The dual complex $\Delta({\mathcal{X}})$ of an snc model ${\mathcal{X}}$ is defined as the dual complex of the snc divisor ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$, as in §2.1. It is equipped with a natural integral affine structure, in which the face $\sigma$ corresponding to a component $Y$ of a non-empty $E_{J}$ is identified with the simplex

$$\sigma=\left\{w\in{\mathbb{R}}_{+}^{J}\mid\sum_{i\in J}b_{i}w_{i}=1\right\},$$

in such a way that $M_{\sigma}={\mathbb{Z}}^{J}$.

As explained in [BFJ16, §3] and [MN15, §3], there is a natural embedding

$$\operatorname{emb}_{\mathcal{X}}\colon\Delta({\mathcal{X}})\to{\mathcal{X}}^{\mathrm{an}}$$

that takes a point $w\in\sigma$ to the corresponding monomial valuation. In particular, the vertex corresponding to $E_{i}$ is sent to the divisorial valuation $v_{E_{i}}=b_{i}^{-1}\operatorname{ord}_{E_{i}}$. The value group of a valuation $v=\operatorname{emb}_{\mathcal{X}}(w)$, $w\in\sigma$, is given by

$$v(F(X)^{*})=M_{\sigma}(w):=\{f(w)\mid f\in M_{\sigma}\}.$$

Further, if $w\in\mathring{\sigma}$, then $Y_{\sigma}$ is the closure of the center of $\operatorname{emb}_{\mathcal{X}}(w)$.

The resulting subspace $\operatorname{Sk}({\mathcal{X}}):=\operatorname{emb}_{\mathcal{X}}(\Delta_{X})\subset{\mathcal{X}}^{\mathrm{an}}\subset X^{\mathrm{an}}$ is called the skeleton of ${\mathcal{X}}$. It is naturally a ${\mathbb{Z}}$-PA space, the ${\mathbb{Z}}$-PA functions on $\operatorname{Sk}({\mathcal{X}})$ being precisely the restrictions of model functions $\phi_{D}$ determined by a Cartier divisor $D$ on some proper modification ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$.

We further have a natural retraction $r_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to\operatorname{Sk}({\mathcal{X}})$, mapping a valuation $v$ centered on ${\mathcal{X}}_{0}$ to the monomial valuation $r_{\mathcal{X}}(v)$ taking the same values on the $E_{i}$’s. These retractions induce a homeomorphism

$$X^{\mathrm{an}}\overset{\sim}{\to}\varprojlim_{\mathcal{X}}\operatorname{Sk}({\mathcal{X}}),$$

where ${\mathcal{X}}$ runs over all proper (or projective) snc models, compare (4.3).

If ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is a proper morphism of snc models, then, by [MN15, 3.1.7],

$$\operatorname{Sk}({\mathcal{X}})\subset\operatorname{Sk}({\mathcal{X}}^{\prime})\subset{\mathcal{X}}^{\prime\mathrm{an}}={\mathcal{X}}^{\mathrm{an}},$$

the first inclusion being ${\mathbb{Z}}$-PA. Further,

$$\bigcup_{{\mathcal{X}}\ \text{snc}}\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$$

coincides with the set of (quasi)monomial, or Abhyankar, valuations.

For a dlt model ${\mathcal{X}}$, the dual complex $\Delta({\mathcal{X}})$ and skeleton $\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$ are simply defined as those of ${\mathcal{X}}_{\mathrm{snc}}$, cf. [NX13]. The retraction $r_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to\operatorname{Sk}({\mathcal{X}})$ can be defined as above when ${\mathcal{X}}$ is ${\mathbb{Q}}$-factorial, but its existence is otherwise unclear (at least to us!).

By [KKMS], any toroidal model ${\mathcal{X}}$ has a dual complex $\Delta({\mathcal{X}})$ endowed with a natural integral affine structure. This dual complex is canonically realized as a subspace $\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$, for instance by setting $\operatorname{Sk}({\mathcal{X}}):=\operatorname{Sk}({\mathcal{X}}^{\prime})$ for any toroidal modification ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ with ${\mathcal{X}}^{\prime}$ snc. Thus $\operatorname{Sk}({\mathcal{X}})$ is equipped with a ${\mathbb{Z}}$-PA structure.