We refer to [CL10] for a general discussion of metrized line bundles in a non-Archimedean context. Suffice it to say that a continuous metric $\|\cdot\|$ on a line bundle $L$ on $X$ is a way to produce a continuous function $\|s\|$ on (the Berkovich space) $X$ from any local section $s$ of $L$. Given a continuous metric $\|\cdot\|$, any other continuous metric on $L$ is of the form $\|\cdot\|e^{-\varphi}$, with $\varphi\in C^{0}(X)$. If we in this expression allow an arbitrary function $\varphi:X\to[-\infty,+\infty[$, then we obtain a singular metric on $L$.

Let $\mathcal{X}$ be a model and $\mathcal{L}$ a line bundle on $\mathcal{X}$ such that $\mathcal{L}|_{X}=L$. To this data one can associate a unique metric $\|\cdot\|_{\mathcal{L}}$ on $L$ with the following property: if $s$ is a nonvanishing local section of $\mathcal{L}$ on an open set $\mathcal{U}\subset\mathcal{X}$, then $\|s\|_{\mathcal{L}}\equiv 1$ on $U:=\mathcal{U}\cap X$. This makes sense since such a section $s$ is uniquely defined up to multiplication by an element of $\Gamma(\mathcal{U},\mathcal{O}_{\mathcal{X}}^{*})$ and such elements have norm 1.

More generally, any $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ such that $\mathcal{L}|_{X}=L$ in $\Pic(X)_{\mathbf{Q}}$ induces a metric $\|\cdot\|_{\mathcal{L}}$ on $L$ by setting $\|s\|_{\mathcal{L}}=\|s^{\otimes m}\|_{m\mathcal{L}}^{1/m}$ for any non-zero $m\in\mathbf{N}$ such that $m\mathcal{L}$ is an actual line bundle. By definition, a model metric⁴⁴ 4 Model metrics are called smooth metrics in [CL10] and formal metrics in [Gub98]. on $L$ is a metric of the form $\|\cdot\|_{\mathcal{L}}$ with $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ for some model $\mathcal{X}$ such that $\mathcal{L}|_{X}=L$. Model metrics are clearly continuous. If $\|\cdot\|$ is a model metric, then $\|\cdot\|e^{-\varphi}$ is a model metric iff $\varphi$ is a model function.

If we denote by $\widehat{\Pic}(X)$ the group of isomorphism classes of line bundles on $X$ endowed with a model metric then it is easy to check that there is a natural isomorphism

(4.1)

$$\varinjlim_{\mathcal{X}\in\mathcal{M}_{X}}\Pic(\mathcal{X})_{\mathbf{Q}}\simeq\widehat{\Pic}(X)_{\mathbf{Q}}$$

and that the natural sequence

(4.2)

$$0\to\mathcal{D}(X)\to\widehat{\Pic}(X)\to\Pic(X)\to 0$$

is exact.