Let $X$ be a topological space together with a sheaf of local rings $\mathscr{O}_{X}$ (“analytic functions”) ; let also $\mathscr{C}_{X}$ be the sheaf of continuous functions on $X$. In analytic geometry, local functions have an absolute value which is a real valued continuous function, satisfying the triangle inequality. Let us thus assume that we have a morphism of sheaves $\mathscr{O}_{X}\rightarrow\mathscr{C}_{X}$, written $f\mapsto\left|{f}\right|$, such that $\left|{fg}\right|=\left|{f}\right|\left|{g}\right|$, $\left|{1}\right|=1$, and $\left|{f+g}\right|\leq\left|{f}\right|+\left|{g}\right|$.

A line bundle on $(X,\mathscr{O}_{X})$ is a sheaf $L$ of $\mathscr{O}_{X}$-modules which is locally isomorphic to $\mathscr{O}_{X}$. In other words, $X$ is covered by open sets $U$ such that $\mathscr{O}_{U}\simeq L|U$ ; such an isomorphism is equivalent to a non-vanishing section $\varepsilon_{U}\in\Gamma(U,L)$, also called a local frame of $L$.

If $s$ is a section of a line bundle $L$ on an open set $U$, the value of $s$ at a point $x\in U$ is only well-defined as an element of the stalk $L(x)$, which is a $\kappa(x)$-vector space of dimension $1$. (Here, $\kappa(x)$ is the residue field of $\mathscr{O}_{X}$ at $x$.) Prescribing a metric on $L$ amounts to assign, in a coherent way, the norms of these values. Formally, a metric on $L$ is the datum, for any open set $U\subset X$ and any section $s\in\Gamma(U,L)$, of a continuous function $\left\|{s}\right\|_{U}\colon U\rightarrow{\mathbf{R}}_{+}$, satisfying the following properties :

1. (1)

   for any open set $V\subset U$, $\left\|{s}\right\|_{V}$ is the restriction to $V$ of the function $\left\|{s}\right\|_{U}$ ;

2. (2)

   for any function $f\in\mathscr{O}_{X}(U)$, $\left\|{fs}\right\|=\left|{f}\right|\left\|{s}\right\|$ ;

3. (3)

   if $s$ is a local frame on $U$, then $\left\|{s}\right\|$ doesn’t vanish on $U$.

One usually writes $\overline{L}$ for the pair $(L,\left\|{\cdot}\right\|)$ of a line bundle $L$ and a metric on it.

Observe that the trivial line bundle $\mathscr{O}_{X}$ has a natural “trivial” metric, for which $\left\|{1}\right\|=1$. In fact, a metric on the trivial line bundle $\mathscr{O}_{X}$ is equivalent to the datum of a continuous function $h$ on $X$, such that $\left\|{1}\right\|=e^{-h}$.

Isomorphism of metrized line bundles are isomorphisms of line bundles which respect the metrics ; they are called isometries. Constructions from tensor algebra extend naturally to the framework of metrized line bundles, compatibly with isometries. The tensor product of two metrized line bundles $\overline{L}$ and $\overline{M}$ has a natural metrization such that $\left\|{s\otimes t}\right\|=\left\|{s}\right\|\left\|{t}\right\|$, if $s$ and $t$ are local sections of $L$ and $M$ respectively. Similarly, the dual of a metrized line bundle has a metrization, and the obvious isomorphism $L\otimes L^{\vee}\simeq\mathscr{O}_{X}$ is an isometry. Consequently, isomorphism classes of metrized line bundles on $X$ form an Abelian group $\overline{\operatorname{Pic}}(X)$. This group fits in an exact sequence

$$0\rightarrow\mathscr{C}(X)\rightarrow\overline{\operatorname{Pic}}(X)\rightarrow\operatorname{Pic}(X)\rightarrow 0,$$

where the first map associates to a real continuous function $h$ on $X$ the trivial line bundle endowed with the metric such that $\left\|{1}\right\|=e^{-h}$, and the second associates to a metrized line bundle the underlying line bundle. It is surjective when any line bundle has a metric (this certainly holds if $X$ has partitions of unity).

