Let $\mathrm{X}$ be a $K$-analytic space in the sense of Berkovich [11]. For simplicity, we will assume that $\mathrm{X}$ is the analytic space associated to a proper scheme over $K$. In that context, the general definition of continuous metrized line bundles given above makes sense.

Let us detail the example of the line bundle $\mathscr{O}(1)$ on the projective space $\mathrm{P}^{n}_{K}$. A point $x\in\mathrm{P}^{n}_{K}$ possesses a complete residue field $\mathscr{H}(x)$ which is a complete extension of $K$ and homogeneous coordinates $[x_{0}:\dots:x_{n}]$ in the field $\mathscr{H}(x)$. As in complex geometry, the projective space $\mathrm{P}^{n}_{K}$ is obtained by glueing $n+1$ copies $\mathrm{U}_{0},\dots,\mathrm{U}_{n}$ of the affine space $\mathrm{A}^{n}_{K}$, where $\mathrm{U}_{i}$ corresponds to those points $x$ such that $x_{i}\neq 0$. Recall also that $\mathrm{A}^{n}_{K}$ is the space of multiplicative semi-norms on the $K$-algebra $K[T_{1},\dots,T_{n}]$ which induce the given absolute value on $K$, together with the coarsest topology such that for any semi-norm $x\in\mathrm{A}^{n}_{K}$, the map $K[T_{1},\dots,T_{n}]\rightarrow{\mathbf{R}}$ defined by $f\mapsto x(f)$ is continuous. The kernel of a semi-norm $x$ is a prime ideal $\mathfrak{p}_{x}$ of $K[T_{1},\dots,T_{n}]$ and $x$ induces a norm on the quotient ring $K[T_{1},\dots,T_{n}]/\mathfrak{p}_{x}$, hence on its field of fractions $K(x)$. The completion of $K(x)$ with respect to this norm is denoted $\mathscr{H}(x)$ and is called the complete residue field of $x$. The images in $\mathscr{H}(x)$ of the intederminates $T_{i}$ are denoted $T_{i}(x)$, more generally, the image in $\mathscr{H}(x)$ of any polynomial $f\in K[T_{1},\dots,T_{n}]$ is denoted $f(x)$ ; one has $x(f)=\left|{f(x)}\right|$.

Let $f$ be a rational function on $\mathrm{P}^{n}_{K}$, that is an element of $\in K(T_{1},\dots,T_{n})$. It defines an actual function on the open set $\mathrm{U}$ of ${\mathbf{P}}^{n}_{K}$ where its denominator does not vanish ; its value at a point $x\in U$ is an element of $\mathscr{H}(x)$. More generally, Berkovich defines an analytic function on an open set $\mathrm{U}$ of $\mathrm{P}^{n}_{K}$ as a function $f$ on $U$ such that $f(x)\in\mathscr{H}(x)$ for any $x\in\mathrm{U}$, and such that any point $x\in\mathrm{U}$ possesses a neighbourhood $\mathrm{V}\subset\mathrm{U}$ such that $f|_{\mathrm{V}}$ is a uniform limit of rational functions without poles on $\mathrm{V}$.

The line bundle $\mathscr{O}(1)$ can also be defined in a similar way to the classical case ; by a similar GAGA theorem, its global sections are exactly the same as in algebraic geometry and are described by homogeneous polynomials of degree $1$ with coefficients in $K$. If $P$ is such a polynomial and $s_{P}$ the corresponding section, then

$$\left\|{s_{P}}\right\|(x)=\frac{\left|{P(x_{0},\dots,x_{n})}\right|}{\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|)}$$

where $[x_{0}:\dots:x_{n}]$ is a system of homogeneous coordinates in $\mathscr{H}(x)$ for the point $x$. The function $\left\|{s_{P}}\right\|$ is continuous on $\mathrm{P}^{n}_{K}$, by the very definition of the topology on $\mathrm{P}^{n}_{K}$. Using the fact that $\mathscr{O}(1)$ is generated by its global sections, one deduces the existence of a continuous metric on $\mathscr{O}(1)$ satisfying the previous formula.

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.

