Let $\mathbb{K}$ be a field and $\mathfrak{M}_{\mathbb{K}}$ a family of absolute values on $\mathbb{K}$ with real weights. For each $v\in\mathfrak{M}_{\mathbb{K}}$ we denote by $|\cdot|_{v}$ the corresponding absolute value, by $n_{v}\in\mathbb{R}$ the weight, and by $\mathbb{K}_{v}$ the completion of $\mathbb{K}$ with respect to $|\cdot|_{v}$. We say that $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ is an adelic field if

1. (1)

   for each $v\in\mathfrak{M}_{\mathbb{K}}$, the absolute value $|\cdot|_{v}$ is Archimedean or associated to a nontrivial discrete valuation;

2. (2)

   for each $\alpha\in\mathbb{K}^{\times}$, $|\alpha|_{v}=1$ except a for a finite number of $v$.

Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be an adelic field. For $\alpha\in\mathbb{K}^{\times}$, the defect of $\alpha$ is

$$\operatorname{def}(\alpha)=\sum_{v\in\mathfrak{M}_{\mathbb{K}}}n_{v}\log|\alpha|_{v}.$$

Since $\operatorname{def}\colon\mathbb{K}^{\times}\to\mathbb{R}$ is a group homomorphism, we have that $\operatorname{def}(\mathbb{K}^{\times})$ is a subgroup of $\mathbb{R}$. If $\operatorname{def}(\mathbb{K}^{\times})=0$, then $\mathbb{K}$ is said to satisfy the product formula. The group of global heights of $\mathbb{K}$ is $\mathbb{R}/\!\operatorname{def}(\mathbb{K}^{\times})$.

The adelic fields in examples 2.49 and 2.50 will be called global fields. For a finite subset $S\subset\mathfrak{M}_{\mathbb{K}}$ containing the Archimedean places, we consider the Noetherian ring $\mathbb{K}^{\circ}_{S}=\{\alpha\in\mathbb{K}\,|\,|\alpha|_{v}\leq 1,\forall v\notin S\}$.

Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be an adelic field. Let $X$ be a proper variety over $\mathbb{K}$ and $L$ a line bundle on $X$. For each $v\in\mathfrak{M}_{\mathbb{K}}$ set $X_{v}=X\times\operatorname{Spec}(K_{v})$ and $L_{v}=L\times\operatorname{Spec}(K_{v})$.

1. (1)

   A metric on $L$ is a family of metrics $\|\cdot\|_{v}$, $v\in\mathfrak{M}_{\mathbb{K}}$, where $\|\cdot\|_{v}$ is a metric on $L_{v}^{{\text{\rm an}}}$. We will denote by ${\overline{L}}=(L,(\|\cdot\|_{v})_{v})$ the corresponding metrized line bundle. The metric is said to be approachable (respectively integrable) if the metrics $\|\cdot\|_{v}$ are approachable (respectively integrable) for all $v\in\mathfrak{M}_{\mathbb{K}}$.

2. (2)

   Suppose that $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ is a global field. A metric on $L$ is called quasi-algebraic if there exists a finite subset $S\subset\mathfrak{M}_{\mathbb{K}}$ containing the Archimedean places, an integer $e\geq 1$ and a proper model $({\mathcal{X}},{\mathcal{L}},e)$ over $\mathbb{K}^{\circ}_{S}$ of $(X,L)$ such that, for each $v\notin S$, the metric $\|\cdot\|_{v}$ is induced by the localization of this model at $v$.

Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be an adelic field, $X$ a proper variety over $\mathbb{K}$ and ${\overline{L}}_{i}$, $i=0,\dots,d$, a family of integrable metrized line bundles on $X$. Let $Y$ be a $d$-dimensional cycle of $X$. We say that $Y$ is integrable with respect to ${\overline{L}}_{0},\dots,{\overline{L}}_{d}$ if there is a proper map $\varphi\colon X^{\prime}\to X$, a cycle $Y^{\prime}$ of $X^{\prime}$ such that $\varphi_{\ast}Y^{\prime}=Y$, and rational sections $s_{i}$ of $\varphi^{\ast}L_{i}$, $i=0,\dots,d$, that intersect $Y^{\prime}$ properly and such that for all but a finite number of $v\in\mathfrak{M}_{\mathbb{K}}$,

(2.54)

$$\operatorname{h}_{v,{\overline{\varphi}}^{\ast}L_{0},\dots,{\overline{\varphi}}^{\ast}L_{d}}(Y^{\prime};s_{0},\dots,s_{d})=0,$$

where $\operatorname{h}_{v}$ denotes the local height function on $X_{v}$.

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 $\mathbb{R}/\!\operatorname{def}(\mathbb{K}^{\times})$.