3.2. Embedding the dual complex in the Berkovich space [01EF]

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3.2. Embedding the dual complex in the Berkovich space

Theorem 3.1.

Let $\mathcal{X}$ be any SNC model of $X$ .

(i)

The image of the evaluation map $\ev_{\mathcal{X}}:X\to\Div_{0}(\mathcal{X})^{*}_{\mathbf{R}}$ coincides with $\Delta_{\mathcal{X}}$ .
(ii)
There exists a unique continuous (injective) map $\emb_{\mathcal{X}}:\Delta_{\mathcal{X}}\to X$ such that:
- (a)
  
  $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}$ is the identity on $\Delta_{\mathcal{X}}$ ;
- (b)
  
  for $s\in\Delta_{\mathcal{X}}$ , the center of $\emb_{\mathcal{X}}(s)$ on $\mathcal{X}$ is the generic point $\xi_{J}$ of $E_{J}$ for the unique subset $J\subset I$ such that $s$ is contained in the relative interior of $\sigma_{J}$ .

The proof is given in §3.3. Let us derive some consequences.

For any two models $\pi:\mathcal{X}\to\mathcal{Y}$ , observe that the natural map ${}^{t}\pi^{*}:\Div_{0}(\mathcal{X})^{*}\to\Div_{0}(\mathcal{Y})^{*}$ maps $\Delta_{\mathcal{X}}$ onto $\Delta_{\mathcal{Y}}$ since ${}^{t}\pi^{*}\circ\ev_{\mathcal{X}}=\ev_{\mathcal{Y}}$ by definition. We may thus form the projective limit $\varprojlim_{\mathcal{X}\ \text{SNC}}\Delta_{\mathcal{X}}$ , and we have

Corollary 3.2.

The maps $\ev_{\mathcal{X}}:X\to\Delta_{\mathcal{X}}\subset\Div_{0}(\mathcal{X})_{\mathbf{R}}^{*}$ induce a homeomorphism

(3.1)

\ev:X\to\varprojlim_{\mathcal{X}\ \text{SNC}}\Delta_{\mathcal{X}}.

Proof.

The map $\ev$ is well-defined by Theorem 3.1 (i). It is a homeomorphism onto its image by Corollary 2.5 and the fact that any model is dominated by an SNC model. As $X$ is compact, we only need to show that $\ev(X)$ is dense in $\varprojlim\Delta_{\mathcal{X}}$ . Pick $s=(s_{\mathcal{X}})_{\mathcal{X}}\in\varprojlim\Delta_{\mathcal{X}}$ and fix an SNC model $\mathcal{X}$ . If $\mathcal{Y}$ is an SNC model dominated by $\mathcal{X}$ , then $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}=\id$ yields $\ev_{\mathcal{Y}}(\emb_{\mathcal{X}}(s_{\mathcal{X}}))=s_{\mathcal{Y}}$ . Hence $s=\lim_{\mathcal{X}}\ev(\emb_{\mathcal{X}}(s_{\mathcal{X}}))\in\overline{\ev(X)}$ . ∎

Definition 3.3.

For any SNC model $\mathcal{X}$ we define a continuous map $p_{\mathcal{X}}:X\to X$ by

p_{\mathcal{X}}:=\emb_{\mathcal{X}}\circ\ev_{\mathcal{X}}.

It follows from Theorem 3.1 that $p_{\mathcal{X}}$ satisfies $p_{\mathcal{X}}\circ p_{\mathcal{X}}=p_{\mathcal{X}}$ and $p_{\mathcal{X}}(x)=x$ iff $x\in\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$ . Hence we view $p_{\mathcal{X}}$ as a retraction of $X$ onto the image of the embedding $\emb_{\mathcal{X}}:\Delta_{\mathcal{X}}\to X$ .

Lemma 3.4.

The retraction map $p_{\mathcal{X}}$ satisfies the following properties:

(i)

$c_{\mathcal{X}}(x)\in\overline{\{c_{\mathcal{X}}(p_{\mathcal{X}}(x))\}}$ for all $x\in X$ .
(ii)

$\varphi_{D}\circ p_{\mathcal{X}}=\varphi_{D}$ for all $D\in\Div_{0}(\mathcal{X})_{\mathbf{R}}$ .

Proof.

By definition of $c_{\mathcal{X}}$ we have $c_{\mathcal{X}}(x)\in E_{i}$ for a given $i\in I$ iff $\langle\ev_{\mathcal{X}}(x),E_{i}\rangle>0$ , and it follows that $\ev_{\mathcal{X}}(x)$ lies in the relative interior of the simplex $\sigma_{J}$ for the maximal $J\subset I$ such that $c_{\mathcal{X}}(x)\in E_{J}$ . Property (b) in Theorem 3.1 then shows that $c_{\mathcal{X}}(p_{\mathcal{X}}(x))$ is the generic point of $E_{J}$ , which proves (i).

