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. ∎