Let $D$ be an effective divisor with simple normal crossing (snc) support in a complex manifold ${\mathcal{X}}$. By definition, $D=\sum_{i\in I}b_{i}E_{i}$ with $b_{i}\in{\mathbb{N}}^{*}$ and $(E_{i})_{i\in I}$ a finite family of smooth irreducible divisors such that

$$E_{J}:=\bigcap_{i\in J}E_{i}$$

is either empty or smooth of codimension $|J|$ (with finitely many connected components) for each $\emptyset\neq J\subset I$. A connected component $Y$ of a non-empty $E_{J}$ is called a stratum. Together with ${\mathcal{X}}\setminus D=E_{\emptyset}$, the locally closed submanifolds $\mathring{Y}:=Y\setminus\bigcup_{i\in I\setminus J}E_{i}$ define a partition of ${\mathcal{X}}$.

The dual complex $\Delta(D)$ is the simplicial complex⁴⁴ 4 This is understood in the slightly generalized sense that the intersection of two faces is a union of common faces. defined as follows: to each stratum $Y$ corresponds a simplex

$$\sigma_{Y}=\left\{w\in{\mathbb{R}}_{+}^{J}\mid\sum_{i\in J}b_{i}w_{i}=1\right\},$$

and $\sigma_{Y}$ is a face of $\sigma_{Y^{\prime}}$ if and only if $Y^{\prime}\subset Y$. This description equips $\Delta(D)$ with an integral affine structure, by which we mean a compatible choice of integral affine structures on each simplex $\sigma$. This further induces a ${\mathbb{Z}}$-PA structure on $\Delta(D)$.

We write $Y_{\sigma}$ for the stratum of a face $\sigma$. Each point $\xi\in D$ belongs to $\mathring{Y_{\xi}}$ for a unique stratum $Y_{\xi}$, obtained as the connected component of $E_{J_{\xi}}$ containing $\xi$, with $J_{\xi}=\{i\in I\mid\xi\in E_{i}\}$. We denote by $\sigma_{\xi}:=\sigma_{Y_{\xi}}$ the corresponding face of $\Delta(D)$.

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.

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

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 $\Delta(D)$.

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 $K\setminus D$.