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)