Let $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ be an snc degeneration, i.e. a proper, surjective holomorphic map from a connected complex manifold to the unit disc in ${\mathbb{C}}$, whose restriction to $X:=\pi^{-1}({\mathbb{D}}^{*})$ is a submersion and such that ${\mathcal{X}}_{0}:=\pi^{-1}(0)=\sum_{i\in I}b_{i}E_{i}$ has snc support. Note that $X_{t}:=\pi^{-1}(t)$ is non-singular for $t\in{\mathbb{D}}^{*}$. The dual complex $\Delta({\mathcal{X}})$ is defined as that of ${\mathcal{X}}_{0}$; it is equipped with its natural ${\mathbb{Z}}$-PA structure. The logarithmic canonical bundle of ${\mathcal{X}}$ is

$$K^{\mathrm{log}}_{\mathcal{X}}:=K_{\mathcal{X}}+{\mathcal{X}}_{0,\mathrm{red}}.$$

Setting $K^{\mathrm{log}}_{{\mathbb{D}}}:=K_{\mathbb{D}}+[0]$, we define the relative logarithmic canonical bundle as

$$K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}:=K^{\mathrm{log}}_{\mathcal{X}}-\pi^{*}K^{\mathrm{log}}_{{\mathbb{D}}}=K_{{\mathcal{X}}/{\mathbb{D}}}+{\mathcal{X}}_{0,\mathrm{red}}-{\mathcal{X}}_{0}.$$

Now suppose we are given a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$. We then have a unique decomposition

$$K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}={\mathcal{L}}+\sum_{i\in I}a_{i}E_{i}$$

with $a_{i}\in{\mathbb{Q}}$. Set $\kappa_{i}:=a_{i}/b_{i}$ and $\kappa_{\min}:=\min_{i}\kappa_{i}$.

In general, $\Delta({\mathcal{L}})$ is neither connected nor pure dimensional. We say that a face of $\Delta({\mathcal{L}})$ is maximal if it is not contained in a larger face of $\Delta({\mathcal{L}})$.

If $\psi$ is a continuous metric on ${\mathcal{L}}$, $\psi|_{Y}$ may thus be viewed as a metric on $K_{(Y,B^{\mathcal{L}}_{Y})}$. When $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$, the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt, and Lemma 1.1 applies. This leads to the following notion.

This measure can be more explicitly described as follows. At each point $\xi\in Y$, pick local coordinates $(z_{0},\dots,z_{n})$ such that $z_{0},\dots,z_{p}$ are local equations for the components $E_{0},\dots,E_{p}$ of ${\mathcal{X}}_{0}$ that pass through $\xi$, indexed so that $J=\{0,\dots,d\}$, where $0\leq d\leq p$, and such that $t=\prod_{j=0}^{p}z_{j}^{b_{j}}$ The logarithmic form

$$\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}$$

is a local trivialization of $K^{\mathrm{log}}_{{\mathcal{X}}}$, and hence induces a local trivialization $\Omega^{\mathrm{rel}}=\Omega\otimes(dt/t)^{-1}$ of $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}$. We may then view $\tau:=\prod_{i=0}^{p}z_{i}^{a_{i}}\Omega^{\mathrm{rel}}$ as a local ${\mathbb{Q}}$-generator of ${\mathcal{L}}$. Under the identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$, we have

$$\tau|_{Y}=\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\operatorname{Res}_{Y}(\Omega)$$

with

$$\operatorname{Res}_{Y}(\Omega)=\frac{dz_{d+1}}{z_{d+1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\bigg|_{Y}.$$

We infer

$$\operatorname{Res}_{Y}(\psi)=|\tau|^{-2}_{\psi}\prod_{i=d+1}^{p}|z_{i}|^{2(a_{i}-\kappa_{\min}b_{i}-1)}\bigg|\bigwedge_{i=d+1}^{n}dz_{i}\bigg|^{2}.$$

(3.1)