Let $k$ be a field of characteristic zero and set $R=k[\negthinspace[t]\negthinspace]$ and $K=k(\negthinspace(t)\negthinspace)$. We fix a $t$-adic absolute value on $K$ by setting $|t|_{K}=1/e$. Let $X$ be a geometrically connected, smooth and proper $K$-variety. Then one can associate to $X$ a $K$-analytic space $X^{\mathrm{an}}$ in the sense of [Be90]. Each point of this space can be interpreted as a real valuation on the residue field of a point of $X$, extending the $t$-adic valuation on $K$. Thus $X^{\mathrm{an}}$ is naturally related to the birational geometry of $R$-models of $X$.

An $snc$-model of $X$ is a regular flat separated $R$-scheme of finite type $\mathscr{X}$, endowed with an isomorphism of $K$-schemes $\mathscr{X}_{K}\to X$, such that the special fiber $\mathscr{X}_{k}$ is a (not necessarily reduced) divisor with strict normal crossings. Each $snc$-model $\mathscr{X}$ of $X$ gives rise to a so-called skeleton $\mathrm{Sk}(\mathscr{X})$, a finite simplicial space embedded in the $K$-analytic space $X^{\mathrm{an}}$, canonically homeomorphic to the dual intersection complex $\mathcal{D}(\mathscr{X}_{k})$ of $\mathscr{X}_{k}$ [MN13, §3]. If $\mathscr{X}$ is proper over $R$, then $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $X^{\mathrm{an}}$ (see Theorem 3.1.3 and (3.1)). Results of this type are fundamental tools in the study of the homotopy type of $K$-analytic spaces, for instance in Berkovich’s proof of local contractibility of smooth $K$-analytic spaces [Be99]. On the other hand, the fact that the space $X^{\mathrm{an}}$ does not depend on any choice of model implies that the homotopy type of $\mathrm{Sk}(\mathscr{X})$ does not depend on the choice of $\mathscr{X}$; see [Th07] for a similar result in the context of embedded resolutions of pairs of varieties over a perfect field.

If $X$ is a curve of genus $\geq 1$, then it is well-known that $X$ has a minimal $snc$-model, which gives rise to a canonical skeleton in $X^{\mathrm{an}}$. However, in higher dimensions, no such distinguished $snc$-model exists, and one can wonder if it is still possible to construct a canonical skeleton inside the space $X^{\mathrm{an}}$. In this paper, we study two such constructions. Although they look quite different at first sight, we prove that they indeed yield the same result.

The first one is the so-called essential skeleton from [MN13, 4.6.2], a generalization of a construction of Kontsevich and Soibelman in [KS06] motivated by homological mirror symmetry. Its definition is quite natural: for every non-zero regular pluricanonical form $\omega$ on $X$ and every proper $snc$-model $\mathscr{X}$ of $X$, the form $\omega$ singles out certain faces of the skeleton $\mathrm{Sk}(\mathscr{X})$ corresponding to intersections of irreducible components where $\omega$ has minimal weight in a suitable sense; see [MN13, 4.5.5] for a precise statement. Taking the union of such faces as $\omega$ varies, we obtain a simplicial subspace $\mathrm{Sk}(X)$ of $\mathrm{Sk}(\mathscr{X})$ that can be characterized intrinsically on $X$ and thus no longer depends on any choice of an $snc$-model. This space $\mathrm{Sk}(X)$ was called the essential skeleton of $X$ in [MN13, 4.6.2]. If $X$ has trivial canonical sheaf, then $\mathrm{Sk}(X)$ coincides with the Kontsevich-Soibelman skeleton from [KS06] associated to any volume form $\omega$ on $X$.

A second construction appears in the context of the Minimal Model Program, specifically in the paper [dFKX12]. If we enlarge our class of models from $snc$-models to so-called $dlt$-models (2.2), then the relative minimal models over ${\rm Spec}(R)$ exist in any dimension, provided that the canonical divisor $K_{X}$ of the generic fiber is semi-ample (see Theorem 2.2.6 – for technical reasons, we are obliged to assume that $X$ is defined over an algebraic $k$-curve and to work with models over the base curve, because the results from MMP that we use have only been proven for $k$-schemes of finite type). Such a minimal $dlt$-model is not unique, but any two of them are crepant birational, which implies that their skeleta are the same (Corollary 3.2.7). Moreover, we prove that this canonical skeleton is still a strong deformation retract of $X^{\mathrm{an}}$ (Corollary 3.2.9).

Our main result, Theorem 3.3.4, states that these two constructions are equivalent: if $K_{X}$ is semi-ample, then the essential skeleton $\mathrm{Sk}(X)$ coincides with the skeleton of any minimal $dlt$-model.

We present two applications of this equivalence. First, as an immediate corollary of the above results, we obtain that the essential skeleton $\mathrm{Sk}(X)$ is a strong deformation retract of $X^{\mathrm{an}}_{K}$ when $K_{X}$ is semi-ample (see Corollary 3.3.6). Second, in Section 4, we study the topological properties of the essential skeleton of a Calabi-Yau variety $X$ over $K$. Using [KK10, Ko11], we show that $\mathrm{Sk}(X)$ is a pseudo-manifold with boundary, and even a closed pseudo-manifold when $\mathrm{Sk}(X)$ has maximal dimension and $k$ is algebraically closed. Moreover, using logarithmic geometry, we show that $\mathrm{Sk}(X)$ only depends on the reduction modulo $t^{2}$ of any proper $snc$-model of $X$, which allows us to remove the technical assumption that $X$ is defined over a curve (Theorem 4.1.4).