Let $X$ be a proper variety over $\mathbb{K}$, ${\overline{L}}_{0},\dots,{\overline{L}}_{d}$ integrable metrized line bundles on $X$, and $Y$ an integrable $d$-dimensional cycle of $X$. Let $X^{\prime}$, $Y^{\prime}$ and $s_{0},\dots,s_{d}$ be as in Definition 2.53. The global height of $Y$ with respect to $s_{0},\dots,s_{d}$ is defined as

$$\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})=\sum_{v\in\mathfrak{M}_{\mathbb{K}}}n_{v}\operatorname{h}_{v,\varphi^{\ast}{\overline{L}}_{0},\dots,\varphi^{\ast}{\overline{L}}_{d}}(Y^{\prime};s_{0},\dots,s_{d})\in\mathbb{R}.$$

The global height of $Y$, denoted $\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)$, is the class of $\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})$ in the quotient group $\mathbb{R}/\!\operatorname{def}(\mathbb{K}^{\times})$.