This kind of metrics are interesting when studying the Kähler geometry of toric varieties. Given a Delzant polytope $\Delta\subset M_{\mathbb{R}}$, Guillemin has constructed a “canonical” Kähler structure on the associated symplectic toric variety [Gui95]. The corresponding symplectic potential is the function $-\vartheta_{\Delta,\ell,c}$, for the case when $r$ is the number of facets of $\Delta$, $c_{i}=1/2$ for all $i$, and $u_{i}$ is a primitive vector in $N$ and $\lambda_{i}$ is an integer such that $\Delta=\{x\in M_{\mathbb{R}}|\langle u_{i},x\rangle\geq\lambda_{i},i=1,\dots,r\}$, see [Gui95, Appendix 2, (3.9)].

In this case, the metric $\|\cdot\|_{\Delta,\ell,c}$ on the line bundle ${\mathcal{O}}(D_{\Psi})^{{\text{\rm an}}}$ is smooth and positive and, as explained in Remark 5.74, its Chern form gives this canonical Kähler form.