Similarly, we can consider pull-backs of metrized line bundle. Let $\varphi\colon Y\rightarrow X$ be a morphism of locally ringed spaces such that $\left|{\varphi^{*}f}\right|=\left|{f}\right|\circ\varphi$ for any $f\in\mathscr{O}_{X}$. Let $\overline{L}$ be a metrized line bundle on $X$. Then, there is a canonical metric on $\varphi^{*}L$ such that $\left\|{\varphi^{*}s}\right\|=\left\|{s}\right\|\circ\varphi$ for any section $s\in\Gamma(U,L)$. This induces a morphism of Abelian groups $\varphi^{*}\colon\overline{\operatorname{Pic}}(X)\rightarrow\overline{\operatorname{Pic}}(Y)$.

In complex analytic geometry, metrics are a very well established tool. Let us first consider the case of the projective space $\mathrm{X}={\mathbf{P}}^{n}({\mathbf{C}})$ ; a point $x\in X$ is a $(n+1)$-tuple of homogeneous coordinates $[x_{0}:\dots:x_{n}]$, not all zero, and up to a scalar. Let $\pi\colon{\mathbf{C}}^{n+1}_{*}\rightarrow X$ be the canonical projection map, where the index $*$ means that we remove the origin $(0,\dots,0)$. The fibers of $\pi$ have a natural action of ${\mathbf{C}}^{*}$. The tautological line bundle $\mathscr{O}(1)$ has for sections $s$ over an open set $\mathrm{U}\subset{\mathbf{P}}^{n}({\mathbf{C}})$ the analytic functions $F_{s}$ on the open set $\pi^{-1}(\mathrm{U})\subset{\mathbf{C}}^{n+1}_{*}$ which are homogeneous of degree $1$. The Fubini-Study metric of $\mathscr{O}(1)$ assigns to the section $s$ the norm $\left\|{s}\right\|_{\mathrm{FS}}$ defined by

$$\left\|{s}\right\|_{{\mathrm{FS}}}([x_{0}:\dots:x_{n}])=\frac{\left|{F_{s}(x_{0},\dots,x_{n})}\right|}{\left(\left|{x_{0}}\right|^{2}+\dots+\left|{x_{n}}\right|^{2}\right)^{1/2}}.$$

It is more than continuous ; indeed, if $s$ is a local frame on an open set $\mathrm{U}$, then $\left\|{s}\right\|$ is a $\mathscr{C}^{\infty}$-function on $\mathrm{U}$ ; such metrics are called smooth.

Line bundles with smooth metrics on smooth complex analytic spaces allow to perform differential calculus. Namely, the curvature of a smooth metrized line bundle $\overline{L}$ is a differential form $c_{1}(\overline{L})$ of type $(1,1)$ on $\mathrm{X}$. Its definition involves the differential operator

$$\mathop{\mathrm{d}\mathrm{d}^{c}}=\frac{i}{\pi}\partial\overline{\partial}.$$

When an open set $\mathrm{U}\subset\mathrm{X}$ admits local coordinates $(z_{1},\dots,z_{n})$, and $s\in\Gamma(\mathrm{U},L)$ is a local frame, then

$$c_{1}(\overline{L})|_{\mathrm{U}}=\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s}\right\|^{-1}=\frac{i}{\pi}\sum_{1\leq j,k\leq n}\frac{\partial^{2}}{\partial z_{j}\partial\overline{z}_{k}}\log\left\|{s}\right\|^{-1}\mathrm{d}z_{j}\wedge\mathrm{d}\overline{z}_{k}.$$

Cauchy-Riemann equations ($\partial f/\partial\overline{z}=0$ for any holomorphic function $f$ of the variable $z$) imply that this formula does not depend on the choice of a local frame $s$. Consequently, these differential forms defined locally glue to a well-defined global differential form on $\mathrm{X}$.

