A closely related result is proven in [Th07, 3.26]. We will explain how our statement can be deduced from that result. Following the notation in [Th07], we denote by $\mathscr{X}^{\beth}$ the $k$-analytic space associated to the toroidal embedding $X\hookrightarrow\mathscr{X}$, where $k$ is endowed with the trivial absolute value. By definition, $\mathscr{X}^{\beth}$ is the generic fiber of the formal $t$-adic completion $\widehat{\mathscr{X}}$ of $\mathscr{X}$, viewed as a special formal $k$-scheme by forgetting the $k[\negthinspace[t]\negthinspace]$-structure [Be96, §1].

The relation between $\mathscr{X}^{\beth}$ and $X_{K}^{\mathrm{an}}$ is explained in detail at the beginning of Section 4 in [Ni11]; let us recall the main idea. Considering the morphism of special formal $k$-schemes $\widehat{\mathscr{X}}\to\mathrm{Spf}\,k[\negthinspace[t]\negthinspace]$ and passing to the generic fibers, we obtain a morphism of $k$-analytic spaces from $\mathscr{X}^{\beth}$ to the open unit disc $D$ over $k$. We can identify the underlying topological space of $D$ with $[0,1[$ by means of the homeomorphism

$$D\to[0,1[\,:x\mapsto|t(x)|.$$

The residue field of $D$ at the point $1/e$ in $[0,1[$ is $K$ with our chosen $t$-adic absolute value $|\cdot|_{K}$, and the $K$-analytic space $X_{K}^{\mathrm{an}}$ is canonically isomorphic to the fiber of $\mathscr{X}^{\beth}$ over $1/e$. Thus we can view $X_{K}^{\mathrm{an}}$ as the subspace of $\mathscr{X}^{\beth}$ consisting of the points $x$ such that $|t(x)|=1/e$.

In [Th07, 3.13], Thuillier constructs a retraction $p_{\mathscr{X}}$ of $\mathscr{X}^{\beth}$ onto a certain subspace $\mathcal{S}(\mathscr{X})$, the skeleton of the toroidal embedding. Moreover, in [Th07, 3.26], he shows that $p_{\mathscr{X}}$ can be extended to a strong deformation retraction $H$ of $\mathscr{X}^{\beth}$ onto $\mathcal{S}(\mathscr{X})$. Going through the definitions, one observes that $p_{\mathscr{X}}$ and $H$ commute with the morphism $\mathscr{X}^{\beth}\to D$ and that the restriction of

$$p_{\mathscr{X}}:\mathscr{X}^{\beth}\to\mathcal{S}(\mathscr{X})$$

over the point $1/e$ of $D$ is precisely the retraction

$$\rho_{\mathscr{X}}:X_{K}^{\mathrm{an}}\to\mathrm{Sk}(\mathscr{X}).$$

Thus by restricting $H$ over $1/e\in D$, we obtain a map that satisfies all the properties in the statement. ∎

By the definition of a $dlt$-model, we know that $\Delta\leq(\mathscr{Y}_{s})_{\mathrm{red}}$. Thus it suffices to show that these divisors are different locally at $\mathrm{red}_{\mathscr{Y}}(x)$. Since $x$ lies on $\mathrm{Sk}(\mathscr{Y})$, its reduction $\mathrm{red}_{\mathscr{Y}}(x)$ is a generic point of the intersection of the irreducible components of $\mathscr{Y}_{s}$ that contain $\mathrm{red}_{\mathscr{Y}}(x)$. Thus if we denote by $h^{\prime}:\mathscr{Y}^{\prime}\to\mathscr{Y}$ the blow-up of $\mathscr{Y}$ at the closure of $\mathrm{red}_{\mathscr{Y}}(x)$, then $\mathscr{Y}^{\prime}$ is again an $snc$-model of $X$.

We denote by $\Delta^{\prime}$ the log pullback of $\Delta$ to $\mathscr{Y}^{\prime}$. The image of the exceptional divisor $E$ of $h^{\prime}$ in $\mathscr{X}$ is the closure of $\mathrm{red}_{\mathscr{X}}(x)=h(\mathrm{red}_{\mathscr{Y}}(x))$ and thus disjoint from $\mathscr{X}^{\mathrm{snc}}$. By the definition of a $dlt$-model, we know that the multiplicity of $E$ in $\Delta^{\prime}$ is strictly smaller than $1$. Since the log pullback of $(\mathscr{Y}_{s})_{\mathrm{red}}$ to $\mathscr{Y}^{\prime}$ is equal to $(\mathscr{Y}^{\prime}_{s})_{\mathrm{red}}$, we see that $\Delta<(\mathscr{Y}_{s})_{\mathrm{red}}$ locally at $\mathrm{red}_{\mathscr{Y}}(x)$. ∎

