Let $L$ be a line bundle on $\mathrm{X}$. Let $0$ be the neutral element of $\mathrm{X}$ and let us fix a trivialization $L_{0}$ of $L$ at $0$.

The line bundle $L^{\otimes 2}$ is canonically decomposed as the tensor product of an even and an odd line bundle :

$$L^{\otimes 2}=(L\otimes[-1]^{*}L)\otimes(L\otimes[-1]^{*}L^{-1}).$$

By the theorem of the cube, an even line bundle $L$ satisfies $[m]^{*}L\simeq L^{\otimes m^{2}}$, while for an odd line bundle $L$, one has $[m]^{*}L\simeq L^{\otimes m}$ ; moreover, there are in each case a unique isomorphism compatible with the trivialization at the origin. By Lemma 2.4.1, an even (resp. an odd) line bundle possesses a canonical continuous metric making this isomorphism an isometry. This furnishes a canonical metric on $L^{\otimes 2}$, hence on $L$. According to this lemma, this metric is semi-positive if $L$ is ample and even. Using a Lemma of Künnemann, ([17], Lemme 2.3), one proves that this also holds if $L$ is algebraically equivalent to $0$. In any case, the canonical metrics are admissible.

When the variety $\mathrm{X}$ has good reduction, the canonical metrics and the associated measures are fairly easy to describe. Indeed, let $\mathfrak{X}$ be the Néron model of $\mathrm{X}$ over $K^{\circ}$, an Abelian scheme. For any line bundle $L$ on $\mathrm{X}$ there is a unique line bundle $\mathfrak{L}$ on $\mathfrak{X}$ which extends $L$ and which admits a trivialization at the $0$ section extending the given one over $K$. By the theorem of the cube for the Abelian scheme $\mathfrak{X}$, the isomorphism $[m]^{*}L\simeq L^{\otimes m^{a}}$ (with $a=1$ or $2$, according to whether $L$ is odd or even) extends uniquely to an isomorphism $[m]^{*}\mathfrak{L}\simeq\mathfrak{L}^{\otimes m^{a}}$. This implies that the canonical metrics are algebraic, induced by these models.

The description of the canonical measures on $\mathfrak{X}$ follows at once. Let $\xi$ be the point of $\mathrm{X}$ whose reduction is the generic point of the special fiber of $\mathfrak{X}$. Then, for any family $(L_{1},\dots,L_{n})$ of line bundles on $\mathrm{X}$, one has

$$c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{n})=\deg(c_{1}(L_{1})\dots c_{1}(L_{n}))\delta_{\xi}.$$

We see in particular that they only depend on the classes of the line bundles $L_{j}$ modulo numerical equivalence.

In the case of bad reduction, the description of the canonical measures has been established by W. Gubler [39].

Up to replacing $K$ by a finite extension, we assume that $\mathrm{X}$ has split semi-stable reduction. Raynaud’s uniformization involves an analytic group $\mathrm{E}$ which is an extension of an abelian variety with good reduction $\mathrm{Y}$ by a split torus $\mathrm{T}\simeq{\mathbf{G}_{\mathrm{m}}}^{t}$, where $t\in\{1,\dots,n\}$ — the so-called Raynaud extension of $\mathrm{X}$. One has $t\geq 1$ since we assume bad reduction ; moreover, $\operatorname{dim}\mathrm{Y}=\operatorname{dim}\mathrm{E}-t=n-t$. There is a morphism $p\colon\mathrm{E}\rightarrow\mathrm{X}$, whose kernel is a discrete subgroup $M$ of $\mathrm{E}(K)$, so that the induced map $\mathrm{E}/\Lambda\rightarrow\mathrm{X}$ is an isomorphism. When $t=n$, one says that $\mathrm{X}$ has totally degenerate reduction, and the morphism $p$ is the rigid analytic uniformization of the abelian variety $\mathrm{X}$.

Moreover, $\mathrm{E}$ is constructed as a contracted product $(\mathrm{E}_{1}\times\mathrm{T})/\mathrm{T}_{1}$ from an extension $\mathrm{E}_{1}$ of $\mathrm{Y}$ by the “unit subtorus” $\mathrm{T}_{1}$ of $\mathrm{T}$ (defined by the equalities $\left|{T_{j}(x)}\right|=1$ for $j\in\{1,\dots,t\}$ and $x\in\mathrm{T}$). The natural map $\lambda_{\mathrm{T}}\colon\mathrm{T}\rightarrow{\mathbf{R}}^{t}$ defined by

$$x\mapsto(-\log\left|{T_{1}(x)}\right|,\dots,-\log\left|{T_{t}(x)}\right|)$$

is continuous and surjective ; it admits a canonical section $\iota_{\mathrm{T}}$ which maps a point $(u_{1},\dots,u_{t})\in{\mathbf{R}}^{t}$ to the semi-norm

$$f\mapsto\sup_{\mathbf{m}\in{\mathbf{Z}}^{t}}a_{\mathbf{m}}e^{-m_{1}u_{1}-\dots-m_{t}u_{t}},\qquad\text{for $f=\sum_{\mathbf{m}}a_{\mathbf{m}}T_{1}^{m_{1}}\dots T_{t}^{m_{t}}\in\mathscr{O}(\mathrm{T}).$}$$

The map $\lambda_{\mathrm{T}}$ extends uniquely to a morphism $\lambda\colon\mathrm{E}\rightarrow{\mathbf{R}}^{t}$ whose kernel contains $\mathrm{E}_{1}$. The image $\Lambda=\lambda(M)$ is a lattice of ${\mathbf{R}}^{t}$, and the morphism $p$ induces a continuous proper morphism $\rho\colon\mathrm{X}\rightarrow{\mathbf{R}}^{t}/\Lambda$. Composing the section $\iota_{\mathrm{T}}$ with the projection $p$ furnishes a section $\iota\colon{\mathbf{R}}^{t}/\Lambda\rightarrow\mathrm{X}$ of $\rho$. Its image is the skeleton of $\mathrm{X}$. Gubler’s theorem ([39], Cor. 7.3) is the following :