(3.1.4) Theorem 3.1.3 can be extended to the case where $X$ is defined over $K$ instead of $C$ and $\mathscr{X}$ is a proper $snc$-model of $X$ over $R$. The general proof technique is the same as in [Th07], but one replaces the formalism of toroidal embeddings by the more flexible language of logarithmic geometry. Details will appear in [Ni13]. We will only use this generalization in the proof of Theorem 4.2.4.