Applying [MN13, 3.1.7] to the proper morphism $h^{-1}(\mathscr{X}^{\mathrm{snc}})\to\mathscr{X}^{\mathrm{snc}}$, we see that $\mathrm{Sk}(\mathscr{X})$ is contained in $\mathrm{Sk}(\mathscr{Y})$. Moreover, it follows from Lemma 3.2.3 that for every point $x$ of $S$, the reduction $\mathrm{red}_{\mathscr{X}}(x)$ must be contained in $\mathscr{X}^{\mathrm{snc}}$. Now let $x$ be any point in $\mathrm{Sk}(\mathscr{Y})$ such that $\mathrm{red}_{\mathscr{X}}(x)$ lies in $\mathscr{X}^{\mathrm{snc}}$. We must show that $v_{x}(\Delta)=v_{x}((\mathscr{Y}_{s})_{\mathrm{red}})$ if and only if $x$ lies in $\mathrm{Sk}(\mathscr{X})$, or, equivalently, $x$ is equal to its projection

$$x^{\prime}=\rho_{\mathscr{X}}(x)$$

to the skeleton of $\mathscr{X}$. Let $\omega$ be a local generator of $\omega_{\mathscr{X}^{\mathrm{snc}}/\mathscr{C}}$ at $\mathrm{red}_{\mathscr{X}}(x)$. It induces a rational section of the canonical bundle $\omega_{X_{K}/K}$ by base change. By [MN13, 4.4.5], we know that $x=x^{\prime}$ if and only if

$$\mathrm{wt}_{\omega}(x)=\mathrm{wt}_{\omega}(x^{\prime}).$$

Since the divisor of $\omega$ is zero in a neighbourhood of $\mathrm{red}_{\mathscr{X}}(x^{\prime})$, we have

$$\mathrm{wt}_{\omega}(x^{\prime})=v_{x^{\prime}}((\mathscr{X}_{s})_{\mathrm{red}})=v_{x}((\mathscr{X}_{s})_{\mathrm{red}}).$$

On the other hand, computing $\mathrm{wt}_{\omega}(x)$ on the model $\mathscr{Y}$ we get

$$\mathrm{wt}_{\omega}(x)=v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+(\mathscr{Y}_{s})_{\mathrm{red}})=v_{x}((\mathscr{X}_{s})_{\mathrm{red}})+v_{x}((\mathscr{Y}_{s})_{\mathrm{red}}-\Delta).$$

Thus we see that $\mathrm{Sk}(\mathscr{X})=S$. ∎

This follows immediately from Proposition 3.2.4. ∎

This follows from Theorem 2.2.6(2) and Corollary 3.2.5. ∎

For the definition of an elementary collapse in a simplicial topological space, we refer to Definition 18 in [dFKX12]. By Corollary 3.2.7, we can assume that the good minimal $dlt$-model $\mathscr{X}$ is the result of running MMP for $(\mathscr{Y},(\mathscr{Y}_{s})_{\mathrm{red}})$. Now the statement follows from Corollary 22 in [dFKX12]. When $k$ is not algebraically closed, see also §31 in [dFKX12]. ∎

Let $\mathscr{Y}\to\mathscr{X}$ be a log resolution of $(\mathscr{X},\mathscr{X}_{s})$. By Theorem 3.2.8, the skeleton $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $\mathrm{Sk}(\mathscr{Y})$. By Theorem 3.1.3, $\mathrm{Sk}(\mathscr{Y})$ is a strong deformation retract of $X_{K}^{\mathrm{an}}$. ∎

Let $x$ be a point of $\mathrm{Sk}(X_{K},\omega)$. If $\mathrm{red}_{\mathscr{X}}(x)$ is contained in $\mathscr{X}^{\mathrm{snc}}$, then $x$ lies in $\widehat{\mathscr{X}}_{\eta}$ and [MN13, 4.4.5] implies that $x$ must lie in $\mathrm{Sk}(\mathscr{X})$, since the restriction of $\mathrm{wt}_{\omega}$ to $\widehat{\mathscr{X}}_{\eta}$ can reach its minimal values only at points of $\mathrm{Sk}(\mathscr{X})$.

Now suppose that $\mathrm{red}_{\mathscr{X}}(x)$ is not contained in $\mathscr{X}^{\mathrm{snc}}$. We will deduce a contradiction with the assumption that $x$ belongs to $\mathrm{Sk}(X_{K},\omega)$. Let $E$ be an irreducible component of $\mathscr{X}^{\mathrm{snc}}_{s}$ whose closure contains $\mathrm{red}_{\mathscr{X}}(x)$, let $\xi$ be the generic point of $E$ and denote by $x^{\prime}$ the unique point in $\mathrm{red}_{\mathscr{X}}^{-1}(\xi)$. We will prove that $\mathrm{wt}_{\omega}(x^{\prime})<\mathrm{wt}_{\omega}(x)$. Then $x$ cannot belong to the locus $\mathrm{Sk}(X_{K},\omega)$ where $\mathrm{wt}_{\omega}$ reaches its minimal value. Note that, since $\mathscr{X}$ is $\mathbb{Q}$-factorial, we have

(3.3.3)

$$|f(x^{\prime})|\geq|f(x)|$$

for every element $f$ of the local ring of $\mathscr{X}$ at $x$.