Taking the curvature form of a metrized line bundle is a linear operation : $c_{1}(\overline{L}\otimes\overline{M})=c_{1}(\overline{L})+c_{1}(\overline{M})$. It also commutes to pull-back : if $f\colon Y\rightarrow X$ is a morphism, then $f^{*}c_{1}(\overline{L})=c_{1}(f^{*}\overline{L})$.

In the case of the Fubini-Study metric over the projective space ${\mathbf{P}}^{n}({\mathbf{C}})$, the curvature is computed as follows. The open subset $\mathrm{U}_{0}$ where the homogeneous coordinate $x_{0}$ is non-zero has local coordinates $z_{1}=x_{1}/x_{0}$, …, $z_{n}=x_{n}/x_{0}$ ; the homogeneous polynomial $X_{0}$ defines a non-vanishing section $s_{0}$ of $\mathscr{O}(1)$ on $\mathrm{U}_{0}$ and

$$\log\left\|{s_{0}}\right\|^{-1}_{\mathrm{FS}}=\frac{1}{2}\log\left(1+\sum_{j=1}^{n}\left|{z_{j}}\right|^{2}\right).$$

Consequently, over $\mathrm{U}_{0}$,

	$\displaystyle c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})$	$\displaystyle=\frac{i}{\pi}\partial\overline{\partial}\log\left\|{s_{0}}\right\|^{-1}_{\mathrm{FS}}$
		$\displaystyle=\frac{i}{2\pi}\partial\left(\sum_{k=1}^{n}\frac{z_{k}}{1+\left\|{z}\right\|^{2}}\mathrm{d}\overline{z}_{k}\right)$
		$\displaystyle=\frac{i}{2\pi}\sum_{j=1}^{n}\frac{1}{1+\left\|{z}\right\|^{2}}\mathrm{d}z_{j}\wedge\mathrm{d}\overline{z}_{j}-\frac{i}{2\pi}\sum_{j,k=1}^{n}\frac{z_{k}\overline{z}_{j}}{(1+\left\|{z}\right\|^{2})^{2}}\mathrm{d}z_{j}\wedge\mathrm{d}\overline{z}_{k}.$

In this calculation, we have abbreviated $\left\|{z}\right\|^{2}=\sum_{j=1}^{n}\left|{z_{j}}\right|^{2}$.

Taking the product of $n$ factors equal to this differential form, we get a differential form of type $(n,n)$ on the $n$-dimensional complex space $\mathrm{X}$. Such a form can be integrated on $\mathrm{X}$ and the Wirtinger formula asserts that

$$\int_{\mathrm{X}}c_{1}(\overline{L})^{n}=\deg(L)$$

is the degree of $L$ as computed by intersection theory. As an example, if $\mathrm{X}={\mathbf{P}}^{1}({\mathbf{C}})$, we have seen that

$$c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})=\frac{i}{2\pi(1+\left|{z}\right|^{2})^{2}}\mathrm{d}z\wedge d\overline{z},$$

where $z=x_{1}/x_{0}$ is the affine coordinate of $X\setminus\{\infty\}$. Passing in polar coordinates $z=re^{i\theta}$, we get

$$c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})=\frac{1}{2\pi(1+r^{2})^{2}}\mathrm{d}r\wedge\mathrm{d}\theta$$

whose integral over ${\mathbf{C}}$ equals

$$\int_{{\mathbf{P}}^{1}({\mathbf{C}})}c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})=\int_{0}^{\infty}\frac{1}{2\pi(1+r^{2})^{2}}2r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta=\int_{0}^{\infty}\frac{1}{(1+u)^{2}}\mathrm{d}u=1.$$

An important formula is the Poincaré–Lelong equation. For any line bundle $L$ with a smooth metric, and any section $s\in\Gamma(\mathrm{X},L)$ which does not vanish identically on any connected component of $\mathrm{X}$, it asserts the following equality of currents¹¹ 1 The space of currents is the dual to the space of differential forms, with the associated grading; in the orientable case, currents can also be seen as differential forms with distribution coefficients. :

$$\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s}\right\|^{-1}+\delta_{\operatorname{div}(s)}=c_{1}(\overline{L}),$$

where $\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s}\right\|^{-1}$ is the image of $\log\left\|{s}\right\|^{-1}$ under the differential operator $\mathop{\mathrm{d}\mathrm{d}^{c}}$, taken in the sense of distributions, and $\delta_{\operatorname{div}(s)}$ is the current of integration on the cycle $\operatorname{div}(s)$ of codimension $1$, $\operatorname{div}(s)$.

