The notions of plurisubharmonic (psh) functions and positive currents lie at the heart of complex analysis. The study of these objects is usually referred to as pluripotential theory, and it has become apparent in recent years that pluripotential theory should admit an analogue in the context of non-Archimedean analytic spaces in the sense of Berkovich.

Potential theory on non-Archimedean curves is by now well-established thanks to the work of Thuillier [Thu05] (see also [FJ04, BR10]). In the higher dimensional case, it is in principle possible to mimick the complex case and define a plurisubharmonic function as an upper semicontinuous function whose restriction to each curve is subharmonic. While this approach is yet to be developed, a global notion of semipositive metric has already appeared several times in the literature [Zha95, Gub98, CL06]. Such metrics are in particular continuous by definition. We propose a general (global) definition of (not necessarily continuous) semipositive metrics, and prove basic compactness and regularization results for these metrics.

In order to better explain our construction, let us briefly recall some facts from the complex case [Dem90]. Let $X$ be (the analytication of) a smooth projective complex variety and let $L$ be an ample line bundle on $X$. A smooth metric $\|\cdot\|$ on $L$ is given in every local trivialization of $L$ by $|\cdot|\,e^{-\varphi}$ for some local smooth function $\varphi$, called the local weight of $h$. The curvature of $\|\cdot\|$ is given in this chart by $dd^{c}\varphi$, and $\|\cdot\|$ is said to be semipositive if its curvature is a semipositive $(1,1)$-form, which means that $\varphi$ is psh. More generally, one defines the notion of singular semipositive metrics by allowing locally $\varphi$ to be a general psh function, in which case the curvature of $h$ is a closed positive $(1,1)$-current.

It is a basic fact that every psh function is locally the decreasing limit of a sequence of smooth psh functions. The global analogue of this result for singular semipositive metrics fails for general line bundles, but a deep result of Demailly shows that every singular semipositive metric on an ample line bundle $L$ is indeed a monotone limit of smooth semipositive metrics (cf. [Dem92] or [GZ05, Theorem 8.1] for a more recent account). In particular, every continuous semipositive metric on $L$ is a uniform limit on $X$ of smooth semipositive metrics, thanks to Dini’s lemma.

A fundamental aspect of singular semipositive metrics is that they form a compact space modulo scaling. This fact can be conveniently understood in terms of global weights as follows. Fixing a smooth metric on $L$ with curvature $\theta$ allows one to identify the set of singular semipositive metrics with the set $\PSH(X,\theta)$ of $\theta$-psh functions. The latter are upper semicontinuous (usc) functions $\varphi:X\to[-\infty,+\infty)$ such that $\theta+dd^{c}\varphi$ is a positive closed $(1,1)$-current. Modulo scaling, $\PSH(X,\theta)$ endowed with the $L^{1}$-topology is homeomorphic to the space of closed positive $(1,1)$-currents lying in the cohomology class $c_{1}(L)$ (with its weak topology), hence is compact.

Let us now turn to the non-Archimedean case. Fix a complete, discrete valuation field $K$ with valuation ring $R$ and residue field $k$ and set $S:=\spec R$. We assume that $k$ (and hence $K$) has characteristic zero, which means concretely that $R$ is (non-uniquely) isomorphic to $k[\![t]\!]$. Let $X$ be a smooth projective $K$-analytic space in the sense of Berkovich, so that $X$ is the analytification of a smooth projective $K$-variety by the GAGA principle. Recall that the underlying topological space of $X$ is compact Hausdorff. A model $\mathcal{X}$ of $X$ is a normal flat projective $S$-scheme such that the analytification of its generic fiber $\mathcal{X}_{K}$ is isomorphic to $X$. Each line bundle $L$ on $X$ is the analytification of a line bundle on $\mathcal{X}_{K}$ (that we also denote by $L$), and a model metric is a metric $h_{\mathcal{L}}$ on $L$ that is naturally induced by the choice of a $\mathbf{Q}$-line bundle $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ such that $\mathcal{L}|_{\mathcal{X}_{K}}=L$ in $\Pic(\mathcal{X}_{K})_{\mathbf{Q}}$. A model function $\varphi$ on $X$ is a function such that $e^{-\varphi}$ is a model metric on the trivial line bundle. The set $\mathcal{D}(X)$ of model functions is then dense in $C^{0}(X)$, a well-known consequence of the Stone-Weierstrass theorem.

