2.2. The hybrid topology [0155]

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2.2. The hybrid topology

Next we define a natural topology on the disjoint union

{\mathcal{X}}^{\mathrm{hyb}}:=({\mathcal{X}}\setminus D)\coprod\Delta(D).

Consider a connected open set ${\mathcal{U}}\subset{\mathcal{X}}$ meeting $D$ and local coordinates $z=(z_{0},\dots,z_{n})$ on ${\mathcal{U}}$ . We say that the pair $({\mathcal{U}},z)$ is adapted (to $D$ ) if the following conditions hold:

(i)

if $E_{0},\dots,E_{p}$ are the irreducible components of $D$ intersecting ${\mathcal{U}}$ , then we have ${\mathcal{U}}\cap E_{0}\cap\dots\cap E_{p}={\mathcal{U}}\cap\mathring{Y}$ for a component $Y$ of $E_{0}\cap\dots\cap E_{p}$ ;
(ii)

$z_{i}$ is an equation of $E_{i}\cap{\mathcal{U}}$ with $|z_{i}|<1$ , $0\leq i\leq p$ .

We call $Y=Y_{\mathcal{U}}$ the stratum of ${\mathcal{U}}$ , and denote by

\sigma_{\mathcal{U}}=\left\{w\in{\mathbb{R}}^{p+1}\mid\sum_{i=0}^{p}b_{i}w_{i}=1\right\}

the corresponding face of $\Delta(D)$ . The function $f_{{\mathcal{U}},z}:=\prod_{i=0}^{p}z_{i}^{b_{i}}$ is an equation of $D$ in ${\mathcal{U}}$ , with $|f_{{\mathcal{U}},z}|<1$ , and we get a continuous map $\operatorname{Log}_{{\mathcal{U}}}\colon{\mathcal{U}}\setminus D\to\sigma_{Y}$ by setting

\operatorname{Log}_{{\mathcal{U}}}=\left(\frac{\log|z_{i}|}{\log|f_{\mathcal{U}}|}\right)_{0\leq i\leq p}.

For any two adapted coordinate charts $({\mathcal{U}},z)$ , $({\mathcal{U}}^{\prime},z^{\prime})$ , with the same stratum $Y$ , we have $z^{\prime}_{i}=u_{i}z_{i}$ with $u_{i}$ nonvanishing on ${\mathcal{U}}\cap{\mathcal{U}}^{\prime}$ , for $i=0,\dots,p$ (after a possible reindexing); it follows that

\operatorname{Log}_{{\mathcal{U}}^{\prime}}=\operatorname{Log}_{{\mathcal{U}}}+O\left(\frac{1}{\log|f_{{\mathcal{U}},z}|^{-1}}\right)

(2.1)

locally uniformly on ${\mathcal{U}}\cap{\mathcal{U}}^{\prime}$ . We next show how to globalize this construction.

Proposition 2.1.

There exists an open neighborhood ${\mathcal{V}}\subset{\mathcal{X}}$ of $D$ and a continuous map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}\setminus D\to\Delta(D)$ such that for each adapted coordinate chart $({\mathcal{U}},z)$ with ${\mathcal{U}}\subset{\mathcal{V}}$ we have $\operatorname{Log}_{\mathcal{V}}({\mathcal{U}}\setminus D)\subset\sigma_{\mathcal{U}}$ and

\operatorname{Log}_{\mathcal{V}}=\operatorname{Log}_{{\mathcal{U}}}+O\left(\frac{1}{\log|f_{{\mathcal{U}},z}|^{-1}}\right)

(2.2)

uniformly on compact subsets of ${\mathcal{U}}$ .

This will be accomplished by means of a partition of unity, using the following elementary special case of [Cle77, Theorem 5.7].

Lemma 2.2.

There exists a family $(({\mathcal{V}}_{\alpha},z_{\alpha}))_{\alpha\in A}$ of adapted coordinate charts, such that $({\mathcal{V}}_{\alpha})_{\alpha}$ forms a locally finite covering of $D$ and such that the strata $Y_{\alpha}$ of the ${\mathcal{V}}_{\alpha}$ satisfy

\bigcap_{\beta\in B}{\mathcal{V}}_{\beta}\neq\emptyset\Longrightarrow\bigcap_{\beta\in B}Y_{\beta}\neq\emptyset

(2.3)

for every finite $B\subset A$ .

Proof of Proposition 2.1.