Metrized line bundles and their associated curvature forms are a basic tool in Arakelov geometry, invented by Arakelov in [2] and developped by Faltings [29], Deligne [23] for curves, and by Gillet-Soulé [32] in any dimension. For our concerns, they allow for a definition of height functions for algebraic cycles on algebraic varieties defined over number fields. As explained by Gubler [33, 34], they also permit to develop a theory of archimedean local heights.

For simplicity, let us assume that $\mathrm{X}$ is proper, smooth, and that all of its connected components have dimension $n$.

Let $\overline{L}_{0},\dots,\overline{L}_{n}$ be metrized line bundles with smooth metrics. For $j\in\{0,\dots,n\}$, let $s_{j}$ be a regular meromorphic section of $L_{j}$ and let $\operatorname{div}(s_{j})$ be its divisor. The given metric of $L_{j}$ furnishes moreover a function $\log\left\|{s_{j}}\right\|^{-1}$ on $X$ and a $(1,1)$-form $c_{1}(\overline{L}_{j})$, related by the Poincaré–Lelong equation $\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s_{j}}\right\|^{-1}+\delta_{\operatorname{div}(s_{j})}=c_{1}(\overline{L}_{j})$. In the terminology of Arakelov geometry, $\log\left\|{s_{j}}\right\|^{-1}$ is a Green current (here, function) for the cycle $\operatorname{div}(s_{j})$ ; we shall write $\mathop{\widehat{\operatorname{div}}}(s_{j})$ for the pair $(\operatorname{div}(s_{j}),\log\left\|{s_{j}}\right\|^{-1})$.

Let $\mathrm{Z}\subset\mathrm{X}$ be a $k$-dimensional subvariety such that the divisors $\operatorname{div}(s_{j})$, for $0\leq j\leq k$, have no common point on $\mathrm{Z}$. Then, one defines inductively the local height pairing by the formula :

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

(1.2.1)

The second hand of this formula requires two comments. 1) The divisor $\operatorname{div}(s_{k}|_{\mathrm{Z}})$ is a formal linear combination of $(k-1)$-dimensional subvarieties of $\mathrm{X}$, and its local height pairing is computed by linearity from the local height pairings of its components. 2) The integral of the right hand side involves a function with singularities ($\log\left\|{s_{k}}\right\|^{-1}$) to be integrated against a distribution : in this case, this means restricting the differential form $c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})$ to the smooth part of $\mathrm{Z}$, multiplying by $\log\left\|{s_{k}}\right\|^{-1}$, and integrating the result. The basic theory of closed positive currents proves that the resulting integral converges absolutely ; as in [32], one can also resort to Hironaka’s resolution of singularities.

It is then a non-trivial result that the local height pairing is symmetric in the involved $\mathop{\widehat{\operatorname{div}}}$isors ; it is also multilinear. See [35] for more details, as well as [32] for the global case.

Consideration of the curvature allows to define positivity notions for metrized line bundles. Namely, one says that a smooth metrized line bundle $\overline{L}$ is positive (resp. semi-positive) if its curvature form is a positive (resp. a non-negative) $(1,1)$-form. This means that for any point $x\in\mathrm{X}$, the hermitian form $c_{1}(\overline{L})_{x}$ on the complex tangent space $\mathrm{T}_{x}\mathrm{X}$ is positive definite (resp. non-negative). As a crucial example, the line bundle $\mathscr{O}(1)$ with its Fubini-Study metric is positive. The pull-back of a positive metrized line bundle by an immersion is positive. In particular, ample line bundles can be endowed with a positive smooth metric ; Kodaira’s embedding theorem asserts the converse : if a line bundle possesses a positive smooth metric, then it is ample.

