Let ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ be a residually metrized model of $K_{X}$, determined on a proper dlt model ${\mathcal{X}}$. If $Y$ is a stratum corresponding to a top-dimensional face of $\Delta({\mathcal{L}})$, Lemma 5.12 shows that the restriction of $\psi_{0}$ to ${\mathcal{L}}|_{Y}$ induces a Hermitian metric $\psi_{Y}$ on $K_{(Y,B^{\mathcal{L}}_{Y})}:=K_{Y}+B^{\mathcal{L}}_{Y}$, with $(Y,B^{\mathcal{L}}_{Y})$ subklt. By Lemma 1.1, we may thus introduce:

This definition is of course compatible with one in §3.1, and can be more explicitly described as follows. Let $\xi$ be a (closed) point of $Y\cap{\mathcal{X}}_{\mathrm{snc}}$, index the irreducible components $E_{0},\dots,E_{p}$ passing through $\xi$ so that $Y$ is a component of $\bigcap_{0\leq i\leq d}E_{i}$ with $d=\dim\Delta({\mathcal{L}})\leq p$. In the notation of Example 5.1, the Poincaré residue

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

is a generator of $K_{(Y,B_{Y})}=K^{\mathrm{log}}_{{\mathcal{X}}/S}\big|_{Y}$. Setting $a_{i}:=\kappa(v_{E_{i}})b_{i}\in{\mathbb{Q}}$, we have

$$K^{\mathrm{log}}_{{\mathcal{X}}/S}={\mathcal{L}}+\sum_{i}a_{i}E_{i},$$

and we may thus view

$$\tau:={t}^{\kappa_{\min}}\prod_{i=0}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}={t}^{\kappa_{\min}}\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}$$

as a local ${\mathbb{Q}}$-generator of ${\mathcal{L}}$. Further, $B^{\mathcal{L}}_{Y}=\sum_{i=d+1}^{p}(1-(a_{i}-\kappa_{\min}b_{i}))E_{i}|_{Y}$, and $\tau|_{Y}$ corresponds to

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

under the identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$. We arrive at

	$\displaystyle\operatorname{Res}_{Y}({\mathcal{L}}^{\#})$	$\displaystyle=\frac{\prod_{i=d+1}^{p}|z_{i}|^{2(a_{i}-\kappa_{\min}b_{i})}}{\left|{t}^{\kappa_{\min}}\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}\right|^{2}_{\psi_{0}}}\left|\operatorname{Res}_{Y}(\Omega)\right|^{2}$
		$\displaystyle=\frac{\prod_{i=d+1}^{p}|z_{i}|^{2(a_{i}-\kappa_{\min}b_{i}-1)}}{\left|{t}^{\kappa_{\min}}\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}\right|^{2}_{\psi_{0}}}|dz_{d+1}\wedge\dots\wedge dz_{n}|^{2}.$		(6.1)