Lemma 21.13.22A11981674229

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. $\square$