The pull-back of a semi-positive metrized line bundle by any morphism is still semi-positive. If $\overline{L}$ is semi-positive, then the measure $c_{1}(\overline{L})^{n}$ is a positive measure.

More generally, both the curvature and the Poincaré–Lelong equation make sense for metrized line bundles with arbitrary (continuous) metrics, except that $c_{1}(\overline{L})$ has to be considered as a current. The notion a semi-positivity can even be extended to this more general case, because it can be tested by duality : a current is positive if its evaluation on any nonnegative differential form is nonnegative. Alternatively, semi-positive (continuous) metrized line bundles are characterized by the fact that for any local frame $s$ of $\overline{L}$ over an open set $\mathrm{U}$, the continuous function $\log\left\|{s}\right\|^{-1}$ is plurisubharmonic on $\mathrm{U}$. In turn, this means that for any morphism $\varphi\colon\overline{D}\rightarrow\mathrm{U}$, where $\overline{D}=\overline{D}(0,1)$ is the closed unit disk in ${\mathbf{C}}$,

$$\log\left\|{s}\right\|^{-1}(\varphi(0))\leq\frac{1}{2\pi}\int_{0}^{2\pi}\log\left\|{s}\right\|^{-1}(\varphi(e^{i\theta}))\mathrm{d}\theta.$$

Assume that $\overline{L}$ is semi-positive. Although products of currents are not defined in general (not more than products of distributions), the theory of Bedford–Taylor [8, 7] and Demailly [24, 25] defines a current $c_{1}(\overline{L})^{n}$ which then is a positive measure on $\mathrm{X}$. There are two ways to define this current. The first one works locally and proceeds by induction : if $u=\log\left\|{s}\right\|^{-1}$, for a local non-vanishing section $s$ of $L$, one defines a sequence $(T_{k})$ of closed positive currents by the formulae $T_{0}=1$, $T_{1}=\mathop{\mathrm{d}\mathrm{d}^{c}}u$,…, $T_{k+1}=\mathop{\mathrm{d}\mathrm{d}^{c}}(uT_{k})$ and $c_{1}(\overline{L})^{n}=\mathop{\mathrm{d}\mathrm{d}^{c}}(u)^{n}$ is defined to be $T_{n}$. What makes this construction work is the fact that at each step, $uT_{k}$ is a well defined current (product of a continuous function and of a positive current), and one has to prove that $T_{k+1}$ is again a closed positive current. The other way, which shall be the one akin to a generalization in the ultrametric framework, consists in observing that if $L$ is a line bundle with a continuous semi-positive metric $\left\|{\cdot}\right\|$, then there exists a sequence of smooth semi-positive metrics $\left\|{\cdot}\right\|_{k}$ on the line bundle $L$ which converges uniformly to the initial metric : for any local section $s$, $\left\|{s}\right\|_{k}$ converges uniformly to $\left\|{s}\right\|$ on compact sets. The curvature current $c_{1}(\overline{L})$ is then the limit of the positive currents $c_{1}(\overline{L}_{k})$, and the measure $c_{1}(\overline{L})^{n}$ is the limit of the measures $c_{1}(\overline{L}_{k})^{n}$. (We refer to [43] for the global statement ; to construct the currents, one can in fact work locally in which case a simple convolution argument establishes the claim.)

