Subsection [04XG]

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(3.4) If $f\colon Y\to B$ is an $n$ -dimensional affinoid torus fibration, then $f$ induces an integral affine structure on the base $B$ [KS06, §4.1]. For every open $U$ in $B$ as in the definition, and every invertible analytic function $h$ on $f^{-1}(U)$ , the absolute value of $h$ is constant on the fibers of $f$ by the maximum modulus principle. Thus $h$ induces a continuous function $|h|\colon U\to\mathbb{R}_{>0}$ . The integral affine functions on $U$ are, by definition, the functions of the form $-\ln|h|$ . If $U$ is connected, then it is proven in Theorem 1 of [KS06, §4.1] that under the homeomorphism $U\to V$ , the ring of integral affine functions on $U$ is identified with the ring of polynomial functions of degree one with $\mathbb{Z}$ -coefficients on $V\subset N_{\mathbb{R}}$ , so that this construction indeed defines an integral affine structure on $B$ (to be precise, in [KS06] the authors consider affine functions with constant term in $\mathbb{R}$ , rather than $\mathbb{Z}$ , but since $K$ is discretely valued in our case, we get a slightly stronger result).

Example 3.5.

We use the tropicalization map to identify the canonical skeleton $\Delta(T)$ with $N_{\mathbb{R}}$ . We denote by $C$ the open cone $(N_{\mathbb{R}}\times\mathbb{R}_{>0})\cup\{0\}$ in $N_{\mathbb{R}}\oplus\mathbb{R}$ . Let $\Sigma$ be a locally finite fan of strongly convex rational polyhedral cones in $C$ . We denote by $\Sigma_{1}$ the rational polyhedral complex in $N_{\mathbb{R}}$ obtained by intersecting the cones in $\Sigma$ with $N_{\mathbb{R}}\times\{1\}$ . Consider the torus embedding $T\to\mathscr{X}$ over $R$ associated with $\Sigma$ as in [Kü98, 1.13]. The $R$ -scheme $\mathscr{X}$ is separated and locally of finite type, and it is quasi-compact if and only if $\Sigma$ is finite. Since $\Sigma$ is supported in $C$ , the generic fiber of $\mathscr{X}$ is canonically isomorphic to the split $K$ -torus $T$ . Assume that $\mathscr{X}$ is regular; this is equivalent to the property that the fan $\Sigma$ is simple, and it implies that the special fiber $\mathscr{X}_{k}$ is a strict normal crossings divisor. Denote by $\mathfrak{X}$ the formal $t$ -adic completion of $\mathscr{X}$ . The generic fiber $\mathfrak{X}_{\eta}$ is a $K$ -analytic space endowed with a natural injective morphism of $K$ -analytic spaces $i:\mathfrak{X}_{\eta}\to T^{\mathrm{an}}$ . The morphism $i$ embeds $\mathfrak{X}_{\eta}$ as an analytic domain in $T^{\mathrm{an}}$ .

The construction of the Berkovich skeleton $\mathrm{Sk}(\mathscr{X})$ and the retraction map $\rho_{\mathscr{X}}$ in [MN15, §3] are local on $\mathscr{X}$ , so that they extend immediately to schemes that are locally of finite type. This yields a canonical embedding of the dual intersection complex $\Delta(\mathscr{X})$ of $\mathscr{X}_{k}$ into $\mathfrak{X}_{\eta}$ . The image of this embedding is called the Berkovich skeleton of $\mathscr{X}$ . The embedding has a canonical retraction $\rho_{\mathscr{X}}:\mathfrak{X}_{\eta}\to\Delta(\mathscr{X})$ . It follows directly from the definitions that $\Delta(\mathscr{X})$ is contained in $\Delta(T)=N_{\mathbb{R}}$ and coincides with the support of $\Sigma_{1}$ . In particular, if $\Sigma$ is a subdivision of $C$ , then $\Delta(\mathscr{X})=\Delta(T)$ . Moreover, the $\Delta$ -structure on $\Delta(\mathscr{X})$ is precisely the polyhedral decomposition $\Sigma_{1}$ . We have $\mathfrak{X}_{\eta}=\rho_{T}^{-1}(|\Sigma_{1}|)$ , and the retraction map $\rho_{\mathscr{X}}$ is the restriction of $\rho_{T}$ to $\mathfrak{X}_{\eta}$ .