By Corollary 4.6, the model $\mathscr{X}$ is snc along every one-dimensional stratum $C$ of $\mathscr{X}$. By means of a finite sequence of blow-ups at zero-dimensional strata, we can moreover arrange that, for every prime component $E$ of $\mathscr{X}_{k}$ that contains $C$, the intersection number $(C\cdot E)$ is negative. This may destroy the property that $\mathscr{X}_{k}$ is reduced, but it preserves the properties that $\mathscr{X}$ is snc along every one-dimensional stratum, $\mathscr{X}$ is a good minimal dlt-model, and $\mathscr{X}$ satisfies assumption (2). Moreover, the sequence of blow-ups has no effect on the map $\rho_{\mathscr{X}}$, by [MN15, 3.1.7]; the effect on the skeleton $\Delta(\mathscr{X}^{\mathrm{snc}})$ is a sequence of star subdivisions of the faces corresponding to the zero-dimensional strata [MN15, 3.1.9].

Thus it suffices to prove the theorem under the following alternative assumptions on the model $\mathscr{X}$:

• •

  $\mathscr{X}$ is a good minimal dlt-model satisfying (2);

• •

  for every one-dimensional stratum $C$ of $\mathscr{X}_{k}$, the model $\mathscr{X}$ is snc along $C$;

• •

  for every one-dimensional stratum $C$ of $\mathscr{X}_{k}$ and every prime component $E$ of $\mathscr{X}_{k}$ that contains $C$, the component $E$ has multiplicity one in $\mathscr{X}_{k}$, and the intersection number $(C\cdot E)$ is negative.

Let $Z$ be the union of the faces of codimension $\geq 2$ in $\Delta(\mathscr{X}^{\mathrm{snc}})$. We will prove that $\rho_{\mathscr{X}}$ is an $n$-dimensional affinoid torus fibration over $\mathrm{Sk}(X)\setminus Z$.

Let $C$ be a one-dimensional stratum of $\mathscr{X}_{k}$. By adjunction, the model $\mathscr{X}$ is log Calabi-Yau along $C$ in the sense of (5). Thus $\mathscr{X}$ is toric along $C$, by Proposition 5.4. More precisely, The proof of Proposition 5.4 gives an explicit description of the formal completion $\widehat{\mathscr{X}_{/C}}$ of $\mathscr{X}$ along $C$. Note that, under our assumptions and with the notations in that proof, the number $\iota$ is equal to one and $N_{j}=1$ for every $j\in J$, so that we can make the construction of the fan $\Sigma$ more explicit: we choose a bijection of $J$ with $\{1,\ldots,n\}$. Then we can take for $(u_{0},\ldots,u_{n-1})$ the standard basis of $\mathbb{Z}^{n}$, and set $u_{n}=0$. The vector $v_{\infty}$ is now given by $(-1,b_{1},\ldots,b_{n-1},N_{\infty})$. Let $\Sigma$ be the fan with maximal cones $\sigma_{0}$ and $\sigma_{\infty}$. Then the toric scheme $\mathscr{Y}$ constructed in the proof of Proposition 5.4 is precisely the torus embedding associated with $\Sigma$ in the sense of Example 3.5.

Let $U$ be the union in $\Delta(\mathscr{X}^{\mathrm{snc}})$ of the open faces corresponding to the strata $c_{0}$, $c_{\infty}$ and $C$ in $\mathscr{X}_{k}$. This is an open subset of $\mathrm{Sk}(X)$ and, as $C$ varies, these open sets cover $\mathrm{Sk}(X)\setminus Z$. Thus it suffices to show that $\rho_{\mathscr{X}}$ is an $n$-dimensional affinoid torus fibration over $U$, and that the induced integral affine structure on $U$ is compatible with the piecewise integral affine structure on $\Delta(\mathscr{X}^{\mathrm{snc}})$.

Set $T=\mathbb{G}^{n}_{m,K}$ and let $V$ be the interior of the intersection of $|\Sigma|$ with $\mathbb{R}^{n}\times\{1\}$. It follows directly from the construction of $\rho_{\mathscr{X}}$ that $\rho_{\mathscr{X}}^{-1}(U)$ is the generic fiber of $\widehat{\mathscr{X}_{/C}}$, and that the restriction of $\rho_{\mathscr{X}}$ over $U$ only depends on the formal $R$-scheme $\widehat{\mathscr{X}_{/C}}$. If $D$ is the torus orbit in $\mathscr{Y}_{k}$ corresponding to the codimension one cone $\sigma_{0}\cap\sigma_{\infty}$ in $\Sigma$, then we have shown in the proof of Proposition 5.4 that $\widehat{\mathscr{X}_{/C}}$ is isomorphic to $\widehat{\mathscr{Y}_{/D}}$. Thus, by Example 3.5, we can identify the restriction of $\rho_{\mathscr{X}}$ over $U$ with the restriction of $\rho_{T}$ over $V$, which is an $n$-dimensional affinoid torus fibration by definition.

It remains to show that the induced integral affine structure on $V$ is compatible with the piecewise integral affine structure on $U$. We will check this on the open face $\tau_{0}$ corresponding to $\sigma_{0}$; the result for $\sigma_{\infty}$ then follows by switching the roles of $c_{0}$ and $c_{\infty}$. We have labelled the rays of $\sigma_{0}$ by $0,\ldots,n$; this induces a labelling of the vertices of $\tau_{0}$ and thus defines a system of barycentric coordinates $(w_{0},\ldots,w_{n})$ on the $n$-simplex $\tau_{0}$. By definition [MN15, 3.2.1], a real-valued function on a connected open subset of $\tau_{0}$ is integral affine if we can write it as a degree one polynomial with $\mathbb{Z}$-coefficients in the variables $(w_{0}/N_{0},w_{1},\ldots,w_{n})$. This coincides with the notion of an integral affine function on the $n$-simplex $\sigma_{0}\cap(\mathbb{R}^{n}\times\{1\})$, which is the convex hull of the points

This concludes the proof. ∎