As we mentioned above, this result is essentially contained in [KK10, Ko11]. Using the terminology there, properties (1)-(3) of a pseudo-manifold all follow from the fact that two minimal log canonical centers of a log crepant structure are $\mathbb{P}^{1}$-linked in the sense of Definition 9 in [Ko11]. We will now explain this in more detail. We denote by $n$ the relative dimension of $X$ over $C$.

By Theorem 2.2.6(1), there exists a a good minimal $dlt$-model $\mathscr{X}$ of $X$ over $\mathscr{C}$. By Theorem 3.3.4, we have $\mathrm{Sk}(X_{K})=\mathrm{Sk}(\mathscr{X})$. As a triangulation on $\mathrm{Sk}(\mathscr{X})$, we take the first barycentric subdivision of the simplicial structure on $\mathrm{Sk}(\mathscr{X})$. This barycentric subdivision is necessary to guarantee that the intersection of two faces is a codimension one face of both, rather than a union of faces (think of a type $I_{2}$ degeneration of elliptic curves, whose skeleton consists of two vertices joined by two edges).

We choose an integer $m>0$ such that $mK_{X}\sim 0$. Since the divisor $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ is semi-ample over $\mathscr{C}$ and trivial over $C$, we see that $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ must be a multiple of $\mathscr{X}_{s}$ and thus trivial over $\mathscr{C}$. Thus we can apply Theorem 10 in [Ko11] to the $dlt$-pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ over $\mathscr{C}$. It states that every two minimal log canonical centers $D$ and $D^{*}$ of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ are $\mathbb{P}^{1}$-linked. This means, in particular, that they have the same dimension, say $n-d$, and that there exist a sequence of $(n-d+1)$-dimensional log canonical centers $E_{1},E_{2},\ldots,E_{\ell}$ and a sequence of $(n-d)$-dimensional log canonical centers $D=D_{0},D_{1},\ldots,D_{\ell}=D^{*}$ such that $D_{i-1},D_{i}\subset E_{i}$ for $1\leq i\leq\ell$. In this way, we obtain properties (1) and (3) of a pseudo-manifold with boundary.

If we have two minimal log canonical centers $D_{1},\,D_{2}$ of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$, contained in an $(n-d+1)$-dimensional log canonical center $E$, and if we write

for some $\Delta\geq 0$, then $(E,D_{1}+D_{2}+\Delta)$ is again a $dlt$-pair [Ko13, 4.19]. Moreover, $D_{1}$ cannot intersect $D_{2}$ or $\lfloor\Delta\rfloor$ because the intersection would be a union of log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$, which contradicts the minimality of $D_{1}$. Thus we are in the situation of the second part of the proof of Theorem 10 in [Ko11]. That proof shows that $D_{1}$ and $D_{2}$ are the only log canonical centers of $(E,D_{1}+D_{2}+\Delta)$. Property (2) follows. ∎

We first notice that in the above statement, $U$ can be replaced by any smaller open set that meets all the log canonical centers: the closure of the restriction of $\sum^{j}_{i=\ell+1}\Delta_{i}|_{U}$ to $U\cap D$ will yield the same divisor on $D$.

Then by induction, we only need to treat the case where $D$ is a component of $\Delta$, say $\Delta_{1}$. If we take a log resolution $f:Y\to(Z,\Delta)$ and let $D^{\prime}=\Delta^{\prime}_{1}$ be the birational transform of $D$, then $\Delta_{D}$ can be computed as follows: if we write $f^{*}(K_{Z}+\Delta)|_{D^{\prime}}=K_{D^{\prime}}+\Delta_{D^{\prime}}$ then $\Delta_{D}=(f|_{D^{\prime}})_{*}(\Delta_{D^{\prime}})$. In particular, as $K_{Z}+\Delta$ is Cartier, we know that $\Delta_{D}$ is a integral divisor. Since it is effective, and all the components of $\Delta_{D}$ are log canonical centers of $(X,\Delta)$, we see that $\Delta_{D}$ must be equal to the closure of the restriction of $\sum^{j}_{i=2}\Delta_{i}|_{U}$. ∎

By running MMP for $\mathscr{Y}$ over $\mathscr{C}$, we know that $X$ has a good minimal dlt model $\mathscr{X}$ with reduced special fiber (see [Fu11] or [HX13]). Then one sees as in the proof of Theorem 4.1.4 that $K_{\mathscr{X}}$ is trivial over $\mathscr{C}$. Our assumption on the dimension of $\mathrm{Sk}(X_{K})$ implies that the minimal log canonical centers of $(\mathscr{X},\mathscr{X}_{s})$ are points. Let $D$ be a one-dimensional log canonical center, and let $D_{i}$ ($1\leq i\leq\ell$) be the 0-dimensional log canonical centers contained in $D$. From Lemma 4.1.6, we know that

Thus $D$ is a rational curve and $\ell=2$, which means that $\mathrm{Sk}(X_{K})$ is closed. ∎

We may assume that $K^{\prime}$ is Galois over $K$. Let $d$ be the ramification index of $K^{\prime}$ over $K$. We will prove that

for every divisorial point $y$ on $(Y^{\prime})^{\mathrm{an}}$ (see [MN13, 2.4.10] for the notion of divisorial point). This immediately implies the statement in the lemma, since $\mathrm{Sk}(X,\omega)$ is the closure of the set of divisorial points where the weight function reaches its minimal value [MN13, 4.5.1].

