Let $R$ be a complete discrete valuation ring with fraction field $K$ and residue field $k$. We shall assume that $k$ has characteristic zero. We let $t\in R$ be a uniformizing parameter and normalize the corresponding absolute value on $K$ by $\log|t|^{-1}=1$. Note that $R\simeq k[\![t]\!]$ and $K\simeq k(\!(t)\!)$, see for instance [Ser68]. Write $S:=\spec R$.

Let $X$ be a smooth projective $K$-variety, i.e. an integral (but not necessarily geometrically integral) smooth projective $K$-scheme. A model of $X$ is a normal, flat and projective $S$-scheme $\mathcal{X}$ with $X$ as its generic fiber. We denote by $\mathcal{X}_{0}$ its special fiber, and by $\Div_{0}(\mathcal{X})$ the group of vertical Cartier divisors, i.e. those supported in $\mathcal{X}_{0}$. We write $\Div_{0}(\mathcal{X})_{\mathbf{R}}$ accordingly.

Let $\mathcal{M}_{X}$ be the set of all isomorphism classes of models of $X$. Given $\mathcal{X}^{\prime},\mathcal{X}$ in $\mathcal{M}_{X}$ we write $\mathcal{X}^{\prime}\geq\mathcal{X}$ if there exists a morphism $\mathcal{X}^{\prime}\to\mathcal{X}$ obtained by blowing up an ideal sheaf co-supported on the special fiber of $\mathcal{X}$. This turns $\mathcal{M}_{X}$ into a directed set.

Given a model $\mathcal{X}$, let $(E_{i})_{i\in I}$ be the set of irreducible components of the special fiber. For each subset $J\subset I$ set $E_{J}:=\bigcap_{j\in J}E_{j}$. A regular model $\mathcal{X}$ is an SNC model if the special fiber has simple normal crossing support and $E_{J}$ is irreducible (or empty) for each $J\subset I$.

As a topological space, the Berkovich space $X^{\mathrm{an}}$ attached to the given smooth projective $K$-variety $X$ is compact and can be described as follows (cf. [Ber90, Theorem 3.4.1]). Choose a finite cover of $X$ by affine open subsets of the form $U=\spec A$ where $A$ is a $K$-algebra of finite type. The Berkovich space $U^{\mathrm{an}}$ is defined as the set of all multiplicative seminorms $|\cdot|:A\to\mathbf{R}_{+}$ extending the given absolute value of $K$, endowed with the topology of pointwise convergence. The space $X^{\mathrm{an}}$ is obtained by gluing the open sets $U^{\mathrm{an}}$.

There is a natural equivalence of categories between projective $K$-analytic spaces and projective $K$-schemes, see [Ber90, §3.4]. In the sequel we shall therefore always identify a projective $K$-scheme with its associated Berkovich space and write $X^{\mathrm{an}}=X$.

Let $\mathcal{X}$ be a model of $X$. To each irreducible component $E$ of the special fiber is associated a divisorial valuation $\ord_{E}$ of the function field of $X$. After rescaling and exponentiating, this gives rise to an element $x_{E}\in X$ called a divisorial point. The set $X^{\mathrm{div}}$ of divisorial points is dense in $X$.

When $\mathcal{X}$ is an SNC model, we can refine this construction. Write the special fiber as $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$. The dual complex $\Delta_{\mathcal{X}}$ of $\mathcal{X}$ is the simplicial complex whose vertices correspond to the irreducible components $E_{i}$ and whose simplices correspond to nonempty intersections $E_{J}$. We can equip $\Delta_{\mathcal{X}}$ with an (integral) affine structure and embed it in the Berkovich space $X$ as follows.

Consider a subset $J\subset I$ with $E_{J}\neq\emptyset$ and pick $w=(w_{j})_{j\in J}$ with $w_{j}\geq 0$ and $\sum_{j\in J}b_{j}w_{j}=1$. Let $\xi_{J}$ be the generic point of $E_{J}$ and pick a system $(z_{j})_{j\in J}$ of regular parameters for $\mathcal{O}_{\mathcal{X},\xi_{J}}$ with $z_{j}$ defining $E_{j}$. By Cohen’s structure theorem, $\widehat{\mathcal{O}}_{\mathcal{X},\xi_{J}}\simeq\kappa(\xi_{J})[[z_{j},j\in J]]$. Let $v_{J,w}$ be the restriction to $\mathcal{O}_{\mathcal{X},\xi}$ of the monomial valuation on this power series ring, taking value $w_{j}$ on $z_{j}$, i.e. $v_{J,w}\left(\sum_{\alpha\in\mathbf{N}^{J}}c_{\alpha}z^{\alpha}\right)=\min\left\{\sum_{j\in J}w_{j}\alpha_{j}\mid c_{\alpha}\neq 0\right\}$. Then $e^{-v_{J,w}}\in X$. This defines an embedding $\emb_{\mathcal{X}}:\Delta_{\mathcal{X}}\to X$, and the parameters $w$ equip $\Delta_{\mathcal{X}}$ with an affine structure.

