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

$$\mathrm{wt}_{\omega^{\prime}}(y)=d\cdot\mathrm{wt}_{\omega}(\pi(y))-d+1$$

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

$$\mathrm{wt}_{\omega^{\prime}}(y)=d\cdot\mathrm{wt}_{\omega}(\pi(y))-d+1$$

(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

$$\mathbf{H}^{i}=\mathbb{H}^{i}(\mathscr{Y}_{s},R\psi_{\mathscr{Y}}(\mathbb{Z}))\cong H^{i}(F(\mathbb{C}),\mathbb{Z})$$

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

$$H^{0}(\mathscr{Y}_{s},\omega_{\mathscr{Y}/\mathscr{C}}(\log\mathscr{Y}_{s})\otimes\mathcal{O}_{\mathscr{Y}_{s}})$$

whose image under the Poincaré residue map

$$\mathcal{R}:H^{0}(\mathscr{Y}_{s},\omega_{\mathscr{Y}/\mathscr{C}}(\log\mathscr{Y}_{s})\otimes\mathcal{O}_{\mathscr{Y}_{s}})\to H^{0}(\mathscr{Y}_{s},\mathrm{Gr}_{-n}^{W}(\omega_{\mathscr{Y}/\mathscr{C}}(\log\mathscr{Y}_{s})\otimes\mathcal{O}_{\mathscr{Y}_{s}}))\cong H^{0}(D,\mathcal{O}_{D})$$

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

$$\mathrm{Gr}_{F}^{0}\mathbf{H}^{i}_{\mathbb{C}}\cong H^{i}(\mathscr{Y}_{s},\mathcal{O}_{\mathscr{Y}_{s}})=0$$

for $0<i<n$ and

$$\mathrm{Gr}_{F}^{0}\mathbf{H}^{i}_{\mathbb{C}}\cong H^{i}(\mathscr{Y}_{s},\mathcal{O}_{\mathscr{Y}_{s}})\cong\mathbb{C}$$

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. ∎