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.