Let us prove (ii). For each $x\in X$ we have

\varphi_{D}(p_{\mathcal{X}}(x))=\langle D,\ev_{\mathcal{X}}(p_{\mathcal{X}}(x))\rangle=\langle D,\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}\circ\ev_{\mathcal{X}}(x)\rangle=\langle D,\ev_{\mathcal{X}}(x)\rangle=\varphi_{D}(x),

using the identity $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}=\id$ .

∎

Proposition 3.5.

If $\mathcal{X}\geq\mathcal{Y}$ are two SNC models, then

(i)

$\ev_{\mathcal{Y}}\circ p_{\mathcal{X}}=\ev_{\mathcal{Y}}$ and $p_{\mathcal{Y}}\circ p_{\mathcal{X}}=p_{\mathcal{Y}}$ .
(ii)

$p_{\mathcal{X}}\circ\emb_{\mathcal{Y}}=\emb_{\mathcal{Y}}$ .

Note that (ii) says that the image in $X$ of $\Delta_{\mathcal{Y}}$ is contained in the image of $\Delta_{\mathcal{X}}$ .

Proof.

(i) amounts to the fact that $\varphi_{D}\circ p_{\mathcal{X}}=\varphi_{D}$ for all $D\in\Div_{0}(\mathcal{Y})$ , which is a special case of Lemma 3.4.

Let us now prove (ii). The map $\emb^{\prime}_{\mathcal{Y}}:=p_{\mathcal{X}}\circ\emb_{\mathcal{Y}}:\Delta_{\mathcal{Y}}\to X$ is continuous, and the previous identity implies that $\ev_{\mathcal{Y}}\circ\emb^{\prime}_{\mathcal{Y}}=\ev_{\mathcal{Y}}\circ\emb_{\mathcal{Y}}=\id$ . By the uniqueness part of Theorem 3.1 it suffices to prove that $c_{\mathcal{Y}}\circ\emb^{\prime}_{\mathcal{Y}}=c_{\mathcal{Y}}\circ\emb_{\mathcal{Y}}$ on $\Delta_{\mathcal{Y}}$ . Pick $s\in\Delta_{\mathcal{Y}}$ and set $x:=\emb_{\mathcal{Y}}(s)$ , $x^{\prime}:=\emb^{\prime}_{\mathcal{Y}}(s)$ . On the one hand (i) shows that

p_{\mathcal{Y}}(x^{\prime})=p_{\mathcal{Y}}\circ p_{\mathcal{X}}\circ\emb_{\mathcal{Y}}(s)=p_{\mathcal{Y}}\circ\emb_{\mathcal{Y}}(s)=x,

so $c_{\mathcal{Y}}(x^{\prime})\in\overline{\{c_{\mathcal{Y}}(x)\}}$ by (i) of Lemma 3.4. On the other hand $p_{\mathcal{X}}(x)=x^{\prime}$ by definition, so $c_{\mathcal{X}}(x)\in\overline{\{c_{\mathcal{X}}(x^{\prime})\}}$ and hence $c_{\mathcal{Y}}(x)\in\overline{\{c_{\mathcal{Y}}(x^{\prime})\}}$ by continuity of the map $\mathcal{X}\to\mathcal{Y}$ for the Zariski topology. ∎

Definition 3.6.

We define the subset $X^{\mathrm{qm}}\subset X$ of quasimonomial points as

X^{\mathrm{qm}}:=\bigcup_{\mathcal{X}}\emb_{\mathcal{X}}(\Delta_{\mathcal{X}}),

where $\mathcal{X}$ ranges over SNC models of $\mathcal{X}$ .

Corollary 3.7.

We have $\lim_{\mathcal{X}}p_{\mathcal{X}}=\id$ pointwise on $X$ . Hence $X^{\mathrm{qm}}$ is dense in $X$ .

Of course, we already knew from Corollary 2.4 that $X^{\mathrm{div}}\subset X^{\mathrm{qm}}$ is dense in $X$ .

Proof.

By Corollary 3.2 it suffices to show that $\lim_{\mathcal{X}}\ev\circ p_{\mathcal{X}}=\ev$ , which amounts to proving $\lim_{\mathcal{X}}\ev_{\mathcal{Y}}\circ p_{\mathcal{X}}=\ev_{\mathcal{Y}}$ for each $\mathcal{Y}$ . This follows from (i) of Proposition 3.5. ∎