We refer to [CL10] for a general account of metrized line bundles in a non-Archimedean context. Suffice it to say that a metric $\|\cdot\|$ on a line bundle $L$ on $X$ is a way to produce a local continuous function $\|s\|$ on (the Berkovich space) $X$ from any local section $s$ of $L$.

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 $m\in\mathbf{N}^{*}$ such that $m\mathcal{L}$ is an actual line bundle. Such a metric is called a model metric on $L$.

Given a model metric $\|\cdot\|$, any continuous metric on $L$ is of the form $\|\cdot\|e^{-\varphi}$, with $\varphi\in C^{0}(X)$. This is a model metric iff $\varphi$ is a model function. By a singular metric on $L$ we mean an expression of the form $\|\cdot\|e^{-\varphi}$ with $\varphi:X\to[-\infty,+\infty[$ an arbitrary function.

Fix a model metric $\|\cdot\|_{\mathcal{L}}$ on $L$ associated to $\mathcal{L}\in\Pic_{\mathbf{Q}}(\mathcal{X})$. The numerical class associated to $\mathcal{L}$ in $N^{1}(\mathcal{X}/S)$ induces a form on $X$ in the sense of §2.3. It does not depend on the choice of model $\mathcal{L}$ defining the metric. We call it the curvature form of the metric and denote it by $c_{1}(L,\|\cdot\|)$. By construction, its de Rham class is given by

(2.1)

$$\{c_{1}(L,\|\cdot\|)\}=c_{1}(L)\in N^{1}(X).$$

If $\varphi\in\mathcal{D}(X)$ is a model function, then

$$c_{1}(L,\|\cdot\|\,e^{-\varphi})=c_{1}(L,\|\cdot\|)+dd^{c}\varphi,$$

where the form $dd^{c}\varphi\in\mathcal{Z}^{1,1}(X)$ is defined in §2.3.

The results in §2.4 have obvious counterparts for singular metrics. In particular, we have:

This result implies that our definition of continuous semipositive metric coincides with that of Zhang and others. Unfortunately, the terminology is not uniform across the literature, see Table 1 below.