Let now $X$ be a scheme of finite type over $A_{r}$. We will associate to $X$ three kinds of analytic spaces.

First, since $X$ is obtained by gluing together finitely many affine schemes cut out by polynomials with coefficients holomorphic on ${\mathbb{D}}^{*}_{r}\subset{\mathbb{C}}$ and meromorphic at $0$, we can associate to $X$ in a functorial way a complex analytic space $X^{\operatorname{hol}}$ over ${\mathbb{D}}^{*}_{r}$, which we call its holomorphic analytification.

Second, since $A_{r}$ is contained in ${\mathbb{C}}(\!({t})\!)$, we may also consider the base change $X_{{\mathbb{C}}(\!({t})\!)}$ and its non-Archimedean analytification $X^{\mathrm{an}}_{{\mathbb{C}}(\!({t})\!)}$ with respect to the non-Archimedean absolute value $r^{\operatorname{ord}_{0}}$ on ${\mathbb{C}}(\!({t})\!)$.

Third, we denote by $X^{\mathrm{hyb}}$ the analytification of $X$ as a scheme of finite type over the Banach ring $A_{r}$, and call it the hybrid analytification of $X$. In view of Proposition A.4, it comes with a continuous structure map

$$\pi\colon X^{\mathrm{hyb}}\to\overline{{\mathbb{D}}}_{r}\simeq{\mathcal{M}}(A_{r}),$$

Recall further that $X^{\mathrm{hyb}}$ is locally compact, Hausdorff if $X$ is separated, and compact if $X$ is proper over $A_{r}$. The discussion above implies:

In §4 we give a topological description of $X^{\mathrm{hyb}}$.