Let ${\mathcal{X}}$ be a semistable model, i.e. ${\mathcal{X}}$ is snc with ${\mathcal{X}}_{0}$ reduced. By (7.1), we have $\kappa_{\min}\in{\mathbb{Z}}$. Since some non-empty $E_{J}$ might have several components, the dual complex $\Delta({\mathcal{X}})$ is possibly not a triangulation of $\operatorname{Sk}(X)$. However, the barycentric subdivision $\Delta^{\prime}$ of $\Delta({\mathcal{X}})$ is a triangulation; the corresponding toroidal modification ${\mathcal{X}}^{\prime}$ is snc, with ${\mathcal{X}}^{\prime}_{0}$ is possibly non-reduced, but $b_{\sigma}=1$ for each $n$-simplex $\sigma$ of $\Delta^{\prime}$. Applying the above discussion to ${\mathcal{X}}^{\prime}$, we infer

with $\sigma$ ranging over the $n$-dimensional faces of $\Delta^{\prime}$, with corresponding strata $y_{\sigma}\in{\mathcal{X}}^{\prime}_{0}$ reduced to single points. It will thus be enough to show that $|\operatorname{Res}_{y_{\sigma}}(\omega)|$ is independent of $\sigma$.

By the strong connectedness property, any two $n$-simplices $\sigma$, $\sigma^{\prime}$ of $\Delta^{\prime}$ can be joined by a chain of $n$-simplices $\sigma=\sigma_{1},\dots,\sigma_{N}=\sigma^{\prime}$ with $\sigma_{i}$ and $\sigma_{i+1}$ sharing a common $(n-1)$-face $\tau_{i}$. Denoting by $y_{i}=y_{\sigma_{i}}$ and $Y_{i}=Y_{\tau_{i}}$ the corresponding strata in ${\mathcal{X}}^{\prime}$, we thus have $y_{i},y_{i+1}\in Y_{i}$. Further, the Poincaré residue $\operatorname{Res}_{Y_{i}}(\omega)$ has poles precisely at $y_{i},y_{i+1}$, since any other pole would correspond to an $n$-simplex of $\Delta^{\prime}$ containing $\tau_{i}$, contradicting the non-branching property. Since $\operatorname{Res}_{y_{i}}\operatorname{Res}_{Y}(\omega)=\operatorname{Res}_{y_{i}}(\omega)$, the residue theorem applied to the Riemann surface $Y_{i}$ yields $\operatorname{Res}_{y_{i}}(\omega)+\operatorname{Res}_{y_{i+1}}(\omega)=0$, and hence $|\operatorname{Res}_{y_{1}}(\omega)|=\dots=|\operatorname{Res}_{y_{N}}(\omega)|$. ∎