Following [59], we now want to explain the analogues of smooth, and, later, of semi-positive metrics.

Smooth metrics come from algebraic geometry over $K^{\circ}$, and, more generally, over the ring of integers of finite extensions of $K$. Let namely $\mathfrak{X}$ be a formal proper $K^{0}$-scheme whose generic fibre in the sense of analytic geometry is $\mathrm{X}$.²² 2 The reader might want to assume that $\mathrm{X}$ is the analytic space associated to a projective $K$-scheme $X$ and that $\mathfrak{X}$ is a projective $K^{0}$-scheme whose generic fibre equals $X$. This doesn’t make too much a difference for our concerns. Let also $\mathfrak{L}$ be a line bundle on $\mathfrak{X}$ which is model of some power $L^{e}$, where $e\geq 1$. From this datum $(\mathfrak{X},\mathfrak{L},e)$, we can define a metric on $L$ as follows. Let $\mathfrak{U}$ be a formal open subset of $\mathfrak{X}$ over which $\mathfrak{L}$ admits a local frame $\varepsilon_{\mathfrak{U}}$ ; over its generic fibre $\mathrm{U}=\mathfrak{U}_{K}$, for any section $s$ of $L$, one can write canonically $s^{e}=f\varepsilon_{\mathfrak{U}}$, where $f\in\mathscr{O}_{\mathrm{X}}(\mathrm{U})$. We decrete that $\left\|{s}\right\|=\left|{f}\right|^{1/e}$. In other words, the norm of a local frame on the formal model is assigned to be identically one. This makes sense because if $\eta_{\mathfrak{U}}$ is another local frame of $\mathfrak{L}$ on $\mathfrak{U}$, there exists an invertible formal function $f\in\mathscr{O}_{\mathfrak{X}}(\mathfrak{U})^{*}$ such that $\eta_{\mathfrak{U}}=f\varepsilon_{\mathfrak{U}}$ and the absolute value $\left|{f}\right|$ of the associated analytic function on $\mathrm{U}$ is identically equal to $1$. Considering a finite cover of $\mathfrak{X}$ by formal open subsets, their generic fibers form a finite cover of $\mathrm{X}$ by closed subsets and this is enough to glue the local definitions to a continuous metric on $L$.

Metrics on $L$ given by this construction, for some model $(\mathfrak{X},\mathfrak{L},e)$ of some power $L^{e}$ of $L$ will be said to be smooth.