By Hironaka’s version of the Chow theorem (in turn a consequence of the flattening theorem), see [Hir75, Corollary 2], there exists a complex manifold ${\mathcal{X}}^{\prime\prime}$ and a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ such that ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$. Since ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is an isomorphism above $X$, the construction in [Hir75] further guarantees that ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is an isomorphism above $X$. Indeed, the proof proceeds by blowing up well-chosen smooth centers contained in the non-flat locus of ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, see Définition 4.4.3 (2) in loc. cit.

We may therefore assume that ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ itself is projective, and more precisely the blowup of an ideal $I$ cosupported on ${\mathcal{X}}_{0}$. By the principalization theorem for ideals, there exists a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ that is a composition of simple blowups, such that the pullback of $I$ to ${\mathcal{X}}^{\prime\prime}$ is a principal ideal, see [Kol07, Theorem 3.45] or [Wło09, Theorem 2.0.3]. In particular, ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$. ∎

To ease notation, set $r^{\prime}:=r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $r:=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$. Let $\sigma$ be the smallest simplex of $\Delta({\mathcal{X}})$ containing $r(\sigma^{\prime})$. Write $Y$, $Y^{\prime}$ and $Y^{\prime\prime}$ for the strata of ${\mathcal{X}}_{0}$, ${\mathcal{X}}^{\prime}_{0}$ and ${\mathcal{X}}^{\prime\prime}_{0}$ corresponding to $\sigma$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$, respectively. The restrictions $r^{\prime}|_{\sigma^{\prime\prime}}\colon\sigma^{\prime\prime}\to\sigma^{\prime}$ and $r_{\sigma^{\prime}}\colon\sigma^{\prime}\to\sigma$ are given by ${\mathbb{Q}}$-linear maps, and we have induced morphisms $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$.

First suppose that $\sigma^{\prime\prime}$ is active for $r^{\prime}$ and $\sigma^{\prime}$ is active for $r$. Then $r^{\prime}|_{\sigma^{\prime\prime}}$ and $r|_{\sigma^{\prime}}$ are given by ${\mathbb{Q}}$-linear isomorphisms; hence so is the composition $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}|_{\sigma^{\prime\prime}}$. Similarly, the maps $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ are bimeromorphic morphisms; hence so is the composition $Y^{\prime\prime}\to Y$. It follows that $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$.

Conversely, suppose $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$. Since the map $Y^{\prime\prime}\to Y$ is a bimeromorphic morphism, the map $Y^{\prime\prime}\to Y^{\prime}$ (resp. $Y^{\prime}\to Y$) must be injective (resp. surjective). In particular, $\dim Y^{\prime\prime}\leq\dim Y^{\prime}$ and $\dim Y\leq\dim Y^{\prime}$. Similarly, since the ${\mathbb{Q}}$-linear map defining $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}|_{\sigma^{\prime\prime}}=r|_{\sigma^{\prime}}\circ r^{\prime}|_{\sigma^{\prime\prime}}$ is an isomorphism, the ${\mathbb{Q}}$-linear map defining $r^{\prime}|_{\sigma^{\prime\prime}}$ (resp. $r|_{\sigma^{\prime}}$) must be injective (resp. surjective). In particular, $\dim\sigma^{\prime\prime}\leq\dim\sigma^{\prime}$ and $\dim\sigma\leq\dim\sigma^{\prime}$. Now

so we infer that $\dim Y^{\prime\prime}=\dim Y^{\prime}=\dim Y$ and $\dim\sigma=\dim\sigma^{\prime}=\dim\sigma^{\prime\prime}$. This further implies that the maps $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ are bimeromorphic morphisms, and that the ${\mathbb{Q}}$-linear maps defining $r^{\prime}|_{\sigma^{\prime\prime}}$ and $r|_{\sigma^{\prime}}$ are isomorphisms. Hence $\sigma^{\prime\prime}$ and $\sigma^{\prime}$ are active for $r^{\prime}$ and $r$, respectively. ∎

This is formal consequence of the relations $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\cap r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}^{-1}(A_{{\mathcal{X}}{\mathcal{X}}^{\prime}})$. For example, let us prove (a). Pick any point $w\in\Delta({\mathcal{X}})$. The assumption implies that we can find $w^{\prime\prime}\in A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ with $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}(w^{\prime\prime})=w$. Then $w^{\prime}:=r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}(w^{\prime\prime})\in A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ and $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}(w^{\prime})=w$. Thus (a) holds. The proofs of (b)–(d) are similar and left to the reader. ∎

This is well known (see e.g.  [KS06, p.381]) but we supply a proof for the convenience of the reader. To simplify notation, we set $r:=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$, $A:=A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$, $\Delta:=\Delta({\mathcal{X}})$ and $\Delta^{\prime}:=\Delta({\mathcal{X}}^{\prime})$.

