Now consider the case when $X\to{\mathbb{D}}^{*}$ is projective.⁵⁵ 5 In the projective case, the existence of the spaces ${\mathcal{X}}^{\mathrm{hyb}}$ and $X^{\mathrm{hyb}}$ was observed by Kontsevich and Soibelman, see [KS06, p.383]. As we now explain, we can then view $X^{\mathrm{hyb}}$ and its central fiber as analytic spaces.

The projectivity assumption means that $X$ can be viewed as a smooth subspace ${\mathbb{P}}^{N}\times{\mathbb{D}}^{*}$, defined by homogeneous polynomials with coefficients that are holomorphic functions on ${\mathbb{D}}^{*}$ and meromorphic at $0\in{\mathbb{D}}$.

We can view these coefficients as complex formal Laurent series, that is, elements of the field $K:={\mathbb{C}}(\!({t})\!)$. Given $r\in(0,1)$, this field admits a natural non-Archimedean absolute value that is trivial on ${\mathbb{C}}^{*}$ and normalized by $|{t}|=r$. In other words, we have $|\sum_{j}a_{j}t^{j}|=r^{\min\{j\mid a_{j}\neq 0\}}$.

Further, the equations defining $X$ now define a smooth projective variety $X_{K}$ over the field $K$. To this variety we can associate a non-Archimedean space $X_{K}^{\mathrm{an}}$, namely the Berkovich analytification of $X_{K}$ with respect to non-Archimedean norm on $K$. This is a connected and locally connected compact (Hausdorff) space.

We claim that $X^{\mathrm{hyb}}_{0}$ is homeomorphic on $X_{K}^{\mathrm{an}}$. To see this, we note that, for the same reasons as above, every projective snc model ${\mathcal{X}}\to{\mathbb{D}}$ of $X$ defines a projective snc model ${\mathcal{X}}_{R}$ of $X_{K}$ over the valuation ring $R={\mathbb{C}}[\![{t}]\!]$ of $K$. Further, the dual complex $\Delta({\mathcal{X}})$ of ${\mathcal{X}}$ can be identified with the dual complex $\Delta({\mathcal{X}}_{R})$ of ${\mathcal{X}}_{R}$. Now, there exists a canonical retraction map $r_{\mathcal{X}}\colon X_{K}\to\Delta({\mathcal{X}}_{K})$, and we have

$$X_{K}^{\mathrm{an}}\overset{\sim}{\to}\varprojlim_{{\mathcal{X}}\ \text{projective snc}}\Delta({\mathcal{X}}).$$

(4.3)

This was announced in [KS06, Theorem 10, p.383]; see e.g.  [BFJ16, Corollary 3.2] for details. On the other hand, Lemma 4.1 implies that in $X^{\mathrm{hyb}}_{0}=\varprojlim_{\mathcal{X}}\Delta({\mathcal{X}})$, we may take the limit over projective snc models. This implies that $X^{\mathrm{hyb}}_{0}\simeq X_{K}^{\mathrm{an}}$.

Next we analyze the space $X^{\mathrm{hyb}}$ itself, using Appendix A. Fix $0<r<1$ and consider the Banach ring

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

where $\|\cdot\|_{\mathrm{hyb}}$ is the maximum of the usual norm and the trivial norm on ${\mathbb{C}}$. The Berkovich spectrum ${\mathcal{M}}(A_{r})$ of $A_{r}$ is homeomorphic to $\overline{{\mathbb{D}}}_{r}$.

Every function that is holomorphic on ${\mathbb{D}}^{*}$ and meromorphic at $0\in{\mathbb{D}}$ defines an element of $A_{r}$. Hence we can define the base change $X_{A_{r}}\subset{\mathbb{P}}^{N}_{A_{r}}$ using the same homogeneous equations as above. Then $X_{A_{r}}$ is a scheme of finite type over $A_{r}$, so its analytification $X_{A_{r}}^{\mathrm{An}}$ is a compact Hausdorff space with a continuous map $\pi_{r}$ onto $(\operatorname{Spec}A_{r})^{\mathrm{An}}={\mathcal{M}}(A_{r})\simeq\overline{{\mathbb{D}}}_{r}$. (In Appendix A.6, this analytification is denoted by $X^{\mathrm{hyb}}$, but here we use $X_{A_{r}}^{\mathrm{An}}$ for clarity.) We have a homeomorphism

$$\tau\colon\pi_{r}^{-1}(\overline{{\mathbb{D}}}^{*}_{r})\overset{\sim}{\to}X_{\overline{{\mathbb{D}}}^{*}_{r}}\overset{\sim}{\to}X^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}^{*}_{r}}$$

(4.4)

and another homeomorphism

$$\tau_{0}\colon\pi^{-1}(0)\overset{\sim}{\to}X^{\mathrm{hyb}}_{0}\overset{\sim}{\to}X_{K}^{\mathrm{an}}.$$

(4.5)