Let $K$ be a field equipped with a complete multiplicative norm and let $X$ be a smooth projective variety over $K$. To this data we can associate an analytification ${X^{\mathrm{an}}}$. When $K$ is the field of complex numbers with its usual norm, ${X^{\mathrm{an}}}$ is a compact complex manifold. When the norm is non-Archimedean, ${X^{\mathrm{an}}}$ is a $K$-analytic space in the sense of Berkovich [Berk90]. In either case, it is a compact Hausdorff space.

Let $L$ be a line bundle on $X$. It also admits an analytification ${L^{\mathrm{an}}}$. A metric on ${L^{\mathrm{an}}}$ is a rule that to a local section $s:U\to{L^{\mathrm{an}}}$, where $U\subset{X^{\mathrm{an}}}$, associates a function $\|s\|$ on $U$, subject to the condition $\|fs\|=|f|\cdot\|s\|$, for any analytic function $f$ on $U$. The metric is continuous if $\|s\|$ is continuous on $U$ for every $s$.

For our purposes it is convenient to use additive notation for metrics and line bundles. Given an open cover $U_{\alpha}$ of ${X^{\mathrm{an}}}$ and local trivializations of ${L^{\mathrm{an}}}$ on each $U_{\alpha}$, we can identify a section $s$ of $L$ with a collection $(s_{\alpha})_{\alpha}$ of analytic functions. A metric $\phi$ is then a collection of functions $(\phi_{\alpha})_{\alpha}$ in such a way that $\|s\|_{\phi}=|s_{\alpha}|e^{-\phi_{\alpha}}$ on $U_{\alpha}$. With this convention, if $\phi$ is a metric on ${L^{\mathrm{an}}}$, any other metric is of the form $\phi+f$, where $f$ is a function on ${X^{\mathrm{an}}}$. If $\phi_{i}$ is a metric on $L_{i}$, $i=1,2$, then $\phi_{1}+\phi_{2}$ is a metric on $L_{1}+L_{2}$.

Over the complex numbers, smooth metrics $\phi$ (i.e. each $\phi_{\alpha}$ is smooth), play an important role. Of similar status, for $K$ non-Archimedean, are model metrics defined as follows.²² 2 Model metrics are not smooth in the sense of [CD12] but nevertheless, for our purposes, play the same role as smooth metrics in the complex case. Let $R$ be the valuation ring of $K$ and $k$ the residue field. A model of $X$ is a normal scheme ${\mathcal{X}}$, flat and projective over $\operatorname{Spec}R$ and with generic fiber isomorphic to $X$. A model of $L$ is a ${\mathbf{Q}}$-line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ whose restriction to $X$ is isomorphic to $L$. It defines a continuous metric $\phi_{\mathcal{L}}$ on $L$ in such a way that any local nonvanishing section of a muliple of ${\mathcal{L}}$ has norm constantly equal to one. Model functions, that is, model metrics on ${\mathcal{O}}_{X}$, are dense in $C^{0}({X^{\mathrm{an}}})$. We refer to [CL11] or [BFJ12] for a more thorough discussion.

Over ${\mathbf{C}}$, a smooth metric $\phi$ on ${L^{\mathrm{an}}}$ is semipositive (positive) if its curvature form $dd^{c}\phi$ is a semipositive (positive) $(1,1)$-form. Here $dd^{c}\phi=dd^{c}\phi_{\alpha}=\frac{i}{\pi}\partial\overline{\partial}\phi_{\alpha}$ for any $\alpha$. Such metrics only exist when $L$ is nef.

In the non-Archimedean setting we say that a model metric $\phi_{\mathcal{L}}$ on ${L^{\mathrm{an}}}$ is semipositive if the line bundle ${\mathcal{L}}$ is relatively nef, that is, its degree is nonnegative on any proper curve contained in the special fiber ${\mathcal{X}}_{0}$. This implies that $L$ is nef.

In both the complex and non-Archimedean case we say that a continuous metric $\phi$ is semipositive if there exists a sequence $(\phi_{m})_{1}^{\infty}$ of semipositive smooth/model metrics such that $\lim_{m\to\infty}\sup_{{X^{\mathrm{an}}}}|\phi_{m}-\phi|=0$. In the non-Archimedean case, this notion was first introduced by Zhang [Zha95] and Gubler [Gub98]. In the complex case, it is more natural to say that a continuous metric $\phi$ is semipositive if its curvature current $dd^{c}\phi$ is a positive closed current. At least when $L$ is ample, one can then prove (see §6 below) that $\phi$ can be approximated by smooth metrics; such an approximation is furthermore crucial for many arguments in pluripotential theory.

In the non-Archimedean case, Chambert-Loir and Ducros have introduced a notion of forms and currents on Berkovich spaces. However, it is not known whether a continuous metric whose curvature current (in their sense) is semipositive can be approximated by semipositive model metrics.

In both the complex and non-Archimedean case we denote by $\operatorname{PSH}^{0}({L^{\mathrm{an}}})$ the space of continuous semipositive metrics on ${L^{\mathrm{an}}}$. Here the superscript refers to continuity ($C^{0}$) whereas “PSH” reflects the fact that in the complex case, semipositive metrics are global versions of plurisubharmonic functions.