Let $W$ be the center of the blowup $\rho$, and $Z$ the smallest stratum of ${\mathcal{X}}_{0}$ containing $W$. Let $E_{i}$, $i\in I$ be the irreducible components of ${\mathcal{X}}_{0}$, $J\subset I$ the subset such that $Z$ is an component of $E_{J}$, and $\sigma_{Z}$ the simplex defined by $Z$. Let $E^{\prime}_{i}$, $i\in I$ be the strict transform of $E_{i}$ to ${\mathcal{X}}^{\prime}$. Finally, let $E^{\prime}$ be the exceptional divisor of $\rho$. It corresponds to a vertex $v^{\prime}=v^{\prime}_{E^{\prime}}$ of $\Delta^{\prime}$.

First assume $W\subsetneq Z$. In this case, $\Delta^{\prime}$ is obtained from $\Delta$ by “raising a tent over the simplex $\sigma_{Z}$”. Let us be more precise. Consider a simplex $\sigma$ of $\Delta$, corresponding to a stratum $Y$ of ${\mathcal{X}}_{0}$. By the definition of a simple blowup, $W$ meets every irreducible component of ${\mathcal{X}}_{0}$ transversely (if at all). It follows that $Y$ cannot be contained in $W$, so $\rho$ is a biholomorphism above a general point of $Y$. Thus the strict transform $Y^{\prime}$ of $Y$ defines a stratum of ${\mathcal{X}}^{\prime}_{0}$ as well as a simplex $\sigma^{\prime}$ of $\Delta^{\prime}$, whose vertices correspond to the strict transforms of the vertices of $\sigma$. In this case, $r$ maps $\sigma^{\prime}$ onto $\sigma$, and $\rho\colon Y^{\prime}\to Y$ is a bimeromorphic morphism, so $\sigma^{\prime}$ is active for $r$.

This proves that $r\colon A\to\Delta$ is surjective. To prove injectivity, consider a stratum $Y^{\prime}$ of ${\mathcal{X}}^{\prime}_{0}$, with corresponding simplex $\sigma^{\prime}$ of $\Delta^{\prime}$. If $Y^{\prime}$ is not contained in $E^{\prime}$, then $\rho$ is a biholomorphism at the general point of $Y^{\prime}$, $Y:=\rho(Y^{\prime})$ is a stratum of ${\mathcal{X}}_{0}$ of the same dimension as $Y^{\prime}$, and $Y^{\prime}$ is the strict transform of $Y$. Thus we are in the situation above. On the other hand, if $Y^{\prime}$ is contained in $E^{\prime}$, then there exist irreducible components $E_{i}$, $i\in J$ of ${\mathcal{X}}_{0}$, having strict transforms $E^{\prime}_{i}$, $i\in J$, such that $\sigma^{\prime}$ has $v^{\prime}$ and $v^{\prime}_{i}$, $i\in J$ as vertices. Since $W$ is not a stratum of ${\mathcal{X}}_{0}$, the smallest stratum $Y$ containing $\rho(Y^{\prime})$ is cut out by $E_{i}$, $i\in J$. It follows that $r$ maps the simplex $\sigma^{\prime}$ onto the lower-dimensional simplex $\sigma$, so $\sigma^{\prime}$ is not active for $r$. Hence $r\colon A\to\Delta$ is injective.

Now assume $W=Z$ is stratum of ${\mathcal{X}}_{0}$, defining a simplex $\sigma$ with vertices $v_{i}$, $i\in J$. In this case, $\Delta^{\prime}$ is obtained from $\Delta$ by a barycentric subdivision of the simplex $\sigma_{Z}$. Again, let us be more precise. The same argument as above shows that if $Y$ is a stratum of ${\mathcal{X}}_{0}$ that is not contained in $W$, and $Y^{\prime}$ is the strict transform, then the simplex $\sigma^{\prime}_{Y^{\prime}}$ is active for $r$ and $r(\sigma^{\prime}_{Y^{\prime}})=\sigma_{Y}$. Further, $\sigma^{\prime}_{Y^{\prime}}$ is the unique simplex in ${\mathcal{X}}^{\prime}_{0}$ that is active for $r$ and whose image under $r$ meets the interior of $\sigma_{Y}$.