We denote by $R^{\prime}$ the integral closure of $R$ in $K^{\prime}$. Let $\mathscr{Y}^{\prime}$ be a regular separated $R^{\prime}$-scheme of finite type with irreducible special fiber $\mathscr{Y}^{\prime}_{k}$, endowed with an isomorphism of $K^{\prime}$-schemes $\mathscr{Y}^{\prime}_{K^{\prime}}\to Y^{\prime}$. Let $y$ be the unique point in $\mathrm{red}_{\mathscr{Y}^{\prime}}^{-1}(\xi)$, where $\xi$ denotes the generic point of $\mathscr{Y}^{\prime}_{k}$. Removing a closed subset of $\mathscr{Y}^{\prime}_{k}$ if necessary, we can find a regular separated $R$-scheme of finite type $\mathscr{Y}$ and an isomorphism $\mathscr{Y}_{K}\to Y$ such that $\mathscr{Y}^{\prime}$ is an open subscheme of the normalization of $\mathscr{Y}\times_{R}R^{\prime}$. Then $\mathrm{red}_{\mathscr{Y}}(\pi(y))$ is a generic point of $\mathscr{Y}_{k}$.

If we use the notations from (3.2) and denote by $(S^{\prime})^{+}$ the log scheme associated to $R^{\prime}\setminus\{0\}\to R^{\prime}$, then the $(S^{\prime})^{+}$-log scheme $(\mathscr{Y}^{\prime})^{+}$ is isomorphic to an open log subscheme of the $fs$ base change of $\mathscr{Y}^{+}$ from $S^{+}$ to $(S^{\prime})^{+}$. Since log differentials are compatible with $fs$ base change, we can deduce from the description of the weight function in (3.2) that

(the scaling factor $d$ is caused by the renormalization of the discrete valuation on $K^{\prime}$). ∎

By Lemma 4.1.9 and the Semi-Stable Reduction Theorem we can assume that $X$ has a projective $snc$-model $\mathscr{Y}$ over $\mathscr{C}$ such that $\mathscr{Y}_{s}$ is reduced. For every integer $i\geq 0$, we denote by

the degree $i$ nearby cohomology of $\mathscr{Y}$ at $s$; here $R\psi_{\mathscr{Y}}(\mathbb{Z})$ denotes the complex of nearby cycles with $\mathbb{Z}$-coefficients associated to $\mathscr{Y}$. By [St76], the spaces $\mathbf{H}^{i}$ carry a canonical mixed Hodge structure, whose weight filtration coincides with the monodromy filtration. In particular, there exists a Jordan block of monodromy of size $n+1$ on $\mathbf{H}^{n}_{\mathbb{Q}}$ if and only if $W_{0}\mathbf{H}^{n}_{\mathbb{Q}}\neq 0$.

By [Be09, 5.1] and its proof, the $\mathbb{Q}$-vector space $W_{0}\mathbf{H}^{i}_{\mathbb{Q}}$ is canonically isomorphic to the degree $i$ singular cohomology of $X_{K}^{\mathrm{an}}$, for every $i\geq 0$. Since $X_{K}^{\mathrm{an}}$ is homotopy equivalent to $\mathrm{Sk}(X_{K})$ by Corollary 3.3.6, we see that $W_{0}\mathbf{H}^{n}_{\mathbb{Q}}$ can only be different from zero if the dimension of $\mathrm{Sk}(X_{K})$ is equal to $n$. We will now prove the converse implication. Suppose that $\mathrm{Sk}(X_{K})$ has dimension $n$ and let $\omega$ be a relative volume form on $X$ over $C$ such that $\omega$ extends to a global section of $\omega_{\mathscr{Y}/\mathscr{C}}(\log\mathscr{Y}_{s})$ that generates $\omega_{\mathscr{Y}/\mathscr{C}}(\log\mathscr{Y}_{s})$ at at least one generic point of $\mathscr{Y}_{s}$ (modulo shrinking $\mathscr{C}$, such $\omega$ always exists). Then it follows from [MN13, 4.5.5] that $\mathrm{Sk}(X_{K})$ is the simplicial subspace of $\mathrm{Sk}(\mathscr{Y})$ spanned by the vertices corresponding to the irreducible components $E$ of $\mathscr{Y}_{s}$ such that $\omega$ generates $\omega_{\mathscr{Y}/\mathscr{C}}(\log\mathscr{Y}_{s})$ at the generic point of $E$. Since $\mathrm{Sk}(X_{K})$ has dimension $n$, we can find such components $E_{1},\ldots,E_{n}$ that intersect in a point. Denote by $D$ the union of $n$-fold intersection points of components of $\mathscr{Y}_{s}$. Then by reduction modulo $t$, $\omega$ induces an element of

whose image under the Poincaré residue map

is different from zero. However, by the degeneration of the Hodge and weight spectral sequences, the image of $\mathcal{R}$ injects into $W_{0}\mathbf{H}^{n}_{\mathbb{C}}$. Thus $W_{0}\mathbf{H}^{n}_{\mathbb{Q}}$ is non-trivial.

Finally, assume that $\mathrm{Sk}(X_{K})$ has dimension $n$ and that $h^{i,0}(X_{\mathrm{gen}})=0$ for $0<i<n$. Then

for $0<i<n$ and

for $i=0,n$ by the degeneration of the Hodge spectral sequence for the limit mixed Hodge structure. Thus $W_{0}\mathbf{H}^{i}_{\mathbb{Q}}=0$ for $0<i<n$, $W_{0}\mathbf{H}^{0}_{\mathbb{Q}}\cong\mathbb{Q}$ and $W_{0}\mathbf{H}^{n}_{\mathbb{Q}}$ has dimension at most one; it must have dimension one since we have already proven that it is non-zero. It follows that $\mathrm{Sk}(X_{K})$ is a $\mathbb{Q}$-homology sphere. ∎