3.1. Retraction to the skeleton of an s n c -model [04V1]

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3.1. Retraction to the skeleton of an $snc$ -model

(3.1.1) Let $\mathscr{Y}$ be a connected regular flat separated $R$ -scheme of finite type such that the special fiber $\mathscr{Y}_{k}$ is a divisor with strict normal crossings. Then, as explained in [MN13, §3.1], one can associate to $\mathscr{Y}$ its skeleton $\mathrm{Sk}(\mathscr{Y})$ , which is a topological subspace of the generic fiber $\widehat{\mathscr{Y}}_{\eta}$ of the formal $t$ -adic completion of $\mathscr{Y}$ . It is the set of points of $\widehat{\mathscr{Y}}_{\eta}$ that correspond to a real valuation on the function field of $\mathscr{Y}_{K}$ that is monomial with respect to the strict normal crossings divisor $\mathscr{Y}_{k}$ . The skeleton $\mathrm{Sk}(\mathscr{Y})$ is canonically homeomorphic to the dual intersection complex $\mathcal{D}((\mathscr{Y}_{k})_{\mathrm{red}})$ of $\mathscr{Y}_{k}$ , and there exists a canonical continuous retraction

\rho_{\mathscr{Y}}:\widehat{\mathscr{Y}}_{\eta}\to\mathrm{Sk}(\mathscr{Y}).

Moreover, $\mathrm{Sk}(\mathscr{Y})$ carries a canonical piecewise $\mathbb{Z}$ -affine structure [MN13, §3.2].

(3.1.2) We keep the notations from (2.1). For each $\mathscr{C}$ -model $\mathscr{X}$ of $X$ , we define the skeleton of $\mathscr{X}$ by

\mathrm{Sk}(\mathscr{X})=\mathrm{Sk}(\mathscr{X}^{\mathrm{snc}}_{R})\subset\widehat{(\mathscr{X}^{\mathrm{snc}}_{R})}_{\eta}\subset X_{K}^{\mathrm{an}}

and we write $\rho_{\mathscr{X}}$ for $\rho_{\mathscr{X}_{R}^{\mathrm{snc}}}$ . If $\mathscr{X}$ is a proper $snc$ -model of $X$ , one has the following crucial property.

Theorem 3.1.3.

If $\mathscr{X}$ is a proper $snc$ -model of $X$ over $\mathscr{C}$ , then there exists a continuous map

H:[0,1]\times X_{K}^{\mathrm{an}}\to X_{K}^{\mathrm{an}}

such that $H(0,\cdot)$ is the identity, $H(t,x)=x$ for all $x$ in $\mathrm{Sk}(\mathscr{X})$ and all $t$ in $[0,1]$ , and $H(1,\cdot)=\rho_{\mathscr{X}}$ . Thus $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $X_{K}^{\mathrm{an}}$ .

Proof.

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. ∎

(3.1.4) Theorem 3.1.3 can be extended to the case where $X$ is defined over $K$ instead of $C$ and $\mathscr{X}$ is a proper $snc$ -model of $X$ over $R$ . The general proof technique is the same as in [Th07], but one replaces the formalism of toroidal embeddings by the more flexible language of logarithmic geometry. Details will appear in [Ni13]. We will only use this generalization in the proof of Theorem 4.2.4.

3.1. Retraction to the skeleton of an s ​ n ​ c -model [04V1]

3.1. Retraction to the skeleton of an s​n​csnc-model

Theorem 3.1.3.

Proof.

3.1. Retraction to the skeleton of an s n c -model [04V1]

3.1. Retraction to the skeleton of an $snc$ -model