Subsubsection [04W0]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

(4.1.3) For the reader’s convenience, we include some basic facts about adjunction for $dlt$ -pairs. We refer to Chapter 4 of [Ko13] for more background. Let $(Y,\Delta)$ be a $dlt$ -pair over $k$ , and let $D$ be a log canonical center of $(Y,\Delta)$ . Then $D$ is normal, by [Ko13, 4.16]. There is a well defined $\mathbb{Q}$ -divisor $\Delta_{D}$ on $D$ , called the different of $\Delta$ on $D$ [Ko13, 4.18], which is induced by the Poincaré map and satisfies the equation

(K_{Y}+\Delta)|_{D}=K_{D}+\Delta_{D}.

In the sequel, whenever we write such an equation it will be understood that $\Delta_{D}$ is the different of $\Delta$ on $D$ . The pair $(D,\Delta_{D})$ is again a $dlt$ -pair, by [Ko13, 4.19]. Write $\lfloor\Delta\rfloor=\sum_{i\in I}D_{i}$ . If $J$ is a subset of $I$ and $D$ is a component of $\bigcap_{j\in J}{\Delta}_{j}$ , it is not hard to see that for every non-empty subset $J^{\prime}$ of $I\setminus J$ , every irreducible component of the intersection

D\cap\bigcap_{j\in J^{\prime}}\Delta_{j}

is a log canonical center of $(D,\Delta_{D})$ (see [Ko13, 4.19]). Conversely, by repeatedly using inversion of adjunction [Ko13, 4.9], one sees that any log canonical center of $(D,\Delta_{D})$ is a log canonical center of $(Y,\Delta)$ , and thus an irreducible component of an intersection $D\cap\bigcap_{j\in J^{\prime}}\Delta_{j}$ for some non-empty subset $J^{\prime}$ of $I\setminus J$ .

Theorem 4.1.4.

Assume that $K_{X}$ is $\mathbb{Q}$ -linearly equivalent to $0$ over $C$ . Then the underlying topological space of $\mathrm{Sk}(X_{K})$ is a a pseudo-manifold with boundary.

Proof.

As we mentioned above, this result is essentially contained in [KK10, Ko11]. Using the terminology there, properties (1)-(3) of a pseudo-manifold all follow from the fact that two minimal log canonical centers of a log crepant structure are $\mathbb{P}^{1}$ -linked in the sense of Definition 9 in [Ko11]. We will now explain this in more detail. We denote by $n$ the relative dimension of $X$ over $C$ .

By Theorem 2.2.6(1), there exists a a good minimal $dlt$ -model $\mathscr{X}$ of $X$ over $\mathscr{C}$ . By Theorem 3.3.4, we have $\mathrm{Sk}(X_{K})=\mathrm{Sk}(\mathscr{X})$ . As a triangulation on $\mathrm{Sk}(\mathscr{X})$ , we take the first barycentric subdivision of the simplicial structure on $\mathrm{Sk}(\mathscr{X})$ . This barycentric subdivision is necessary to guarantee that the intersection of two faces is a codimension one face of both, rather than a union of faces (think of a type $I_{2}$ degeneration of elliptic curves, whose skeleton consists of two vertices joined by two edges).

We choose an integer $m>0$ such that $mK_{X}\sim 0$ . Since the divisor $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ is semi-ample over $\mathscr{C}$ and trivial over $C$ , we see that $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ must be a multiple of $\mathscr{X}_{s}$ and thus trivial over $\mathscr{C}$ . Thus we can apply Theorem 10 in [Ko11] to the $dlt$ -pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ over $\mathscr{C}$ . It states that every two minimal log canonical centers $D$ and $D^{*}$ of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ are $\mathbb{P}^{1}$ -linked. This means, in particular, that they have the same dimension, say $n-d$ , and that there exist a sequence of $(n-d+1)$ -dimensional log canonical centers $E_{1},E_{2},\ldots,E_{\ell}$ and a sequence of $(n-d)$ -dimensional log canonical centers $D=D_{0},D_{1},\ldots,D_{\ell}=D^{*}$ such that $D_{i-1},D_{i}\subset E_{i}$ for $1\leq i\leq\ell$ . In this way, we obtain properties (1) and (3) of a pseudo-manifold with boundary.

If we have two minimal log canonical centers $D_{1},\,D_{2}$ of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ , contained in an $(n-d+1)$ -dimensional log canonical center $E$ , and if we write

(K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}})|_{E}=K_{E}+D_{1}+D_{2}+\Delta

for some $\Delta\geq 0$ , then $(E,D_{1}+D_{2}+\Delta)$ is again a $dlt$ -pair [Ko13, 4.19]. Moreover, $D_{1}$ cannot intersect $D_{2}$ or $\lfloor\Delta\rfloor$ because the intersection would be a union of log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ , which contradicts the minimality of $D_{1}$ . Thus we are in the situation of the second part of the proof of Theorem 10 in [Ko11]. That proof shows that $D_{1}$ and $D_{2}$ are the only log canonical centers of $(E,D_{1}+D_{2}+\Delta)$ . Property (2) follows. ∎