Pick an open cover $({\mathcal{V}}_{\alpha})_{\alpha}$ as in Lemma 2.2, and denote by $\operatorname{Log}_{\alpha}\colon{\mathcal{V}}_{\alpha}\setminus D\to\sigma_{\alpha}$ the corresponding maps. Set ${\mathcal{V}}:=\bigcup_{\alpha}{\mathcal{V}}_{\alpha}$ , and pick a partition of unity $(\chi_{\alpha})$ subordinate to $({\mathcal{V}}_{\alpha})$ . We claim that for each $\xi\in{\mathcal{V}}$ there exists an open neighborhood $W$ of $\xi$ and a face $\sigma_{W}$ of $\Delta(D)$ such that

W\cap\operatorname{supp}\chi_{\alpha}\neq\emptyset\Longrightarrow\sigma_{\alpha}\subset\sigma_{W}

for any $\alpha\in A$ . Indeed, using (2.3) it is easy to see that

W:=\bigcap_{\alpha\mid\xi\in{\mathcal{U}}_{\alpha}}{\mathcal{V}}_{\alpha}\setminus\bigcup_{\alpha\mid\xi\notin\operatorname{supp}\chi_{\beta}}\operatorname{supp}\chi_{\beta}

satisfies this property. By convexity of $\sigma_{W}$ , it follows that $\operatorname{Log}_{\mathcal{V}}:=\sum_{\alpha}\chi_{\alpha}\operatorname{Log}_{{\mathcal{V}}_{\alpha}}$ is well-defined on $W\setminus D$ , and hence yields a continuous map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}\setminus D\to\Delta(D)$ . The last property is a direct consequence of (2.1). ∎

We extend the previous map as

\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}^{\mathrm{hyb}}:=({\mathcal{V}}\setminus D)\cup\Delta(D)\to\Delta(D)

by setting $\operatorname{Log}_{\mathcal{V}}=\operatorname{id}$ on $\Delta(D)$ .

Definition 2.3.

The hybrid topology on ${\mathcal{X}}^{\mathrm{hyb}}:=({\mathcal{X}}\setminus D)\cup\Delta(D)$ is defined as the coarsest topology such that:

(i)

${\mathcal{X}}\setminus D\hookrightarrow{\mathcal{X}}^{\mathrm{hyb}}$ is an open embedding;
(ii)

For every open neighborhood ${\mathcal{V}}$ of $D$ in ${\mathcal{X}}$ , the set $({\mathcal{V}}\setminus D)\cup\Delta({\mathcal{X}})$ is open in ${\mathcal{X}}^{\mathrm{hyb}}$ ;
(iii)

$\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}^{\mathrm{hyb}}\to\Delta(D)$ is continuous.

Using (2.2), this definition is easily seen to be independent of the choice of map $\operatorname{Log}_{\mathcal{V}}$ . If $D$ is compact and $K\subset{\mathcal{X}}$ is a compact neighborhood of $D$ , then one easily checks that the corresponding subset $K^{\mathrm{hyb}}=(K\setminus D)\cup\Delta(D)$ is compact (Hausdorff). When $D=b_{0}E_{0}$ has only one irreducible component, $K^{\mathrm{hyb}}$ is simply the Tychonoff one-point compactification of $K\setminus D$ .

Example 2.4.

Set ${\mathcal{X}}={\mathbb{D}}^{2}$ and $D=E_{0}+E_{1}$ the union of the coordinate axes, with coordinates $(z_{0},z_{1})$ . Then ${\mathcal{U}}={\mathcal{X}}$ is itself an adapted coordinate chart. In these coordinates, $\operatorname{Log}_{{\mathcal{U}}}\colon{\mathcal{U}}\setminus D\to\sigma_{\mathcal{U}}$ becomes the map $({\mathbb{D}}^{*})^{2}\to[0,1]$ sending $(z_{0},z_{1})$ to $\log|z_{1}|/\log|z_{0}z_{1}|$ . As a consequence, given $t\in{\mathbb{R}}_{+}^{*}$ and $0<\varepsilon\ll 1$ , the closure in ${\mathcal{X}}^{\mathrm{hyb}}$ of the closed subset

F_{\varepsilon}:=\{0<|z_{0}|,|z_{1}|\leq\varepsilon,|z_{0}|^{t+\varepsilon}\leq|z_{1}|\leq|z_{0}|^{t-\varepsilon}\}\subset{\mathbb{D}}^{2}

is given by ${\bar{F}}_{\varepsilon}=F_{\varepsilon}\cup I_{\varepsilon}$ , where $I_{\varepsilon}:=\{t\in[0,1]\mid\frac{t-\varepsilon}{1+t-\varepsilon}\leq t\leq\frac{t+\varepsilon}{1+t+\varepsilon}\}$ . Further, the sets ${\bar{F}}_{\varepsilon}$ , for $0<\varepsilon\ll 1$ form a basis of closed neighborhoods of the point $\frac{t}{1+t}\in[0,1]$ in ${\mathcal{X}}^{\mathrm{hyb}}$ . See Figure 1.

Original source figure — Figure 1. The figure shows the closed subset $F_{\varepsilon}$ in Example 2.4.