Let $\overline{L}$ be a metrized line bundle and let $s$ be a regular meromorphic section of $L$. Its divisor $\operatorname{div}(s)$ is a Cartier divisor in $\mathrm{X}$. The function $\log\left\|{s}\right\|^{-1}$ is defined on the open set $\mathrm{X}\setminus\left|{\operatorname{div}(s)}\right|$ ; by analogy to the complex case, we call it a Green function for the divisor $\operatorname{div}(s)$. When the metric on $\overline{L}$ is smooth, the Green function is said smooth. The same remark applies for the other qualificatives semi-positive, or admissible, that wil be introduced later.

Let us take for $L$ the trivial line bundle, with its canonical trivialization $s=1$, and let us endow it with a smooth metric. By definition, we call $\log\left\|{s}\right\|^{-1}$ a smooth function. More generally, we define the space $\mathscr{C}^{\infty}(\mathrm{X})$ of (real valued) smooth functions to be the real vector space spanned by these elementary smooth functions. Observe that this definition reverses what happens in complex geometry where smooth metrics on the trivial line bundle are defined from the knowledge of smooth functions.

Let us consider the smooth metric on $\mathscr{O}(1)$ associated to the model $(\mathfrak{X},\mathscr{O}(1),1)$ of $(\mathrm{P}^{n}_{K},\mathscr{O}(1))$. Let $\mathfrak{U}_{i}$ be the formal open subset of $\mathrm{P}^{n}_{K}$ defined by the non-vanishing of the homogeneous coordinate $x_{i}$. Over, $\mathfrak{U}_{i}$, $\mathscr{O}(1)$ has a global non-vanishing section, namely the one associated to the homogeneous polynomial $X_{i}$. The generic fiber $U_{i}$ of $\mathfrak{U}_{i}$ in the sense of algebraic geometry is an affine space, with coordinates $z_{j}=x_{j}/x_{i}$, for $0\leq j\leq n$, and $j\neq i$. However, its generic fiber $\mathrm{U}_{i}$ in the sense of rigid geometry is the $n$-dimensional polydisk in this affine space defined by the inequalities $\left|{z_{j}}\right|\leq 1$. We thus observe that for any $x\in(\mathfrak{U}_{i})_{K}$,

$$\left\|{X_{i}}\right\|(x)=1=\frac{1}{\max(\left|{z_{0}}\right|,\dots,\left|{z_{i-1}}\right|,1,\left|{z_{i+1}}\right|,\dots,\left|{z_{n}}\right|)}=\frac{\left|{x_{i}}\right|}{\max(\left|{x_{0}}\right|,\dots,\left|{x_{i}}\right|)}=\left\|{X_{i}}\right\|_{\mathrm{W}}(x).$$

In other words, the Weil metric on $\mathscr{O}(1)$ is a smooth metric.

Let us show that any line bundle has a smooth metric. There is a general theory, due to Raynaud, that shows how to define formal models from rigid analytic objects. In the present case, $X$ being projective, we may assume that $L$ is ample and consider a closed embedding of $\mathrm{X}$ in a projective space $\mathrm{P}^{n}_{K}$ given by some power $L^{e}$. Let $\mathfrak{X}$ be the Zariski closure of $\mathrm{X}$ in $\mathrm{P}^{n}_{K^{\circ}}$ ; in concrete terms, if $I\subset K[X_{0},\dots,X_{n}]$ is the homogeneous ideal of $i(X)$, $I\cap K^{\circ}[X_{0},\dots,X_{n}]$ is the homogeneous ideal of $\mathfrak{X}$. Let then $\mathfrak{L}$ be the restriction to $\mathfrak{X}$ of the line bundle $\mathscr{O}(1)$. The triple $(\mathfrak{X},\mathfrak{L},e)$ is a model of $L$ and induces a smooth metric on $L$.

