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 $\Delta(D)$.