(2.1.1) Let $k$ be a field of characteristic zero. We set $R=k[\negthinspace[t]\negthinspace]$ and $K=k(\negthinspace(t)\negthinspace)$, and we fix a $t$-adic absolute value $|\cdot|_{K}$ on $K$ by setting $|t|_{K}=1/e$. For every $K$-scheme of finite type $Y$, we denote by $Y^{\mathrm{an}}$ the associated $K$-analytic space. For every separated $R$-scheme of finite type $\mathscr{Y}$ we set $\mathscr{Y}_{k}=\mathscr{Y}\times_{R}k$ and $\mathscr{Y}_{K}=\mathscr{Y}\times_{R}K$. Moreover, we will denote by $\widehat{\mathscr{Y}}$ the $t$-adic completion of $\mathscr{Y}$, by $\widehat{\mathscr{Y}}_{\eta}$ the generic fiber of $\widehat{\mathscr{Y}}$ in the category of $K$-analytic spaces and by

$$\mathrm{red}_{\mathscr{Y}}:\widehat{\mathscr{Y}}_{\eta}\to\mathscr{Y}_{k}$$

the canonical reduction map. The generic fiber $\widehat{\mathscr{Y}}_{\eta}$ is an analytic domain in $\mathscr{Y}_{K}^{\mathrm{an}}$, and it is equal to $\mathscr{Y}_{K}^{\mathrm{an}}$ if and only if $\mathscr{Y}$ is proper over $R$.