Following S.-W.Zhang [Zha95, 3.1] (see also [KT, 6.2.1], [Gub98, 7.13], [CL06, 2.2]), we shall say that a model metric $h_{\mathcal{L}}$ on $L$ is semipositive if $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ is nef on the special fiber $\mathcal{X}_{0}$ of $\mathcal{X}$, i.e. $\mathcal{L}\cdot C\geq 0$ for all projective curves $C$ in $\mathcal{X}_{0}$. A continuous metric on $L$ is then semipositive in the sense of Zhang if it can be written as a uniform limit over $X$ of a sequence of semipositive model metrics. The reader may consult [CL10] for a nice survey on these notions.

In order to define a general notion of singular semipositive metrics, we use the finer description of $X$ as a projective limit of dual complexes (called skeletons in Berkovich’s terminology). Since the residue field $k$ of $R$ has characteristic zero, it follows from [Tem06] that each model of $X$ is dominated by a SNC model $\mathcal{X}$, by which we understand a regular model whose special fiber $\mathcal{X}_{0}$ has simple normal crossing support (plus a harmless irreducibility condition that we impose for convenience). To each SNC model $\mathcal{X}$ corresponds its dual complex $\Delta_{\mathcal{X}}$, a compact simplicial complex which encodes the incidence properties of the irreducible components of $\mathcal{X}_{0}$. The dual complex $\Delta_{\mathcal{X}}$ embeds in $X$ and for the purposes of this introduction we shall view $\Delta_{\mathcal{X}}$ as a compact subset of $X$. There is furthermore a retraction $p_{\mathcal{X}}:X\to\Delta_{\mathcal{X}}$. These maps are compatible with respect to domination of models, and we thus get a map

$$X\to\varprojlim_{\mathcal{X}}\Delta_{\mathcal{X}}$$

which is known to be a homeomorphism (compare for instance [KS06, p.77, Theorem 10]).

Following the philosophy of [BGS95], we define the space of closed $(1,1)$-forms on $X$ as the direct limit over all models $\mathcal{X}$ of $X$ of the spaces $N^{1}(\mathcal{X}/S)$ of codimension one numerical equivalence classes. Any model metric gives rise to a curvature form lying in this space. The main reason for working with numerical equivalence (instead of rational equivalence as in [BGS95]) is that we can then adapt a result of [Kün96] to show that a line bundle $L\in\Pic(X)$ has vanishing first Chern class $c_{1}(L)\in N^{1}(X)$ iff it admits a model metric with zero curvature (cf. Corollary 4.5).

Fix a reference model metric $\|\cdot\|$ on $L$ with curvature form $\theta$. Any other metric can be written $\|\cdot\|e^{-\varphi}$ for some function $\varphi$ on $X$. When $\|\cdot\|e^{-\varphi}$ is a semipositive model metric, we say that the model function $\varphi$ is $\theta$-psh.

Here again, the consistency of the definition for model functions will be guaranteed by Theorem 5.11 below. Let $\varphi$ be a $\theta$-psh function. Since $\varphi$ is usc and each $\varphi\circ p_{\mathcal{X}}$ is continuous it follows immediately that $\varphi=\inf_{\mathcal{X}}\varphi\circ p_{\mathcal{X}}$, so that $\varphi$ is uniquely determined by its restriction to the dense subset $\bigcup_{\mathcal{X}}\Delta_{\mathcal{X}}$ of $X$. We may therefore endow the set $\PSH(X,\theta)$ of all $\theta$-psh functions (or equivalently of all singular semipositive metrics on $L$) with the topology of uniform convergence on dual complexes. We view this topology as an analogue of the $L^{1}$-topology in the complex case. Our first main theorem shows that this space is indeed compact modulo additive constants, something that was also announced in the preliminary work [KT].

In other words, the set of singular semipositive metrics on $L$ modulo scaling is compact. For curves, this result is a consequence of the work of Thuillier [Thu05], and follows from basic properties of subharmonic functions on metrized graphs.

Our second main result is the following analogue of Demailly’s global regularization theorem.

When $\dim X=1$, Theorem B is a special case of [Thu05, Théorème 3.4.19]. thanks to Dini’s lemma, Theorem B implies a non-Archimedean version of the Demailly-Richberg theorem, stating that every continuous $\theta$-psh function is a uniform limit over $X$ of $\theta$-psh model functions. In other words, for continuous metrics our definition of semipositivity agrees with Zhang’s.

