Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$, $i=0,\dots,d$, be a family of toric line bundles, with integrable toric metrics. Denote by ${\overline{L}}_{i}^{{\operatorname{can}}}$ the same line bundles equipped with the canonical metric. Let $Y$ be a $d$-dimensional cycle of $X_{\Sigma}$. Then the toric local height of $Y$ with respect to ${\overline{L}}_{0},\dots,{\overline{L}}_{d}$ is

(6.2)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)=\operatorname{h}_{\varphi^{\ast}{\overline{L}}_{0},\dots,\varphi^{\ast}{\overline{L}}_{d}}(Y^{\prime};s_{0},\dots,s_{d})-\\ \operatorname{h}_{\varphi^{\ast}{\overline{L}}_{0}^{{\operatorname{can}}},\dots,\varphi^{\ast}{\overline{L}}_{d}^{{\operatorname{can}}}}(Y^{\prime};s_{0},\dots,s_{d}),$$

where $\Sigma^{\prime}$ is a regular refinement of $\Sigma$ (hence $X_{\Sigma^{\prime}}$ is projective), $\varphi\colon X_{\Sigma^{\prime}}\to X_{\Sigma}$ is the corresponding proper toric morphism, $Y^{\prime}$ is a cycle of $X^{\prime}$ such that $\varphi_{\ast}Y^{\prime}=Y$ and $s_{0},\dots,s_{d}$ are sections meeting $Y^{\prime}$ properly. When ${\overline{L}}_{0}=\dots={\overline{L}}_{d}={\overline{L}}$ we will denote

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y).$$