PomeLemma 21.13.22.A11981674229 There is a smooth function $F_{\theta }\colon N_{\mathbb {R}}\rightarrow \mathbb {R}$ such that
\[ \theta =\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^*F_{\theta }\quad \textup{on }T(\mathbb {C}). \]
21.13.21 Toric partial Okounkov bodiesE81A302A1530
We continue to use the notations in subsection 21.13.1.
In order to study the toric-invariant singular plurisubharmonic metrics on $L$, we need to fix a reference toric-invariant smooth Hermitian metric, so that the psh metrics can be identified with quasi-psh functions. Unlike the ample case studied in section 21.12, in the case of big line bundles, there does not seem to be a natural choice of a smooth Hermitian metric on $L$ similar to Guillemin's metric.
We shall fix a $T_ c$-invariant Hermitian metric $h$ on $L$ so that $\theta =c_1(L,h)$.
The first observation is the following:Proof. Step 1. We first prove the existence of $F_{\theta }$ for a specific choice of $h$.
Write $D$ as the difference of two toric-invariant ample divisors, say $D_1-D_2$. Let $h_1$ and $h_2$ be the associated Guillemin's metrics on $D_1$ and $D_2$ respectively. We define $h=h_1\otimes h_2^{-1}$.
In this case, our assertion follows since it holds in the case of Guillemin's metrics.
Step 2. In general, fix $h_0$ as in Step 1. Then the general $h$ can be written as $h_0\exp (-g)$ for some smooth function $g$ on $X$. Since both metrics are $T_ c$-invariant, so is $g$, and on the open torus we may write $g=\operatorname{Trop}^* r$ for some smooth function $r$ on $N_{\mathbb {R}}$. Hence it suffices to modify $F_{\theta }$ in Step 1 by $r$ to conclude. $\square$
Note that $F_{\theta }$ is well-defined up to a linear term.
Next, we make an additional requirement on $F_{\theta }$ to fix the linear term. Let $s_ D$ be a rational section of $L$ corresponding to $D$. Then $s_ D$ is well-defined up to a non-zero multiple. By Lelong–Poincaré formula proposition 5.20.4, we have
\[ \mathrm{dd}^{\mathrm{c}}\left( \operatorname{Trop}^*F_{\theta }+\log |s_ D|_ h^2 \right)=0 \]
on $T(\mathbb {C})$. Therefore, $\operatorname{Trop}^*F_{\theta }+\log |s_ D|_ h^2$ is the tropicalization of a linear function. Hence, after adding a linear function to $F_{\theta }$, we can guarantee that
CDCD55296D94
21.13.23
\begin{equation} \operatorname{Trop}^*F_{\theta }+\log |s_ D|_ h^2=0 \end{equation}
from now on. Note that a different choice of $s_ D$ means adding a constant to $F_{\theta }$.
Summarizing the situation, we have chosen the following data in addition to the fan $\Sigma $ so far: A divisor $D$, a Hermitian metric $h$ on $L$ and a rational section $s_ D$ of $L$ subject to various conditions.
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