There is also a retraction $p_{\mathcal{X}}:X\to\Delta_{\mathcal{X}}$, defined as follows. Any point $x\in X$ admits a center on $\mathcal{X}$. This is the unique point $\xi=c_{\mathcal{X}}(x)\in\mathcal{X}_{0}$ such that $|\varphi|_{x}\leq 1$ for $\varphi\in\mathcal{O}_{\mathcal{X},\xi}$ and $|\varphi|_{x}<1$ for $\varphi\in\mathfrak{m}_{\mathcal{X},\xi}$. Let $J\subset I$ be the maximal subset such that $\xi\in E_{J}$. Then $p_{\mathcal{X}}(x)\in\Delta_{\mathcal{X}}$ corresponds to the monomial valuation with weight $-\log|z_{j}|_{x}$, $j\in J$.

We have $p_{\mathcal{X}}=\id$ on $\Delta_{\mathcal{X}}$. If $\mathcal{Y}$ dominates $\mathcal{X}$, then $\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{Y}}$ and $p_{\mathcal{X}}\circ p_{\mathcal{Y}}=p_{\mathcal{X}}$. The retractions induce a homeomorphism of $X$ onto the inverse limit $\varprojlim\Delta_{\mathcal{X}}$.

In order to keep notation light, we shall identify $\Delta_{\mathcal{X}}$ with its image in $X$ under $\emb_{\mathcal{X}}$. Note that this convention differs from the one adopted in [BFJ11]. A point in $X$ lying in some dual complex $\Delta_{\mathcal{X}}$ is called quasi-monomial, and the set of such points is denoted by $X^{\mathrm{qm}}$.

Let $\mathcal{X}$ be a model of $X$. A vertical fractional ideal sheaf $\mathfrak{a}$ is a finitely generated $\mathcal{O}_{\mathcal{X}}$-submodule of the function field of $\mathcal{X}$ such that $\mathfrak{a}|_{X}=\mathcal{O}_{X}$. Then $\mathfrak{a}$ defines a continuous function $\log|\mathfrak{a}|\in C^{0}(X)$ by setting

$$\log|\mathfrak{a}|(x):=\max\left\{\log|f|_{x}\mid f\in\mathfrak{a}_{c_{\mathcal{X}}(x)}\right\}.$$

Note that each vertical Cartier divisor $D\in\Div_{0}(\mathcal{X})$ defines a vertical fractional ideal sheaf $\mathcal{O}_{\mathcal{X}}(D)$, hence a continuous function $f_{D}:=\log|\mathcal{O}_{\mathcal{X}}(D)|$. Note that $f_{\mathcal{X}_{0}}$ is the constant function $1$ since $\log|t|^{-1}=1$. The map $D\mapsto f_{D}$ extends by linearity to $\Div_{0}(\mathcal{X})_{\mathbf{R}}\to C^{0}(X)$.

Let $\mathcal{X}$ be a model of $X$. The space $N^{1}(\mathcal{X}/S)$ of (relative, codimension $1$) numerical equivalence classes on $\mathcal{X}$ is defined as the quotient of $\Pic(\mathcal{X})_{\mathbf{R}}$ by the subspace spanned by numerically trivial line bundles, i.e. those $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{R}}$ such that $\mathcal{L}\cdot C=0$ for all projective curves contained in a fiber of $\mathcal{X}\to S$. It is in fact enough to consider vertical curves, i.e. those contained in the special fiber $\mathcal{X}_{0}$. A class $\theta\in N^{1}(\mathcal{X}/S)$ is nef if $\theta\cdot C\geq 0$ for all such curves $C$.

We say that a closed $(1,1)$-form $\theta\in\mathcal{Z}^{1,1}(X)$ is determined on a given model $\mathcal{X}$ if it is the image of an element $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$. By definition, two classes $\theta\in N^{1}(\mathcal{X}/S)$ and $\theta^{\prime}\in N^{1}(\mathcal{X}^{\prime}/S)$ define the same element in $\mathcal{Z}^{1,1}(X)$ iff they pull back to the same class on a model dominating both $\mathcal{X}$ and $\mathcal{X}^{\prime}$.

