An arithmetic intersection theory on arithmetic surfaces was introduced by Arakelov and used by Faltings to prove the Mordell conjecture. In higher dimensions, the theory was developed by Gillet and Soulé which proved to be a very useful tool in diophantine geometry. To produce arithmetic intersection numbers from a given line bundle $L$ on a proper variety $X$ over a number field $K$, one has to endow the complexification of $L$ with a smooth hermitian metric and one has to choose an ${\mathcal{O}}_{K}$-model $({\mathscr{X}},{\mathscr{L}})$ for $(X,L)$.

Zhang [Zha95] realized that the contribution of a non-archimedean place $v$ to this arithmetic intersection number is completely determined by a metric on $L(K_{v})$ associated to ${\mathscr{L}}$, where $K_{v}$ is the completion of $K$ at $v$. This adelic point of view is very pleasant as it allows to deal with archimedean and non-archimedean places in a similar way. Motivated by his studies of the Bogomolov conjecture [Zha93], Zhang [Zha95] introduced semipositive adelic metrics as a uniform limit of metrics induced by nef models and he showed that every polarized dynamical system has a canonical metric inducing the canonical height of Call and Silverman.

In [Gub98], it became clear that Zhang’s metrics can be generalized to any non-archimedean field $K$ working with formal models of the line bundle over the valuation ring. It turned out that such metrics are continuous on the Berkovich analytification of the line bundle and so we call them continuous semipositive metrics.

Chambert-Loir introduced measures $c_{1}(L,{\|\hskip 4.30554pt\|})^{\wedge n}$ on the Berkovich space ${X^{\rm an}}$ for a continuous semipositive metric ${\|\hskip 4.30554pt\|}$ of a line bundle $L$ over $X$ ([Cha06], [Gub07a]). These measures are non-archimedean equidistribution measures as in Yuan’s equidistribution theorem [Yua08] over number fields (see also [CT09]). The analogue over function fields was proven in [Fab09], [Gub08] and gave rise to progress for the geometric Bogomolov conjecture [Gub07a], [Yam13, Yam16].

Continuous semipositive metrics played an important role in the study of the arithmetic geometry of toric varieties due to Burgos-Gil, Philippon and Sombra, see [BPS14], [BPS15], [BPS16], [BMPS16] with Moriwaki and [BPRS15] with Rivera-Letelier. Katz–Rabinoff–Zureick-Brown [KRZ15] used semipositive model metrics to give explicit uniform bounds for the number of rational points in situations suitable to the Chabauty–Coleman method.

For the non-archimedean Monge–Ampère problem, continuous semipositive metrics are of central importance. Uniqueness up to scaling was shown by Yuan and Zhang [YZ16]. In case of residue characteristic $0$, a solution was given by Boucksom, Favre and Jonsson [BFJ16, BFJ15] using an algebraicity condition which was removed in [BGJKM].

Semipositive model metrics played also a role in the thesis of Thuillier [Thu05] on potential theory on curves, in the work of Chambert-Loir and Ducros on forms and currents on Berkovich spaces [CD12] and in the study of delta-forms in [GK14, GK15].

Looking at the above references, one observes that the authors work either under the hypothesis that the valuation is discrete or that $K$ is algebraically closed. The reasoning behind the former is that the valuation ring and hence the models are noetherian. If $K$ is algebraically closed, then the valuation ring is not noetherian (unless the valuation is trivial, but we exclude this case here). Working with formal models using Raynaud’s theory, this is not really a problem. The assumption that $K$ is algebraically closed is used to have plenty of formal models which have locally the form ${\rm Spf}({\mathscr{A}}^{0})$, where ${\mathscr{A}}^{0}$ is the subring of power bounded elements in an $K$-affinoid algebra ${\mathscr{A}}$. It has further the advantage that finite base changes are not necessary in the semistable reduction theorem or in de Jong’s alteration theorems. This division has the annoying consequence that many results obtained under one of these hypotheses cannot be used under the other hypothesis. Moreover, there is a growing group of people who would like to use Zhang’s metrics over any non-archimedean base field. The goal of this paper is to remedy this situation and to study these metrics in the utmost generality which is available to us.

From now on, we assume that $K$ is a non-archimedean field which means in this paper that $K$ is a field endowed with a non-trivial non-archimedean complete absolute value. We denote the valuation ring by ${K^{\circ}}$.

We first restrict us to the case of a line bundle $L$ on a proper scheme $X$ over $K$. Later on, we prove many results more generally for paracompact strictly $K$-analytic spaces. We call a metric ${\|\hskip 4.30554pt\|}$ on $L^{\rm an}$ algebraic (resp. formal) if it is induced by a line bundle ${\mathscr{L}}$ on a flat proper scheme ${\mathscr{X}}$ (resp. a line bundle ${\mathfrak{L}}$ on an admissible formal scheme ${\mathfrak{X}}$) over ${K^{\circ}}$ with generic fibre $X$ and with $L={\mathscr{L}}|_{X}$. We use the notation ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{\mathscr{L}}$. Such a metric is called semipositive if ${\mathscr{L}}$ (resp. ${\mathfrak{L}}$) restricts to a nef line bundle on the special fibre of ${\mathscr{X}}$ (resp. ${\mathfrak{X}}$). More generally, we call ${\|\hskip 4.30554pt\|}$ a model metric if there is a non-zero $k\in{\mathbb{N}}$ such that ${\|\hskip 4.30554pt\|}^{\otimes k}$ is an algebraic metric. Then a model metric ${\|\hskip 4.30554pt\|}$ is called semipositive if ${\|\hskip 4.30554pt\|}^{\otimes k}$ is semipositive in the previous sense. We say that ${\|\hskip 4.30554pt\|}$ is a continuous semipositive metric if it is the uniform limit of a sequence of semipositive model metrics on ${L^{\rm an}}$.