An important example of semi-positive metric which is continuous, but not smoth, is furnished by the Weil metric on the line bundle $\mathscr{O}(1)$ on ${\mathbf{P}}^{n}({\mathbf{C}})$. This metric is defined as follows : if $\mathrm{U}\subset{\mathbf{P}}^{n}({\mathbf{C}})$ is an open set, and $s$ is a section of $\mathscr{O}(1)$ on $U$ corresponding to an analytic function $F_{s}$ on $\pi^{-1}(\mathrm{U})\subset{\mathbf{C}}^{n+1}_{*}$ which is homogeneous of degree $1$, then for any $(x_{0},\dots,x_{n})\in\pi^{-1}(\mathrm{U})$, one has

$$\left\|{s}\right\|_{\mathrm{W}}=\frac{\left|{F_{s}(x_{0},\dots,x_{n})}\right|}{\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|}.$$

The associated measure $c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{W}})^{n}$ on ${\mathbf{P}}^{n}({\mathbf{C}})$ is as follows, cf. [58, 43] : the subset of all points $[x_{0}:\dots:x_{n}]\in{\mathbf{P}}^{n}({\mathbf{C}})$ such that $\left|{x_{j}}\right|=\left|{x_{k}}\right|$ for all $j,k$ is naturally identified with the polycircle $\mathbf{S}_{1}^{n}$ (map $[x_{0}:\dots:x_{n}]$ to $(x_{1}/x_{0},\dots,x_{n}/x_{0})$) ; take the normalized Haar measure of this compact group and push it onto ${\mathbf{P}}^{n}({\mathbf{C}})$.

Let us say that a continuous metrized line bundle is admissible if it can be written as $\overline{L}\otimes\overline{M}^{\vee}$, where $\overline{L}$ and $\overline{M}$ are metrized line bundles whose metrics are continuous and semi-positive. Admissible metrized line bundles form a subgroup $\overline{\operatorname{Pic}}_{\text{ad}}(\mathrm{X})$ of $\overline{\operatorname{Pic}}(\mathrm{X})$ which maps surjectively onto $\operatorname{Pic}(\mathrm{X})$ if $\mathrm{X}$ is projective.

The curvature current $c_{1}(\overline{L})$ of an admissible metrized line bundle $\overline{L}$ is a differential form of type $(1,1)$ whose coefficients are signed measures. Its $n$th product $c_{1}(\overline{L})^{n}$ is well-defined as a signed measure on $\mathrm{X}$.

The good analytic properties of semi-positive metrics allow to extend the definition of the local height pairing to the case of admissible line bundles. Indeed, when one approximates uniformly a semi-positive line bundle by a sequence of smooth semi-positive line bundles, one can prove that the corresponding sequence of local height pairings converges, the limit being independent on the chosen approximation.

The proof is inspired by Zhang’s proof of the global case in [59] and goes by induction. Let us consider, for each $j$, two smooth semi-positive metrics on the line bundle $L_{j}$ and assume that they differ by a factor $e^{-h_{j}}$. Then, the corresponding local height pairings differ from an expression of the form

$$\sum_{j=0}^{k}\int_{\mathrm{Z}}h_{j}c_{1}(\overline{L}_{0})\dots\widehat{c_{1}(\overline{L}_{j})}\dots c_{1}(\overline{L}_{k}),$$

where the written curvature forms are associated to the first metric for indices $<j$, and to the second for indices $>j$. This differential forms are positive by assumption, so that the integral is bounded in absolute value by

$$\sum_{j=0}^{k}\left\|{h_{j}}\right\|_{\infty}\int_{\mathrm{Z}}c_{1}(\overline{L}_{0})\dots\widehat{c_{1}(\overline{L}_{j})}\dots c_{1}(\overline{L}_{k})=\sum_{j=0}^{K}\left\|{h_{j}}\right\|_{\infty}(c_{1}(L_{0})\dots\widehat{c_{1}(L_{j}))}\dots c_{1}(L_{k})|{\mathrm{Z}}),$$

where the last expression is essentially a degree. (In these formulae, the factor with a hat is removed.) This inequality means that on the restriction to the space of smooth semi-positive metrics, with the topology of uniform convergence, the local height pairing is uniformly continuous. Therefore, it first extends by continuity. on the space of continuous semi-positive metrics, and then by multilinearity to the space of admissible metrics.

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 $\mathfrak{L}_{0},\dots,\mathfrak{L}_{k-1}$.