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 :