(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)].