Different models can give rise to the same metric. If $\varphi\colon\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$ is a morphism of models, and $\mathfrak{L}^{\prime}=\varphi^{*}\mathfrak{L}$, then $(\mathfrak{X}^{\prime},\mathfrak{L}^{\prime},e)$ defines the same smooth metric on $L$. Moreover, if two models $(\mathfrak{X}_{i},\mathfrak{L}_{i},e_{i})$, for $i\in\{1,2\}$, define the same metric, there exists a third model $(\mathfrak{X},\mathfrak{L})$, with two morphisms $\varphi_{i}\colon\mathfrak{X}\rightarrow\mathfrak{X}_{i}$ such that the pull-backs $\varphi_{i}^{*}\mathfrak{L}_{i}^{e_{1}e_{2}/e_{i}}$ coincide with $\mathfrak{L}$. More precisely, if two models $\mathfrak{L}$ and $\mathfrak{L}^{\prime}$ of some power $L^{e}$ on a normal model $\mathfrak{X}$ define the same metric, then they are isomorphic. (See, e.g., Lemma 2.2 of [19] ; this may be false for non-normal models ; it suffices that $\mathfrak{X}$ be integrally closed in its generic fiber.)

As a consequence, the set $\overline{\operatorname{Pic}}_{\text{sm}}(\mathrm{X})$ of smooth metrized line bundles is a subgroup of the group $\overline{\operatorname{Pic}}(\mathrm{X})$. The group $\overline{\operatorname{Pic}}_{\text{sm}}(\mathrm{X})$ fits within an exact sequence

$$0\rightarrow\mathscr{C}^{\infty}(\mathrm{X})\rightarrow\overline{\operatorname{Pic}}_{\text{sm}}(\mathrm{X})\rightarrow\operatorname{Pic}(\mathrm{X})\rightarrow 0,$$

the last map is surjective because every line bundle admits a model. If $f\colon\mathrm{Y}\rightarrow\mathrm{X}$ is a morphism, then $f^{*}(\overline{\operatorname{Pic}}_{\text{sm}}(\mathrm{X}))\subset\overline{\operatorname{Pic}}_{\text{sm}}(\mathrm{Y})$.

A smooth metric is said to be ample if it is defined by a model $(\mathfrak{X},\mathfrak{L},e)$ such that the restriction $\mathfrak{L}_{\tilde{K}}$ of $\mathfrak{L}$ to the closed fiber $\mathfrak{X}_{\tilde{K}}$ is ample. The Weil metric on the line bundle $\mathscr{O}(1)$ on the projective space is ample. The proof given above of the existence of smooth metrics shows, more precisely, that ample line bundles admit ample metrics, and that the pull-back of a smooth ample metric by an immersion is a smooth ample metric.

A smooth metric is said to be semi-positive if it can be defined on a model $(\mathfrak{X},\mathfrak{L},e)$ such that the restriction $\mathfrak{L}_{\tilde{K}}$ of $\mathfrak{L}$ to the closed fiber $\mathfrak{X}_{\tilde{K}}$ is numerically effective : for any projective curve $C\subset\mathfrak{X}_{\tilde{K}}$, the degree of $\mathfrak{L}_{\tilde{K}}$ is non-negative. Ample metrics are semi-positive.

The pull-back of a smooth semi-positive metric by any morphism is semi-positive.

Let us say that a continuous metric on a line bundle $L$ is semi-positive if it is the uniform limit of a sequence of smooth semi-positive metrics on the same line bundle $L$. As in the complex case, we then say that a metrized line bundle is admissible if it can be written as $\overline{L}\otimes\overline{M}^{\vee}$, for two line bundles $L$ and $M$ with continuous semi-positive metrics.

Let $L$ be a metrized line bundle, and let $\left\|{\cdot}\right\|_{1}$ and $\left\|{\cdot}\right\|_{2}$ be two continuous metrics on $L$. It follows from the definition that the metrics $\left\|{\cdot}\right\|_{\min}=\min(\left\|{\cdot}\right\|_{1},\left\|{\cdot}\right\|_{2})$ and $\left\|{\cdot}\right\|_{\max}=\max(\left\|{\cdot}\right\|_{1},\left\|{\cdot}\right\|_{2})$ are continuous metrics.