It remains to consider strata of ${\mathcal{X}}_{0}$ contained in $Z$. This becomes a toroidal calculation. Let $Y$ be such a stratum, cut out by $E_{i}$, $i\in K$, where $J\subset K$. Then $\rho^{-1}(Y)$ consists of $|J|$ strata $Y^{\prime}_{i}$, $i\in J$, each cut out by $E^{\prime}$ and $E^{\prime}_{j}$, $j\in K\setminus\{i\}$. The restriction $\rho|_{Y^{\prime}_{i}}\colon Y^{\prime}_{i}\to Y$ is a bimeromorphic morphism, and the the corresponding simplex $\sigma^{\prime}_{i}$ is active for $r$ and maps homeomorphically onto a simplex contained in $\sigma_{Y}$. Further, these simplices $r(\sigma^{\prime}_{i})$ have disjoint interiors and cover $\sigma_{Y}$. Finally, if $Y^{\prime}$ is a stratum of ${\mathcal{X}}^{\prime}_{0}$ contained in $E=\rho^{-1}(Z)$, then $Y=\rho(Y^{\prime})$ is a stratum contained in $Z$, hence $Y^{\prime}=Y^{\prime}_{i}$ is one of the strata above. This completes the proof. ∎

Since $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is continuous, $A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is compact, and $\Delta({\mathcal{X}})$ is Hausdorff, it suffices to prove that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is bijective.

Using Lemma 4.6 (c)–(d) and Lemma 4.7, one proves by induction on the number of blowups that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is bijective when ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is a composition of simple blowups.

Now consider the general case. Using Lemma 4.1 we find an snc model ${\mathcal{X}}^{\prime\prime}$ dominating both ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ and such that the morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is a composition of simple blowups. Thus $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is bijective. By Lemma 4.6 (a), it follows that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is surjective. Since ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ were arbitrary snc models with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$, it follows that $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is also surjective. It now follows from Lemma 4.6 (b) that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is injective, which completes the proof. ∎

Surjectivity follows from Proposition 4.3, and continuity from (4.2) after unwinding the definitions. It remains to establish (4.2). Consider any point $\xi^{\prime}\in{\mathcal{X}}_{0}$ and set $\xi=\pi(\xi^{\prime})$. We can find adapted coordinate charts $({\mathcal{U}}^{\prime},z^{\prime})$ at $\xi^{\prime}$ on ${\mathcal{X}}^{\prime}$ and $({\mathcal{U}},z)$ at $\xi$ on ${\mathcal{X}}$ such that $\rho({\mathcal{U}}^{\prime})\subset{\mathcal{U}}$ and such that the following holds: $t=\prod_{i=0}^{p}z_{i}^{b_{i}}$ in ${\mathcal{U}}$, $t=\prod_{j=0}^{p^{\prime}}(z^{\prime}_{j})^{b^{\prime}_{j}}$ in ${\mathcal{U}}^{\prime}$ and $\rho^{*}z_{i}=\prod_{j}(z^{\prime}_{j})^{a_{ij}}$. Since the map $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is given by (4.1), the result now follows from Proposition 2.1. ∎

It follows from (4.4) and (4.5) that $\tau$ is a bijection. Since $X_{A_{r}}^{\mathrm{An}}$ is compact and $X_{\overline{{\mathbb{D}}}_{r}}^{\mathrm{hyb}}$ is Hausdorff, it only remains to prove that $\tau$ is continuous. It suffices to show that the corresponding map $\tau_{\mathcal{X}}\colon X_{A_{r}}^{\mathrm{An}}\to{\mathcal{X}}^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}_{r}}$ is continuous for a given snc model ${\mathcal{X}}$. For this, in turn, it suffices to show that $\operatorname{Log}_{\mathcal{X}}\circ\tau_{\mathcal{X}}$ is continuous near the central fiber.

Consider a coordinate chart $({\mathcal{U}},z)$ adapted to ${\mathcal{X}}_{0}$ in the sense of §2.2. Let $E_{0},\dots,E_{p}$ be the irreducible components of ${\mathcal{X}}_{0}$ intersecting ${\mathcal{U}}$. Let $\hat{\mathcal{U}}\subset X_{A_{r}}^{\mathrm{An}}$ be the set of seminorms satisfying $|z_{i}|<1$ for $0\leq i\leq p$. Then we have

on $\hat{\mathcal{U}}\setminus\pi^{-1}(0)$. Now the function $(\log|z_{i}|^{-1})_{i}$ is continuous on $\hat{\mathcal{U}}$ with values in the simplex $\sigma={\mathbb{R}}_{+}^{p+1}\cap\{\sum_{0}^{p}b_{i}w_{i}=1\}\subset\Delta({\mathcal{X}})$. This completes the proof, since we can cover a neighborhood of the central fiber in $X_{A_{r}}^{\mathrm{An}}$ with sets of the type $\hat{{\mathcal{U}}}$. ∎