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.