Moreover, these metrics $\left\|{\cdot}\right\|_{\min}$ and $\left\|{\cdot}\right\|_{\max}$ are smooth if the initial metrics are smooth. Indeed, there exists a model $\mathfrak{X}$, as well as two line bundles $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ extending the same power $L^{e}$ of $L$ and defining the metrics $\left\|{\cdot}\right\|_{1}$ and $\left\|{\cdot}\right\|_{2}$ respectively. We may assume that $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ have regular global sections $s_{1}$ and $s_{2}$ on $\mathfrak{X}$ which coincide on $X$, with divisors $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ respectively. (The general case follows, by twisting $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ by a sufficiently ample line bundle on $\mathfrak{X}$.) The blow-up $\pi\colon\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$ of the ideal $\mathfrak{I}_{\mathfrak{D}_{1}}+\mathfrak{I}_{\mathfrak{D}_{2}}$ ; it carries an invertible ideal sheaf $\mathfrak{I}_{\mathfrak{E}}=\pi^{*}(\mathfrak{I}_{\mathfrak{D}_{1}}+\mathfrak{I}_{\mathfrak{D}_{2}})$, with corresponding Cartier divisor $\mathfrak{E}$. Since $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ coincide on the generic fiber, $\mathfrak{I}_{\mathfrak{D}_{1}}+\mathfrak{I}_{\mathfrak{D}_{2}}$ is already invertible there and $\pi$ is an isomorphism on the generic fiber.

The divisors $\pi^{*}\mathfrak{D}_{1}$ and $\pi^{*}\mathfrak{D}_{2}$ decompose canonically as sums

$$\pi^{*}\mathfrak{D}_{1}=\mathfrak{D}^{\prime}_{1}+\mathfrak{E},\quad\pi^{*}\mathfrak{D}_{2}=\mathfrak{D}^{\prime}_{2}+\mathfrak{E}.$$

Let us pose

$$\mathfrak{D}^{\prime}=\mathfrak{D}^{\prime}_{1}+\mathfrak{D}^{\prime}_{2}+\mathfrak{E}=\mathfrak{D}^{\prime}_{1}+\pi^{*}\mathfrak{D}_{2}=\pi^{*}\mathfrak{D}_{1}+\mathfrak{D}^{\prime}_{2}.$$

An explicit computation on the blow-up shows that $(\mathfrak{X}^{\prime},\mathfrak{D}^{\prime},e)$ and $(\mathfrak{X}^{\prime},\mathfrak{E},e)$ are models of $\left\|{\cdot}\right\|_{\min}$ and $\left\|{\cdot}\right\|_{\max}$ respectively. In particular, these metrics are smooth.

Assume that the initial metrics are semi-positive, and that some positive power of $L$ is effective. Then, the metric $\left\|{\cdot}\right\|_{\min}$ is semi-positive too. By approximation, it suffices to treat the case where the initial metrics are smooth and semi-positive. Then, the previous construction applies. Keeping the introduced notation, let us show that the restriction to the special fiber of the divisor $\mathfrak{(}D^{\prime})_{\tilde{K}}$ is numerically effective. Let $C\subset\mathfrak{X}^{\prime}_{\tilde{K}}$ be an integral curve and let us prove that $C\cdot\mathfrak{(}D^{\prime})_{\tilde{K}}$ is nonnegative. If $C$ is not contained in $\mathfrak{D}^{\prime}_{1}$, then $C\cdot(\mathfrak{D}^{\prime}_{1})_{\tilde{K}}\geq 0$, and $C\cdot(\pi^{*}\mathfrak{D}_{2})_{\tilde{K}}=\pi_{*}C\cdot\mathfrak{D}_{2}\geq 0$ since $(\mathfrak{D}_{2})_{\tilde{K}}$ is numerically effective ; consequently, $C\cdot\mathfrak{(}D^{\prime})_{\tilde{K}}\geq 0$. Similarly, $C\cdot\mathfrak{(}D^{\prime})_{\tilde{K}}\geq 0$ when $C$ is not contained in $\mathfrak{D}^{\prime}_{2}$. Since $\mathfrak{D}^{\prime}_{1}\cap\mathfrak{D}^{\prime}_{2}=\emptyset$, this shows that $C\cdot\mathfrak{D}^{\prime}_{\tilde{K}}\geq 0$ in any case, hence $(\mathfrak{D}^{\prime})_{\tilde{K}}$ is numerically effective.