The natural map $N^{1}(\mathcal{X}/S)\to N^{1}(X)$ gives rise to a map $\mathcal{Z}^{1,1}(X)\to N^{1}(X)$ which in fact is surjective. We refer to $\{\theta\}$ as the de Rham class of the closed $(1,1)$-form $\theta$. When $\theta$ is semipositive, the de Rham class $\{\theta\}\in N^{1}(X)$ is nef on $X$. In what follows, we shall mainly work with forms having ample de Rham class.

Any model function $f\in\mathcal{D}(X)$ induces a form $dd^{c}f\in\mathcal{Z}^{1,1}(X)$ as follows: for any determination $\mathcal{X}$ of $f$, $dd^{c}f$ is the class of the divisor $\sum_{i\in I}b_{i}f(x_{i})E_{i}$, where $\mathcal{X}_{0}=\sum_{i}b_{i}E_{i}$ and $x_{i}\in X$ is the divisorial point associated to $E_{i}$.

Fix a form $\theta\in\mathcal{Z}^{1,1}(X)$ with ample de Rham class $\{\theta\}\in N^{1}(X)$.

It is a nontrivial fact that if $\varphi$ is a $\theta$-psh model function then the form $\theta+dd^{c}\varphi$ is in fact semipositive, see [BFJ11, Theorem 5.11]. In particular, the zero function is $\theta$-psh iff $\theta$ is semipositive. In this case, $\max\{\varphi,-t\}$ is $\theta$-psh when $\varphi$ is $\theta$-psh and $t\in\mathbf{R}$.

In fact, the continuity statement above can be made uniform in $\varphi$:

We endow $\PSH(X,\theta)$ with the topology of uniform convergence on dual complexes. Notice that the divisorial points are dense on each dual complex $\Delta_{X}$, see [BFJ11, Corollary 3.13] or [JM10, Remark 3.9]. As a consequence of equicontinuity we thus have

Finally we have the following regularization result. Its proof relies on multiplier ideals.

The complex analogue of this result is due to Demailly [Dem92] (see also [GZ05, Appendix] for the case of a line bundle). By Dini’s lemma, we get as a consequence:

Proposition 4.5 below refines Theorem 2.11 and asserts that any $\theta$-psh function is actually the decreasing limit of a sequence of $\theta$-psh model functions (but the proof heavily uses Theorem 2.11).

Let $\theta$ be a form as in §2.4.

Recall that the usc regularization $u^{*}$ of a function $u:X\to[-\infty,+\infty[$ is the smallest usc function such that $u^{*}\geq u$.

Thanks to Proposition 2.13 $P_{\theta}(f)$ is either $-\infty$ or belongs to $\PSH(X,\theta)$. If $f$ is usc, then clearly $P_{\theta}(f)\leq f$ on $X$, and $P_{\theta}(f)$ is then the largest $\theta$-psh function with this property.

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.

The Monge-Ampère operator that we will use arises from intersection theory on models.

Let $\mathcal{X}$ be a model of $X$, and pick numerical classes $\theta_{1,\mathcal{X}},\dots,\theta_{n,\mathcal{X}}\in N^{1}(\mathcal{X}/S)$. For any vertical divisor $D\in\Div_{0}(\mathcal{X})$ we define

$$D\cdot\theta_{1}\cdot\ldots\cdot\theta_{n}:=\sum_{E}\ord_{E}(D)\,(\theta_{1,\mathcal{X}}|_{E}\cdot\ldots\cdot\theta_{n,\mathcal{X}}|_{E}),$$

where $E$ ranges over all irreducible components of the special fiber $\mathcal{X}_{0}$. We obtain a pairing that is linear in each entry and symmetric in the $\theta_{i}$’s.

As a special case, fix $\theta\in\mathcal{Z}^{1,1}(X)$. To any $\theta$-psh model functions $\varphi_{1},\dots,\varphi_{n}$ we then associate a mixed Monge-Ampère measure

$$(\theta+dd^{c}\varphi_{1})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n}).$$

This is an atomic positive measure on $X$ of mass $\{\theta\}^{n}$.

Analogously to the complex case we have the following integration by parts formula:

The next result follows from the Hodge index theorem, compare [YZ10, Theorem 2.1.1].

We shall make frequent use of basic integration and measure theory. Let $X$ be a compact (Hausdorff) space. A Radon measure on $X$ is a positive linear functional $\mu:C^{0}(X)\to\mathbf{R}$. With this definition, it follows from the Riesz representation theorem that Radon measures are in 1-1 correspondence with regular Borel measures on $X$; see [Fol99, §7.1–2].

Since we shall be dealing with (possibly uncountable) nets rather than sequences, one has to be careful using results from integration theory. For example, the monotone convergence theorem is of course not true for general nets. However, as the next results show, integration of semicontinuous functions against Radon measures is often well behaved.

In particular, one has