4.4. The limit hybrid space [0169]

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4.4. The limit hybrid space

Proposition 4.8 allows us to introduce

Definition 4.9.

The hybrid space associated to $X$ is the topological space

X^{\mathrm{hyb}}:=\varprojlim_{\mathcal{X}}{\mathcal{X}}^{\mathrm{hyb}},

where ${\mathcal{X}}$ runs over all snc models of $X$ .

Here $X^{\mathrm{hyb}}$ is equipped with the inverse limit topology. The maps $\pi\colon{\mathcal{X}}^{\mathrm{hyb}}\to{\mathbb{D}}$ define a continuous and proper map

\pi\colon X^{\mathrm{hyb}}\to{\mathbb{D}}

We can identify $X$ with the open subset $\pi^{-1}({\mathbb{D}}^{*})$ . Similarly, the compact subset $X^{\mathrm{hyb}}_{0}:=\pi^{-1}(0)$ can be identified with $\varprojlim_{\mathcal{X}}\Delta({\mathcal{X}})$ . For every snc model ${\mathcal{X}}$ we have, by the definition of the inverse limit, a continuous proper map $r_{\mathcal{X}}\colon X^{\mathrm{hyb}}\to{\mathcal{X}}^{\mathrm{hyb}}$ . We also have an embedding $i_{\mathcal{X}}\colon\Delta({\mathcal{X}})\to X^{\mathrm{hyb}}$ of $\Delta({\mathcal{X}})$ onto a closed subset of $X^{\mathrm{hyb}}_{0}$ . It satisfies $r_{\mathcal{X}}\circ i_{\mathcal{X}}=\operatorname{id}$ on $\Delta({\mathcal{X}})$ .

Remark 4.10.

It is not clear how to define a map $\operatorname{Log}\colon X^{\mathrm{hyb}}\to\varprojlim_{\mathcal{X}}\Delta({\mathcal{X}})$ , since each tropicalization map $\operatorname{Log}_{\mathcal{X}}$ is only defined on $X_{{\mathbb{D}}^{*}(r)}$ , where $r=r_{\mathcal{X}}$ depends on ${\mathcal{X}}$ . See §4.6 for a substitute in the projective case.