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. ∎

It is easy to see that (1) holds, since we can detect regularity by looking at the dimensions of the Zariski tangent spaces at the points of

Moreover, the special fibers of $\mathscr{Y}$ and $\mathscr{Z}$ are isomorphic so that $\mathscr{Z}_{k}$ is a divisor with strict normal crossings. Point (2) is more subtle and follows from [Ki03, 2.6(2)]. ∎

Reducing $f$ modulo $t$, we obtain an isomorphism of $k$-schemes $\mathscr{Y}_{k}\to\mathscr{Z}_{k}$ and, by taking the dual intersection complexes, an isomorphism of simplicial spaces with piecewise $\mathbb{Z}$-affine structure $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$. We will prove that this isomorphism maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$.

We use the notations from Proposition 4.2.2(2) and we set $s^{+}=s_{1}^{+}$. We denote by $\mathscr{Y}^{+}_{k}$ the log scheme $\mathscr{Y}^{+}\times_{S^{+}}s^{+}$ obtained by restricting the log structure on $\mathscr{Y}^{+}$ to the special fiber $\mathscr{Y}_{k}$ of $\mathscr{Y}$. It follows from [IKN05, 7.1] that

is a free $R$-module of rank one and that the reduction map

is an isomorphism. Let $\omega$ be a generator of the $R$-module $\Omega$ and denote by $\omega_{k}$ its image in $\Omega_{k}$. By (3.2), the generic point $\xi$ of an irreducible component $E$ of $\mathscr{Y}_{k}$ is $\omega$-essential in the sense of [MN13, 4.5.4] if and only if $\omega_{k}$ generates $\omega_{\mathscr{Y}^{+}_{k}/s^{+}}$ at the point $\xi$. Moreover, the skeleton $\mathrm{Sk}(\mathscr{Y}_{K})=\mathrm{Sk}(\mathscr{Y}_{K},\omega)$ is the simplicial subspace of $\mathrm{Sk}(\mathscr{Y})$ spanned by the vertices corresponding to such points $\xi$ [MN13, 4.5.5]. However, for every integer $d>0$, the stalk of $\omega_{\mathscr{Y}^{+}_{k}/s^{+}}$ at $\xi$ is generated by global sections if and only if $\omega_{\mathscr{Y}^{+}\times_{S^{+}}s_{d}^{+}/s_{d}^{+}}$ is generated by global sections at any point lying above $\xi$, by the base change property in [IKN05, 7.1]. The analogous statements hold for $\mathscr{Z}$. Thus it follows from Proposition 4.2.2(2) that the isomorphism $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$ maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$. ∎

Let $\mathscr{X}$ be a proper $snc$-model of $X$ over $R$. By a standard argument based on spreading out and Greenberg Approximation (as explained in [MN13, 5.1.2], for instance) we can find a connected smooth $k$-curve $\mathscr{C}$, a $k$-rational point $s$ on $\mathscr{C}$, a uniformizer $t$ in $\mathcal{O}_{\mathscr{C},s}$ and a smooth and proper $\mathscr{C}$-scheme $\mathscr{X}^{\prime}$ with geometrically connected fibers such that there exists an isomorphism

over $R/(t^{2})\cong{O}_{\mathscr{C},s}/(t^{2})$. Inspecting the proof of [MN13, 5.1.2], we see that we can also assume that the relative canonical sheaf of $\mathscr{X}^{\prime}$ is trivial over $C=\mathscr{C}\setminus\{s\}$ (if the generic fiber of a smooth and proper family over an integral scheme has trivial canonical sheaf, then this holds for all fibers over some dense open subscheme of the base).

By (3.1) we know that $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $X_{K}^{\mathrm{an}}$. Thus by Proposition 4.2.3, it suffices to prove assertions (1)–(3) for $\mathscr{X}^{\prime}\times_{\mathscr{C}}\mathrm{Spec}\,K$ instead of $X$. In this case, they follow from Corollary 3.3.6 and Theorems 3.2.8, 3.3.4, 4.1.4 and 4.1.7.

It remains to prove (4). Invoking the Lefschetz Principle, we may assume that $k=\mathbb{C}$. Taking for $\mathscr{X}$ a projective $snc$-model over $R$, we can arrange that $\mathscr{X}^{\prime}$ is projective over $\mathscr{C}$. By Proposition 4.2.2 and the theory of logarithmic nearby cycles [Na98, 3.3] the action of $\sigma$ on

has a Jordan block of size $n+1$ if and only if the corresponding statement holds for $\mathscr{X}^{\prime}_{K}$. By Deligne’s comparison theorem for étale and complex analytic nearby cycles in [SGA7b, Exp.XIV], it is also equivalent to the property that the monodromy action on the degree $n$ singular cohomology of a general fiber of $\mathscr{X}^{\prime}$ has a Jordan block of size $n+1$. If $h^{i,0}(X)=0$ for $0<i<n$, then we can assume that this also holds for a general fiber of $\mathscr{X}^{\prime}$, by the proof of [MN13, 5.1.2] (if this property is satisfied by the generic fiber of a smooth and proper family over an integral scheme, then it holds for all fibers over a dense open subscheme of the base, by semi-continuity). Thus the assertion (4) follows from Theorem 4.1.10. ∎