(2.1) Let $X$ be a Calabi-Yau variety over $K$. The essential skeleton $\mathrm{Sk}(X)$ of $X$ was first defined by Kontsevich and Soibelman in [KS06]. The construction was then refined and generalized in [MN15]. Let $\omega$ be a volume form on $X$. Then one can attach to the pair $(X,\omega)$ a weight function

$$\mathrm{wt}_{\omega}\colon X^{\mathrm{an}}\to\mathbb{R}\cup\{+\infty\}$$

that measures the degeneration of $(X,\omega)$ at $t=0$ along points of $X^{\mathrm{an}}$; see [MN15, §4.5]. The essential skeleton $\mathrm{Sk}(X)$ is the locus of points in $X^{\mathrm{an}}$ where $\mathrm{wt}_{\omega}$ reaches its minimal value. This definition only depends on $X$, and not on $\omega$, because multiplying $\omega$ with a scalar $\lambda\in K^{\ast}$ shifts the weight function by the constant $\mathrm{ord}_{t}\lambda$. The essential skeleton is a non-empty compact subspace of $X^{\mathrm{an}}$, which can be explicitly computed in the following way. Let $\mathscr{X}$ be an snc-model of $X$, with special fiber $\mathscr{X}_{k}=\sum_{i\in I}N_{i}E_{i}$. If we view $\omega$ as a rational section of the line bundle $\omega_{\mathscr{X}/R}(\mathscr{X}_{k,\mathrm{red}})$, then it defines a Cartier divisor on $\mathscr{X}$ that we denote by $\mathrm{div}_{\mathscr{X}}(\omega)$. It is supported on $\mathscr{X}_{k}$ because $\omega$ is nowhere vanishing on $X$; thus we can write $\mathrm{div}_{\mathscr{X}}(\omega)=\sum_{i\in I}\nu_{i}E_{i}$. If we denote by $\Delta(\mathscr{X})$ the dual intersection complex of $\mathscr{X}_{k}$, then $\mathrm{Sk}(X)$ is canonically homeomorphic to the sub-$\Delta$-complex of $\Delta(\mathscr{X})$ spanned by the vertices corresponding to the components $E_{i}$ for which $\nu_{i}/N_{i}$ is minimal (see Theorem 3 in [KS06, §6.6] and Theorem 4.7.5 in [MN15]). In particular, $\mathrm{Sk}(X)$ is homeomorphic to a finite $\Delta$-complex of dimension $\leq\dim(X)$.

(2.2) Kontsevich and Soibelman postulated that $\mathrm{Sk}(X)$ should be the base of the non-archimedean SYZ fibration, but the definition of $\mathrm{Sk}(X)$ does not provide us with a map $X^{\mathrm{an}}\to\mathrm{Sk}(X)$. To construct such a map, we will use an alternative description of the essential skeleton that appeared in [NX16a]. Let $\mathscr{X}$ be a minimal dlt-model of $X$, and denote by $\mathscr{X}^{\mathrm{snc}}$ the open subscheme of $\mathscr{X}$ consisting of the points where $\mathscr{X}$ is regular and $\mathscr{X}_{k}$ has strict normal crossings. Then the dual intersection complex $\Delta(\mathscr{X}^{\mathrm{snc}})$ of $\mathscr{X}^{\mathrm{snc}}_{k}$ can be canonically embedded into $X^{\mathrm{an}}$ (see [MN15, §3]). It follows from [NX16a, 3.3.3] that the image of this embedding is exactly the essential skeleton $\mathrm{Sk}(X)$. To be precise, it is assumed in the statement of [NX16a, 3.3.3] that $\mathscr{X}$ is $\mathbb{Q}$-factorial and defined over an algebraic curve, but these assumptions are not used in the proof. If the minimal dlt-model $\mathscr{X}$ is good, we will now construct a continuous retraction $\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X)$ by generalizing the construction for snc-models in [MN15, 3.1.5].

(2.3) Let $\mathscr{X}$ be a good minimal dlt-model of $X$. We need to make the following technical assumption: the strata of $\mathscr{X}_{k}$ are precisely the log canonical centers of the pair $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ that are contained in $\mathscr{X}_{k}$. By the definition of a dlt-model, every log canonical center of $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ is a stratum. The converse implication is known when $\mathscr{X}$ is defined over an algebraic curve [Ko13, 4.16]. We will prove in Corollary 4.4 that it also holds when $\mathscr{X}_{k}$ is reduced, which is the most important case for our purposes. We expect that the assumption is always satisfied, but the relevant parts of the Minimal Model Program have not been written down for $R$-schemes. In any case, if our technical assumption holds, we can proceed in the following way.