We note that the above definitions are global definitions. It is desirable to have local analytic definitions. Let $V$ be a paracompact strictly $K$-analytic space and $L$ a line bundle on $V$. First, we say that a metric ${\|\hskip 4.30554pt\|}$ on $L$ is a piecewise linear metric if there is a ${\rm G}$-covering $(V_{i})_{i\in I}$ of $V$ (i.e. a covering with respect to the G-topology on $V$) and frames $s_{i}$ of $L$ over $V_{i}$ with $\|s_{i}\|\equiv 1$. Note that such metrics are already considered in [Gub98], but they were called formal there which is a bit confusing. We say that a metric ${\|\hskip 4.30554pt\|}$ is piecewise ${\mathbb{Q}}$-linear if there is a ${\rm G}$-covering $(V_{i})_{i\in I}$ of $V$ and some integers $(k_{i})_{i\in I}$ such that for each $i\in I$, the restriction of ${\|\hskip 4.30554pt\|}^{\otimes k_{i}}$ to $V_{i}$ is a piecewise linear metric on $V_{i}$. We refer to Section 2 for details and properties.

Following a suggestion of Tony Yue Yu, we call a piecewise linear metric ${\|\hskip 4.30554pt\|}$ semipositive in $x\in V$ if $x$ has a strictly $K$-affinoid domain $W$ of $V$ as a neighbourhood (in the Berkovich topology) such that the restriction of ${\|\hskip 4.30554pt\|}$ to $L|_{W}$ is a semipositive formal metric. This notion was studied in [GK15] for $K$ algebraically closed. A semipositive piecewise linear metric on $L$ is a piecewise linear metric which is semipositive in every $x\in V$. Semipositive metrics are studied in Section 3. We highlight here the following result which is useful in comparing the various definitions mentioned above.

As seen in Remark 2.5, the equivalence of (a) and (b) follows from [GK14, Proposition 8.13] (as the argument does not use the assumption that $K$ is algebraically closed). The equivalence of (b) and (c) holds more generally over any paracompact strictly $K$-analytic space as shown in Proposition 2.8. This equivalence was known before only in case of a compact reduced space over an algebraically closed field. Neither base change nor the old argument can be used and so we give an entirely new argument here. In the semipositive case, the equivalence of (a) and (b) follows immediately from Proposition 3.5. Finally, the equivalence of (b) and (c) is shown in Proposition 3.10. It holds more generally for a boundaryless paracompact strictly $K$-analytic space.

We also prove the following result (Theorem 2.15) which generalizes [Gub98, Theorem 7.12] from the compact to the paracompact case.

Let us come back to semipositive metrics. For this, let us consider $X$ a proper scheme over $K$. It is a natural question if the notion of semipositivity is closed in the space of model metrics of a given line bundle $L$ of $X$. First, we look at this question for uniform convergence of metrics. We consider a model metric ${\|\hskip 4.30554pt\|}$ on $L^{\rm an}$ which is semipositive as a continuous metric, which means by definition that it is uniform limit of semipositive model metrics on $L^{\rm an}$. Then the closedness problem is equivalent to show that ${\|\hskip 4.30554pt\|}$ is semipositive as a model metric. By passing to a tensor power, we may assume that ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{\mathscr{L}}$ for a line bundle ${\mathscr{L}}$ on a model ${\mathscr{X}}$ of $X$. By assumption, ${\|\hskip 4.30554pt\|}$ is the uniform limit of semipositive model metrics ${\|\hskip 4.30554pt\|}_{n}$ on $L^{\rm an}$. For every $n\in{\mathbb{N}}$, there is a non-zero $k_{n}\in{\mathbb{N}}$ such that ${\|\hskip 4.30554pt\|}_{n}^{\otimes k_{n}}$ is an algebraic metric associated to a nef line bundle ${\mathscr{L}}_{n}$ living on a proper flat scheme ${\mathscr{X}}_{n}$ over ${K^{\circ}}$ with generic fiber $X$. Since the models ${\mathscr{X}}_{n}$ might be completely unrelated to ${\mathscr{X}}$, it is non-obvious to show that ${\mathscr{L}}$ is nef if all the line bundles ${\mathscr{L}}_{n}$ are nef.

An even more challenging problem is to show that the space of model metrics is closed with respect to pointwise convergence. The solution of this problem is the main result of this paper:

If the residue characteristic of $K$ is zero, then this theorem was proven by Boucksom, Favre and Jonsson [BFJ16] Theorem 5.11 using multiplier ideals. They said in [BFJ16] Remark 5.13 that it would be interesting to have a proof along the lines of Goodman’s paper [Goo69, p.178, Proposition 8]. This is what we provide in Theorem 1.3 with a proof holding for any discretely valued non-archimedean field and hence we obtain as an immediate consequence:

For arbitrary non-archimedean fields, this result was first proven in [GK15, Proposition 8.13] using a lifting theorem for closed subvarieties of the special fibre. Amaury Thuillier told us that he found a similar (unpublished) lifting argument to prove Corollary 1.4.

Theorem 1.3 will follow from Theorem 5.6 which is a slightly more general version about pointwise convergence of $\theta$-plurisubharmonic model functions for a closed $(1,1)$-form $\theta$. These notions from [BFJ16] will be introduced in Section 4.