Let us briefly explain how Theorem A above is proved. The first important fact is that any dual complex $\Delta_{\mathcal{X}}$ comes equipped with a natural affine structure [KS06] such that any $\theta$-psh function is convex on the faces of $\Delta_{\mathcal{X}}$. On this complex we put any euclidean metric compatible with the affine structure. The statement that we actually prove, and which implies Theorem A, is

This result is proved in two steps. Assuming, as we may, that $\sup_{\Delta_{\mathcal{X}}}\varphi=0$, we first bound $\varphi$ from below on the vertices of $\Delta_{\mathcal{X}}$. This is done by exploiting the non-negativity of certain intersection numbers, a direct consequence of the model metric $\|\cdot\|e^{-\varphi}$ being determined by a nef line bundle on some model.

The next step is to prove the uniform Lipschitz bound. Here again, the general idea is to exploit the non-negativity of certain intersection numbers, but the argument is more subtle than in the first step. This time, the intersection numbers are computed on (possibly singular) blow-ups of $\mathcal{X}$ corresponding to carefully chosen combinatorial decompositions of $\Delta_{\mathcal{X}}$, in the spirit of the toroidal constructions of [KKMS].

These techniques can be extended to other geometric contexts such as the space of valuations considered in [BFJ08]. In particular they yield a uniform version of Izumi’s theorem [Izu85]. However, we shall postpone these extensions to a separate note.

The proof of Theorem B is of a different nature. Using Theorem A, we first show that the usc upper envelope of any family of $\theta$-psh functions remains $\theta$-psh, a basic property of $\theta$-psh functions in the complex case. As a consequence, given any continuous function $u\in C^{0}(X)$ the set of all $\theta$-psh functions $\psi$ such that $\psi\leq u$ on $X$ admits a largest element, called the $\theta$-psh envelope of $u$ and denoted by $P_{\theta}(u)$. On the other hand, by density of $\mathcal{D}(X)$ in $C^{0}(X)$, we may write any given function $\varphi\in\PSH(X,\theta)$ as the pointwise limit of a decreasing family of (a priori not $\theta$-psh) model functions $u_{j}$, using only the upper semicontinuity of $\varphi$. It is not difficult to see that $P_{\theta}(u_{j})$ decreases to $\varphi$, and we are thus reduced to showing that the $\theta$-psh envelope of any model function is the uniform limit of a sequence of $\theta$-psh model functions. This is proved using multiplier ideals, in the spirit of [DEL00, ELS03, BFJ08]. The required properties of multiplier ideals are shown to hold on regular models in Appendix B, the key point being to show that the expected version of the Kodaira vanishing theorem holds in this context.

Let us comment on our assumptions on the field $K$. It is expected that pluripotential theory can be developed on Berkovich spaces over arbitrary non-Archimedean fields. We restrict our attention to the discretely valued case in order to avoid the use of formal models over non-noetherian rings. We refer to [Gub98, Gub03] for related works on semi-positive metrics in the general setting of a complete non-Archimedean field.

More importantly, Appendix B relies on the discreteness assumption. Furthermore, we use the assumption that $K$ has residue characteristic zero in two ways, through the existence of SNC models and through the cohomology vanishing properties of multiplier ideals. In positive residue characteristic, the existence of SNC models is not known and it is much harder to construct retractions of $X$ onto suitable embedded complexes. We refer to Berkovich [Ber99], Hrushovski-Loeser [HL10], and Thuillier [Thu11] for important contributions to the understanding of this problem.

In a sequel to this paper [BFJ] we will rely on the results obtained here to adapt to the non-Archimedean case the variational approach to complex Monge-Ampère equations developed in [BBGZ09].

The paper is organized as follows. The first two sections present the necessary background on Berkovich spaces and models. The exposition is largely self-contained (hence perhaps a bit lengthy), as we feel that the easy arguments that our setting allows are worth being explained. Section 3 is devoted to dual complexes. The main technical result is Theorem 3.11, on the existence of blow-ups attached to decompositions of a dual complex. Section 4 deals with closed $(1,1)$-forms, defined using a numerical equivalence variant of the approach of [BGS95]. In Section 5 we prove some basic properties of $\theta$-psh model functions. Section 6 contains the proof of Theorem A. Section 7 is devoted to the first properties of general $\theta$-psh functions. Theorem B is proved in Section 8. Finally, Appendix A contains a technical result on Lipschitz constants of convex functions, while Appendix B establishes the expected cohomology vanishing properties of multiplier ideals in our setting.