(2.4) Let $x$ be a point in $X^{\mathrm{an}}$ and let $\mathrm{red}_{\mathscr{X}}(x)$ be its reduction on $\mathscr{X}_{k}$ (see [MN15, 2.2.2]). Let $Z$ be the unique minimal stratum of $\mathscr{X}_{k}$ that contains $\mathrm{red}_{\mathscr{X}}(x)$. By our assumption (2), $Z$ is a log canonical center of $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$. Then $Z\cap\mathscr{X}^{\mathrm{snc}}$ is a non-empty stratum of $\mathscr{X}^{\mathrm{snc}}_{k}$ by the definition of a dlt pair. Thus, it determines a unique face $\tau$ of the dual intersection complex $\Delta(\mathscr{X}^{\mathrm{snc}})$. Let $E_{1},\ldots,E_{r}$ be the prime components of $\mathscr{X}_{k}$ that contain $Z$, and let $N_{1},\ldots,N_{r}$ be their multiplicities in $\mathscr{X}_{k}$. Then $E_{1},\ldots,E_{r}$ correspond precisely to the vertices $v_{1},\ldots,v_{r}$ of $\tau$. We choose a positive integer $m$ such that $mE_{i}$ is Cartier at the point $\mathrm{red}_{\mathscr{X}}(x)$ for every $i$, and we choose a local equation $f_{i}=0$ for $mE_{i}$ at $\mathrm{red}_{\mathscr{X}}(x)$. Then $\rho_{\mathscr{X}}(x)$ is the point of the simplex $\tau$ with barycentric coordinates

$$\alpha=\frac{1}{m}(-N_{1}\ln|f_{1}(x)|,\ldots,-N_{r}\ln|f_{r}(x)|)$$

with respect to the vertices $(v_{1},\ldots,v_{r})$. Under the embedding of $\Delta(\mathscr{X}^{\mathrm{snc}})$ into $X^{\mathrm{an}}$, the point $\rho_{\mathscr{X}}(x)$ corresponds to the monomial point represented by $(\mathscr{X},(E_{1},\ldots,E_{r}),\xi)$ and the tuple

$$\frac{1}{m}(-\ln|f_{1}(x)|,\ldots,-\ln|f_{r}(x)|),$$

in the terminology of [MN15, 2.4.5]. It is obvious that this definition does not depend on the choices of $m$ and the local equations $f_{i}$. It is also straightforward to check that $\rho_{\mathscr{X}}$ is continuous, and that it is a retraction onto $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$.

(2.6) Beware that, even though the subspace $\mathrm{Sk}(X)$ of $X^{\mathrm{an}}$ only depends on $X$, the $\Delta$-structure on $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$ and the retraction $\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X)$ depend on the choice of the (good) minimal dlt-model $\mathscr{X}$; we will illustrate this in Example 2.7 below. However, the essential skeleton $\mathrm{Sk}(X)$ does carry a canonical piecewise integral affine structure, which is induced by the embedding into the $K$-analytic space $X^{\mathrm{an}}$: see [MN15, §3.2]. If $\mathscr{X}$ is a minimal dlt-model for $X$, then this piecewise integral affine structure coincides with the one induced by the $\Delta$-complex structure on $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$, provided that the barycentric coordinates on the faces of $\Delta(\mathscr{X}^{\mathrm{snc}})$ are weighted by the multiplicities of the prime components in $\mathscr{X}_{k}$ as in [MN15, 3.2.1].

(2.9) Let $X$ be a Calabi-Yau variety over $K$ of dimension $n$. We say that $X$ is maximally degenerate if $X$ has a semistable snc-model over $R$ and the essential skeleton $\mathrm{Sk}(X)$ has dimension $n$. This is the class of Calabi-Yau varieties where we expect the SYZ mirror symmetry picture to appear. If $X$ is projective, then the condition $\dim(\mathrm{Sk}(X))=n$ is equivalent to the property that, for any topological generator $\sigma$ of $\mathrm{Gal}(K^{a}/K)\cong\widehat{\mu}(k)$ and any prime number $\ell$, the action of $\sigma$ on the étale cohomology space

$$H^{n}_{\text{\'{e}t}}(X\times_{K}K^{a},\mathbb{Q}_{\ell})$$

has a Jordan block of rank $n+1$, by [NX16a, 4.2.4(4)]. If $X$ is maximally degenerate and projective, then $\mathrm{Sk}(X)$ is a closed pseudomanifold (see [NX16a, 4.2.4(3)] – in the statement of that result, one should add the assumption that $X$ has a semistable snc-model, like in [NX16a, 4.1.7]). If we assume, moreover, that $X$ is geometrically simply connected and $h^{i,0}(X)=0$ for $0<i<n$, then it is expected that $\mathrm{Sk}(X)$ is homeomorphic to $S^{n}$. This has been proven in [KX16] when $n\leq 3$, and also when $n=4$ and $X$ has a minimal dlt-model that is also a semistable snc-model. One can prove in any dimension that $\mathrm{Sk}(X)$ has the $\mathbb{Q}$-rational homology of $S^{n}$, and that its fundamental group has trivial profinite completion; see [NX16a, 4.2.4(4)] and [HN17, 6.1.3(4)].