The following construction plays a crucial role for the understanding of the limit measure appearing in Corollary B.

Consider a model metric ${\mathcal{L}}$ of $K_{X}$ defined on a proper dlt model ${\mathcal{X}}$. Following §3.1 we explain how to associate a subklt pair $(Y,B^{\mathcal{L}}_{Y})$ to each stratum $Y$ of ${\mathcal{X}}_{0}$ corresponding to a maximal simplex in $\Delta({\mathcal{L}})$.

Let us first recall a few facts about adjunction. When ${\mathcal{X}}$ is an snc model, each stratum $Y$ comes with a boundary $B_{Y}:=\sum_{i\notin J_{Y}}E_{i}\cap Y$. Here $(Y,B_{Y})$ is log smooth, and

$$K^{\mathrm{log}}_{{\mathcal{X}}/S}\big|_{Y}=K_{(Y,B_{Y})}:=K_{Y}+B_{Y},$$

(5.6)

the identification being provided by Poincaré residues. When ${\mathcal{X}}$ is merely dlt, each stratum $Y$ is normal, and comes with a canonically defined effective ${\mathbb{Q}}$-divisor $B_{Y}$ such that $(Y,B_{Y})$ is dlt and still satisfies (5.6) (cf. [Kol13, 4.19]). We have

$$B_{Y}=\sum_{i\notin J_{Y}}E_{i}\cap Y+B^{\prime}_{Y}$$

where $B^{\prime}_{Y}$ is an effective ${\mathbb{Q}}$-divisor supported in the complement of ${\mathcal{X}}_{\mathrm{snc}}$.

Now let $\psi$ be a model metric on $K_{X}^{\mathrm{an}}$, determined by a model ${\mathcal{L}}$ of $K_{X}$ on a proper dlt model ${\mathcal{X}}$ of $X$. Introduce as before the function $\kappa:=A_{X}-\psi$ on $X^{\mathrm{an}}$, and note that the ${\mathbb{Q}}$-Cartier divisor

$$D:=K^{\mathrm{log}}_{{\mathcal{X}}/S}-{\mathcal{L}}-\kappa_{\min}{\mathcal{X}}_{0}=\sum_{i}(\kappa(v_{E_{i}})-\kappa_{\min})b_{i}E_{i}$$

is effective.

We emphasize that $B^{\mathcal{L}}_{Y}$ is not effective in general.