, then $m\in kP_ D\cap M$, and $\chi ^ m$ indeed determines a global section of $L^ k$.

In particular, by the same separation argument as in 21.11.60, for any $k\in \mathbb {Z}_{>0}$ and any $m\in k\Delta (\theta ,\varphi )\cap M$, we have

is bounded from above on $T(\mathbb {C})$. In other words, the section $s$ of $L^ k$ corresponding to $m$ satisfies

Therefore,

where $\Delta _ k$ is defined in section 19.4. Hence,

Letting $k\to \infty $ and applying theorem 19.4.41, we find that

Comparing the volumes of both sides using proposition 21.13.28 and (19.4.20), we conclude that the equality holds and (21.13.40) follows. $\square$