Replacing $\omega$ by its $d$-fold tensor power $\omega^{\otimes d}$, with $d$ a positive integer, has no influence on the skeleton $\mathrm{Sk}(X_{K},\omega)$. Thus we may assume that the divisor

$$mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$$

is Cartier on $\mathscr{X}$ and we denote by $\mathcal{L}$ the associated line bundle. We choose a local generator $\theta$ of $\mathcal{L}$ at the point $\mathrm{red}_{\mathscr{X}}(x)$. Note that the pullback of $\mathcal{L}$ to the regular locus $\mathscr{X}^{\mathrm{reg}}_{R}$ of $\mathscr{X}_{R}$ is isomorphic to

$$\omega_{\mathscr{X}^{\mathrm{reg}}_{R}/R}((\mathscr{X}^{\mathrm{reg}}_{s})_{\mathrm{red}})^{\otimes m}.$$

We fix such an isomorphism. Then we can view $\omega$ as a rational section of $\mathcal{L}$ and write $\omega=g\theta$ locally at $\mathrm{red}_{\mathscr{X}}(x)$, with $g$ an element of

$$\mathcal{O}_{\mathscr{X}_{R},\mathrm{red}_{\mathscr{X}}(x)}\otimes_{R}K.$$

Then $\mathrm{wt}_{\omega}(x^{\prime})=-\ln|g(x^{\prime})|$. By (3.3.3), it is enough to show that

$$\mathrm{wt}_{\omega}(x)>-\ln|g(x)|.$$

Let $h:\mathscr{Y}\to\mathscr{X}$ be a log-resolution of $(\mathscr{X},\mathscr{X}_{s})$. Then $\mathrm{Sk}(X_{K},\omega)$ is contained in $\mathrm{Sk}(\mathscr{Y})$. We denote by $\Delta$ the log pullback of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$. Locally at $\mathrm{red}_{\mathscr{Y}}(x)$, it is explicitly given by

$$\frac{1}{m}(\mathrm{div}(h^{*}g)-\mathrm{div}_{\mathscr{Y}}(\omega)).$$

Since $\mathscr{X}$ is a $dlt$-model and $\mathrm{red}_{\mathscr{X}}(x)$ does not belong to $\mathscr{X}^{\mathrm{snc}}$, we know that $\Delta<(\mathscr{Y}_{s})_{\mathrm{red}}$ locally around $\mathrm{red}_{\mathscr{Y}}(x)$ by Lemma 3.2.3. Therefore, we can write

	$\displaystyle\mathrm{wt}_{\omega}(x)$	$\displaystyle=$	$\displaystyle v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+m(\mathscr{Y}_{s})_{\mathrm{red}})$
		$\displaystyle>$	$\displaystyle v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+m\Delta)$
		$\displaystyle=$	$\displaystyle-\ln|g(x)|.$

∎

By Proposition 3.3.2, it is enough to show that $\mathrm{Sk}(\mathscr{X})$ is contained in the right hand side of (3.3.5). Shrinking $\mathscr{C}$ around $s$ if necessary, we can assume that $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ is generated by global sections $\omega_{1},\ldots,\omega_{r}$. Then for each point $x$ on $\mathscr{X}^{\mathrm{snc}}_{s}$, we can choose an index $i$ in $\{1,\ldots,r\}$ such that $\mathrm{div}_{\mathscr{X}^{\mathrm{snc}}}(\omega_{i})+m(\mathscr{X}_{s})_{\mathrm{red}}$ is an effective divisor on $\mathscr{X}$ and $x$ is not contained in its support. This implies that the weight $\mathrm{wt}_{\omega_{i}}$ of $\omega_{i}$ is zero at all points of $\mathrm{Sk}(\mathscr{X})\cap\mathrm{red}_{\mathscr{X}}^{-1}(x)$ and non-negative at all other points of $X^{\mathrm{an}}_{K}$. Thus $\mathrm{Sk}(\mathscr{X})\cap\mathrm{red}_{\mathscr{X}}^{-1}(x)$ is contained in $\mathrm{Sk}(X_{K},\omega_{i})$. Varying the point $x$, we find that $\mathrm{Sk}(\mathscr{X})$ is contained in

$$\bigcup_{i=1}^{r}\mathrm{Sk}(X_{K},\omega_{i}).$$

∎

This follows from Theorem 2.2.6(1), Corollary 3.2.9 and Theorem 3.3.4. ∎