This last result is the analogue in the ultrametric case to the fact that the maximum of two continuous plurisubharmonic functions is continuous plurisubharmonic. However, observe that in the complex case, the maximum or the minimum of smooth functions are not smooth in general.

In the non-archimedean case, there isn’t yet a purely analytic incarnation of the curvature form (or current) $c_{1}(\overline{L})$ of a metrized line bundle $\overline{L}$, although the non-archimedean Arakelov geometry of [13] should certainly be pushed forward in that direction. However, as I discovered in [18], one can define an analogue of the measure $c_{1}(\overline{L})^{n}$ when the space $\mathrm{X}$ has dimension $n$.

The idea consists in observing the local height pairing (defined by arithmetic intersection theory) and defining the measures so that a formula analogous to the complex one holds.

Let us therefore consider smooth metrized line bundles $\overline{L}_{j}$ (for $0\leq j\leq n$) as well as regular meromorphic sections $s_{j}$ which have no common zero on $X$. There exists a proper model $\mathfrak{X}$ of $\mathrm{X}$ over $K^{\circ}$ and, for each $j$, a line bundle $\mathfrak{L}_{j}$ on $\mathfrak{X}$ which extends some power $L_{j}^{e_{j}}$ of $L_{j}$ and which defines its metric.

Let $\mathrm{Z}\subset\mathrm{X}$ be an algebraic $k$-dimensional subvariety and let $\mathfrak{Z}$ be its Zariski closure in $\mathfrak{X}$ ; this is a $(k+1)$-dimensional subscheme of $\mathfrak{X}$. Let’s replace it by its normalization or, more precisely, by its integral closure in its generic fiber. The local height pairing is then given by intersection theory, as

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z})=(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k}|_{\mathfrak{Z}}))|\mathfrak{Z})\,\log\left|{\pi}\right|^{-1},$$

where $\operatorname{div}(s_{j}|_{\mathfrak{Z}})$ means the divisor of $s_{j}$, viewed as a regular meromorphic section of $\mathfrak{L}_{j}$ over $\mathfrak{Z}$. The right hand side means taking the intersection of the indicated Cartier divisors on $\mathfrak{Z}$, which is a well-defined class of a $0$-cycle supported by the special fiber of $\mathfrak{Z}$ ; then take its degree and multiply it by $\log\left|{\pi}\right|^{-1}$. (Recall that $\pi$ is a fixed uniformizing element of $K$ ; is absolute value does not depend on the actual choice.)

When one views $s_{k}|_{\mathrm{Z}}$ as a regular meromorphic section of $\mathfrak{L}_{k}$ on $\mathfrak{Z}$ its divisor has two parts : the first one, say $H$, is “horizontal” and is the Zariski closure of the divisor $\operatorname{div}(s_{k}|_{Z})$ ; the second one, say $V$, is vertical, i.e., lies in the special fiber of $\mathfrak{Z}$ over the residue field of $K^{\circ}$. This decomposes the local height pairing as a sum

	$\displaystyle(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z})$	$\displaystyle=(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k}|_{\mathfrak{Z}}))|\mathfrak{Z})\log\left|{\pi}\right|^{-1}$
		$\displaystyle=(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|\operatorname{div}(s_{k}|_{\mathfrak{Z}}))\log\left|{\pi}\right|^{-1}$
		$\displaystyle=(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|H)\log\left|{\pi}\right|^{-1}$
		$\displaystyle\qquad{}+(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|V)\log\left|{\pi}\right|^{-1}.$

The first term is the local height pairing of $\operatorname{div}(s_{k}|_{\mathrm{Z}})$. Let us investigate the second one.

