(2.2.1) A $dlt$-model of $X$ is a normal proper $\mathscr{C}$-model $\mathscr{X}$ of $X$ such that $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ is a $dlt$-pair. This means that $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ is log canonical and that each log canonical center of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ has non-empty intersection with $\mathscr{X}^{\mathrm{snc}}$. In particular, every proper $snc$-model of $X$ is a $dlt$-model. An equivalent formulation of the definition is the following: $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is $\mathbb{Q}$-Cartier, and for every log resolution $h:\mathscr{Y}\to\mathscr{X}$ of $(\mathscr{X},\mathscr{X}_{s})$ and every irreducible component $E$ of $\mathscr{Y}_{s}$, the multiplicity of $E$ in the log pullback $\Delta$ of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ is at most $1$. Moreover, if it is equal to $1$, then $h(E)$ must have non-empty intersection with $\mathscr{X}^{\mathrm{snc}}$. In practice, we will apply the $dlt$ property via Lemma 3.2.3 below.

(2.2.2) We say that a $dlt$-model $\mathscr{X}$ of $X$ is a good minimal model if $\mathscr{X}$ is $\mathbb{Q}$-factorial and $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is semi-ample over $\mathscr{C}$.

(2.2.3) For every $dlt$-model $\mathscr{X}$ of $X$, we can define the dual complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ for the $dlt$-pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ by gluing cells corresponding to irreducible components of intersections of irreducible components of $\mathscr{X}_{s}$, as in Definition 8 in [dFKX12]. When $k$ is not algebraically closed, we note that we only glue cells corresponding to irreducible components (instead of geometrically irreducible components). In other words, $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the quotient of the $\mathrm{Gal}(\bar{k}/k)$-equivariant dual complex constructed in [dFKX12, §31].

(2.2.4) For every $dlt$-model $\mathscr{X}$ of $X$, the log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ are the irreducible components of intersections of irreducible components of $(\mathscr{X}_{s})_{\mathrm{red}}$, by [Ko13, 4.16]. These are also precisely the closures in $\mathscr{X}_{s}$ of the connected components of intersections of irreducible components of $\mathscr{X}^{\mathrm{snc}}_{s}$ (since these connected components are the log canonical centers of $(\mathscr{X}^{\mathrm{snc}},(\mathscr{X}_{s}^{\mathrm{snc}})_{\mathrm{red}})$). Thus, the dual intersection complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the same as the dual intersection complex of the strict normal crossings divisor $\mathscr{X}^{\mathrm{snc}}_{s}$, and the cells of this complex correspond bijectively to the log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$. See Section 2 in [dFKX12] for more background.

(2.2.5) Let $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ be two $dlt$-models of $X$ over $\mathscr{C}$. We say that $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ are crepant birational if there exist a normal proper $\mathscr{C}$-model $\mathscr{Y}$ of $X$ and morphisms of $\mathscr{C}$-models $f_{i}:\mathscr{Y}\to\mathscr{X}_{i}$ for $i=1,2$ such that the log pullbacks of $(\mathscr{X}_{1,s})_{\mathrm{red}}$ and $(\mathscr{X}_{2,s})_{\mathrm{red}}$ coincide (see [Ko13, 2.23]). Note that we can always assume that $\mathscr{Y}$ is an $snc$-model, by taking a log resolution of $(\mathscr{Y},\mathscr{Y}_{s})$. The following theorem collects two fundamental results from the Minimal Model Program.