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.