Let $(V_{i})$ be the family of irreducible components of this special fiber ; for each $i$, let $m_{i}$ be its multiplicity in the fiber. Then, the vertical component $V$ of $\operatorname{div}(s_{k}|_{\mathfrak{Z}})$ decomposes as

$$V=\sum_{i}c_{i}m_{i}V_{i},$$

where $c_{i}$ is nothing but the order of vanishing of $s_{k}$ along the special fiber at the generic point of $V_{i}$. Then,

$$(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|V)=\sum_{i}c_{i}m_{i}(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|V_{i}).$$

Since $V_{i}$ lies within the special fiber of $\mathfrak{X}$,

$$(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|V_{i})=(c_{1}(\mathfrak{L}_{0})\dots c_{1}(\mathfrak{L}_{k-1})|V_{i}),$$

the multidegree of the vertical component $V_{i}$ with respect to the restriction on the special fiber of the line bundles $\mathfrak{L}_{0},\dots,\mathfrak{L}_{k-1}$.

One remarkable aspect of Berkovich’s theory is the existence, for each $i$, of a unique point $v_{i}$ in $\mathrm{Z}$ which specializes to the generic point of $V_{i}$. (Here, we use that $\mathfrak{Z}$ is integrally closed in its generic fibre.) Then,

$$\log\left\|{s_{k}}\right\|^{-1}(z_{i})=c_{i}\log\left|{\pi}\right|^{-1}.$$

Finally,

$$(c_{1}(\operatorname{div}(s_{0}|_{\mathfrak{Z}}))\dots c_{1}(\operatorname{div}(s_{k-1}|_{\mathfrak{Z}}))|V)\log\left|{\pi}\right|^{-1}\\ =\sum_{i}\log\left\|{s_{k}}\right\|^{-1}(z_{i})(c_{1}(\mathfrak{L}_{0})\dots c_{1}(\mathfrak{L}_{k-1})|V_{i}).$$

Let us sum up this calculation : we have introduced points $v_{i}\in\mathrm{Z}$ and decomposed the local height pairing as a sum :

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z})=(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k-1})|\operatorname{div}(s_{k}|_{\mathrm{Z}}))\\ +\sum_{i}\log\left\|{s_{k}}\right\|^{-1}(v_{i})m_{i}(c_{1}(\mathfrak{L}_{0})\dots c_{1}(\mathfrak{L}_{k-1})|V_{i}).$$

It now remains to define

$$c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})\delta_{\mathrm{Z}}=\sum_{i}m_{i}(c_{1}(\mathfrak{L}_{0})\dots c_{1}(\mathfrak{L}_{k-1})|V_{i})\delta_{v_{i}},$$

where $\delta_{v_{i}}$ is the Dirac measure at the point $v_{i}\in{\mathrm{Z}}$. This is a measure on $\mathrm{X}$, whose support is contained in $\mathrm{Z}$, and whose total mass equals

$$\sum_{i}(c_{1}(\mathfrak{L}_{0})\dots c_{1}(\mathfrak{L}_{k-1})|V_{i})=(c_{1}(\mathfrak{L}_{0})\dots c_{1}(\mathfrak{L}_{k-1})|V)=(c_{1}(L_{0})\dots c_{1}(L_{k-1})|\mathrm{Z}).$$

One can also check that it does not depend on the choice of the section $s_{k}$.

With this definition, the local height pairing obeys an induction formula totally analogous to the one satisfied in the complex case :

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|{\mathrm{Z}})\\ =(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k-1})|\operatorname{div}(s_{k}|_{\mathrm{Z}}))+\int_{\mathrm{X}}\log\left\|{s_{k}}\right\|^{-1}c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})\delta_{\mathrm{Z}}.$$

(1.3.1)

With the notation of the previous paragraph, observe that the measures we have defined are positive when the smooth metrized line bundles are semi-positive. Indeed, this means that the line bundles $\mathfrak{L}_{j}$ are numerically effective hence, as a consequence of the criterion Nakai–Moishezon, any subvariety of the special fiber has a nonnegative multidegree.

With basically the same argment that the one we sketched in the complex case, we conclude that the local height pairing extends by continuity when semi-positive metrized line bundles are approximated by smooth semi-positive metrized line bundles. By linearity, this extends the local height pairing to admissible metrized line bundles.

