Let $A$ be a Banach ring, that is, a commutative ring that is complete with respect to a submultiplicative norm $\|\cdot\|$. The Berkovich spectrum ${\mathcal{M}}(A)$ is the set of all bounded multiplicative seminorms on $A$. In other words, a point $x\in{\mathcal{M}}(A)$ corresponds to a function $|\cdot|_{x}\colon A\to{\mathbb{R}}_{\geq 0}$ such that $|\cdot|_{x}\leq\|\cdot\|$, $|1|_{x}=1$, $|f+g|_{x}\leq|f|_{x}+|g|_{x}$ and $|fg|_{x}=|f|_{x}|g|_{x}$ for $f,g\in A$. The spectrum is a nonempty, compact Hausdorff space with respect to the topology of pointwise convergence.

For $x\in{\mathcal{M}}(A)$, denote by ${\mathfrak{p}}_{x}$ the kernel of $|\cdot|_{x}$. This is a prime ideal of $A$, and $|\cdot|_{x}$ defines a multiplicative norm on $A/{\mathfrak{p}}_{x}$. The completion of the fraction field of $A/{\mathfrak{p}}_{x}$ with respect to this norm is a valued field ${\mathcal{H}}(x)$. We write $f(x)$ for the image of $f\in A$ in ${\mathcal{H}}(x)$; then $|f(x)|=|f|_{x}$. The assignment $x\mapsto{\mathfrak{p}}_{x}$ yields a map ${\mathcal{M}}(A)\to\operatorname{Spec}(A)$ that is continuous for the Zariski topology $\operatorname{Spec}(A)$.

To any scheme $X$ of finite type over a Banach ring $A$, Berkovich associates an analytification⁹⁹ 9 We use the term analytification even though we shall only consider $X^{\mathrm{An}}$ as a topological space. In particular, $X^{\mathrm{An}}$ only depends on the reduced scheme structure of $X$. $X^{\mathrm{An}}$, a locally compact topological space with a continuous morphism $X^{\mathrm{An}}\to{\mathcal{M}}(A)$, defined as follows.

When $X=\operatorname{Spec}B$ is affine, with $B$ a finitely generated $A$-algebra, $X^{\mathrm{An}}$ is defined as the set of multiplicative seminorms $|\cdot|_{x}$ on $B$ whose restriction to $A$ is bounded by the given norm on $A$, i.e. belongs to ${\mathcal{M}}(A)$. The topology on $X^{\mathrm{An}}$ is the weakest one for which $x\mapsto|f|_{x}=|f(x)|$ is continuous for every $f\in B$.

In the general case, the analytification $X^{\mathrm{An}}$ is defined by gluing together the analytifications of an affine open cover, and yields a covariant functor $X\mapsto X^{\mathrm{An}}$. If $X\hookrightarrow Y$ is an open (resp. closed) embedding, then so is $X^{\mathrm{An}}\hookrightarrow Y^{\mathrm{An}}$. If $X\to Y$ is surjective, then so is $X^{\mathrm{An}}\to Y^{\mathrm{An}}$.

The topological space $X^{\mathrm{An}}$ is Hausdorff (resp. compact) if $X$ is separated (resp. projective). The assignment $x\mapsto{\mathfrak{p}}_{x}$ above globalizes to a continuous map

$$\pi\colon X^{\mathrm{An}}\to X,$$

where $X$ is equipped with the Zariski topology.

When $A$ is a valued field, it is more common to write $X^{\mathrm{an}}$ instead of $X^{\mathrm{An}}$ [Berk90].

Denote by ${\mathbb{C}}_{\mathrm{hyb}}$ the Banach field $({\mathbb{C}},\|\cdot\|_{\mathrm{hyb}})$, where the hybrid norm is defined as

$$\|\cdot\|_{\mathrm{hyb}}:=\max\{|\cdot|_{0},|\cdot|_{\infty}\},$$

with $|\cdot|_{0}$ the trivial absolute value and $|\cdot|_{\infty}$ the usual absolute value.

The elements of the Berkovich spectrum ${\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})$ are of the form $|\cdot|_{\infty}^{\rho}$ for $\rho\in[0,1]$, interpreted as the trivial absolute value $|\cdot|_{0}$ for $\rho=0$. This yields a homeomorphism ${\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})\simeq[0,1]$.

If $X$ is a scheme of finite type over ${\mathbb{C}}$, we denote by $X^{\operatorname{hol}}=X({\mathbb{C}})$ its analytification with respect to the usual absolute value $|\cdot|_{\infty}$, by $X^{\mathrm{an}}_{0}$ its analytification with respect to the trivial absolute value, and by $X^{\mathrm{hyb}}$ its analytification with respect to the hybrid norm $\|\cdot\|_{\mathrm{hyb}}$.

From the structure morphism $X\to\operatorname{Spec}{\mathbb{C}}$ we obtain a continuous map $\lambda\colon X^{\mathrm{hyb}}\to{\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})\simeq[0,1]$. The fiber $\lambda^{-1}(\rho)$ is equal to the analytification of $X$ with respect to the multiplicative norm $|\cdot|_{\infty}^{\rho}$ on ${\mathbb{C}}$. In particular, we have canonical identifications $\lambda^{-1}(1)\simeq X^{\operatorname{hol}}$ and $\lambda^{-1}(0)\simeq X^{\mathrm{An}}_{0}$. For $0<\rho\leq 1$, the fiber $\lambda^{-1}(\rho)$ is also homeomorphic to $X^{\operatorname{hol}}$. In fact, we have a a homeomorphism

$$\lambda^{-1}\left((0,1]\right)\simeq(0,1]\times X^{\operatorname{hol}},$$

see [Berk09, Lemma 2.1].

Now consider the hybrid circle of radius $r\in(0,1)$, that is, $C_{\mathrm{hyb}}(r):=\{|{t}|=r\}\subset{\mathbb{A}}^{1,\mathrm{hyb}}=(\operatorname{Spec}{\mathbb{C}}[{t}])^{\mathrm{hyb}}$. By [Poi10, Prop 2.1.1], this is compact and realized as the Berkovich spectrum of the Banach ring

$$A_{r}:=\left\{f=\sum_{\alpha\in{\mathbb{Z}}}c_{\alpha}{t}^{\alpha}\in{\mathbb{C}}(\!({t})\!)\ \bigg|\ \|f\|_{\mathrm{hyb}}:=\sum_{\alpha\in{\mathbb{Z}}}\|c_{\alpha}\|_{\mathrm{hyb}}r^{\alpha}<+\infty\right\}.$$

Since $\|c_{\alpha}\|_{\mathrm{hyb}}\geq|c_{\alpha}|_{\infty}$, every $f\in A_{r}$ defines a continuous function $f^{\operatorname{hol}}$ on the punctured closed disc $\overline{{\mathbb{D}}}^{*}_{r}$ that is holomorphic on ${\mathbb{D}}^{*}_{r}$ and meromorphic at 0.

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 $X^{\mathrm{hyb}}$.