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 $v$.