Let us now return to semi-positive metrized line bundles $\overline{L}_{0},\dots,\overline{L}_{k-1}$, approximated by smooth semi-positive metrized line bundles $\overline{L}_{j}^{(m)}$. I claim that for any $k$-dimensional variety $\mathrm{Z}\subset\mathrm{X}$, the sequence of measures $c_{1}(\overline{L}_{0}^{(m)})\dots c_{1}(\overline{L}_{k-1}^{(m)})\delta_{\mathrm{Z}}$ converges to a measure on $\mathrm{X}$.

To prove the claim, we may assume that $\overline{L}_{0},\dots,\overline{L}_{k-1}$ have sections $s_{0},\dots,s_{k-1}$ without common zeroes on $\mathrm{Z}$. Let also consider a smooth function $\varphi$ on $\mathrm{X}$ ; let $\overline{L}_{k}$ be the trivial line bundle with the section $s_{k}=1$, metrized in such a way that $\left\|{s_{k}}\right\|=e^{-\varphi}$. Then, one has

$$\int_{\mathrm{X}}\varphi c_{1}(\overline{L}^{(m)}_{0})\dots c_{1}(\overline{L}^{(m)}_{k-1})\delta_{\mathrm{Z}}=(\mathop{\widehat{\operatorname{div}}}(s_{0})^{(m)}\dots\mathop{\widehat{\operatorname{div}}}(s_{k-1})^{(m)}\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z});$$

writing $\overline{L}_{k}$ has the quotient of two ample metrized line bundles, we deduce from the existence of the local height pairing for admissible metrics that these integrals converge when $m\rightarrow\infty$. Consequently, the sequence of measures $(c_{1}(\overline{L}^{(m)}_{0})\dots c_{1}(\overline{L}^{(m)}_{k-1})\delta_{\mathrm{Z}})_{m}$ converges to a positive linear form on the space of smooth functions. By a theorem of Gubler ([34], Theorem 7.12), which builds on the Stone-Weierstraß theorem and the compactness of the Berkovich space $\mathrm{X}$, the space of smooth functions is dense in the space of continuous complex functions on $\mathrm{X}$. A positivity argument, analogous to the proof that positive distributions are measures, then implies that our linear form is actually a positive measure which deserves the notation

$$c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})\delta_{\mathrm{Z}}.$$

We then extend this definition by linearity to the case of arbitrary admissible line bundles. The total mass of this measure is again the multidegree of ${\mathrm{Z}}$ with respect to the line bundles $L_{j}$ (for $0\leq j\leq k-1$).

The definition of the convergence of a sequence of measures is convergence of all integrals against a given continuous compactly supported function. In applications, however, it can be desirable to integrate against more general functions. The inductive formula (1.2.1) for the local height pairing in the complex case, is such an example, as is the interpretation of Mahler measures of polynomials as (the archimedean component of) heights. However, its analogue (Equation 1.3.1) a priori holds only when $\log\left\|{s_{0}}\right\|^{-1}$ is continuous, that is when the section $s_{0}$ has no zeroes nor poles.

The fact that it still holds in the archimedean case is a theorem of Maillot [43] building on the theory of Bedford–Taylor. We proved in [19, Th. 4.1] that this relation holds in the ultrametric case too. The proof (valid both in the ultrametric and archimedean cases) works by induction, and ultimately relies on an approximation lemma according to which any semi-positive Green function $g$ for a divisor $D$ is an increasing limit of smooth functions $(g_{n})$ such that, for any $n$, $g-g_{n}$ is a semi-positive Green function for $D$. In fact, it suffices to pose $g_{n}=\min(g,n\log\left|{\pi}\right|^{-1})$ ; then, $g-g_{n}=\max(0,g-n\log\left|{\pi}\right|^{-1})$ is the maximum of two semi-positive Green functions, hence is semi-positive. (In the archimedean case, one needs to further regularize $g_{n}$ ; see [19] for details.)

The symmetry of the local height pairing then implies the following analogue of the Poincaré–Lelong formula. When $\overline{L}$ is the trivial line bundle, with the metric defined by an admissible function $\varphi$, the factor $c_{1}(\overline{L})$ will be written $\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi$, by analogy to the complex case.