5.7. The skeleton of a metric on K X [016Y]

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5.7. The skeleton of a metric on $K_{X}$

The purpose of this section is to introduce and study a slight generalization of the Kontsevich–Soibelman skeleton introduced in [KS06] and further analyzed in [MN15, NX13].

Definition 5.8.

If $\psi$ is a continuous (or usc) metric on $K_{X}^{\mathrm{an}}$ , set $\kappa:=A_{X}-\psi$ and $\kappa_{\min}:=\inf_{X^{\mathrm{an}}}\kappa$ . The skeleton of $\psi$ is the compact set

\operatorname{Sk}(\psi)=\left\{x\in{X^{\mathrm{an}}}\mid\kappa(x)=\kappa_{\min}\right\}.

Note that $\kappa$ is an lsc function $X^{\mathrm{an}}\to(-\infty,+\infty]$ , and hence achieves its infimum.

Definition 5.9.

Let ${\mathcal{L}}$ be a model of $K_{X}$ determined on a proper dlt model ${\mathcal{X}}$ . We denote by $\Delta({\mathcal{L}})$ the subcomplex of $\Delta({\mathcal{X}})$ such that a face $\sigma$ of $\Delta({\mathcal{X}})$ is in $\Delta({\mathcal{L}})$ if and only if each vertex of $\sigma$ achieves $\min_{i}\kappa(v_{i})$ with $\kappa=A_{X}-\phi_{\mathcal{L}}$ .

Concretely, the values $\kappa(v_{i})$ are computed as follows: we have

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

with $a_{i}\in{\mathbb{Q}}$ , and $\kappa(v_{E_{i}})=a_{i}/b_{i}$ . Note that each face of $\Delta({\mathcal{X}})$ contains at most one maximal face of $\Delta({\mathcal{L}})$ .

Proposition 5.10.

Assume that $\psi$ is 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 $X$ . Then $\operatorname{Sk}(\psi)\subset\operatorname{Sk}({\mathcal{X}})$ , and $\kappa=A_{X}-\psi$ is affine on each face of $\Delta({\mathcal{X}})$ . In particular,

\kappa_{\min}=\min_{i}\kappa(v_{i}),

(5.5)

where $v_{i}$ runs over the vertices in $\Delta({\mathcal{X}})$ , and $\operatorname{Sk}(\psi)$ is the subset of $\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$ corresponding to the subcomplex $\Delta({\mathcal{L}})$ of $\Delta({\mathcal{X}})$ .

Proof.

Since the relative log canonical divisor $K^{\mathrm{log}}_{{\mathcal{X}}/S}$ and ${\mathcal{L}}$ are both models of $K_{X}$ , $D:=K^{\mathrm{log}}_{{\mathcal{X}}/S}-{\mathcal{L}}$ is a ${\mathbb{Q}}$ -Cartier divisor supported on ${\mathcal{X}}_{0}$ . The corresponding model function $\phi_{D}$ satisfies $\kappa=A_{\mathcal{X}}+\phi_{D}$ , which shows that $\kappa|_{\operatorname{Sk}({\mathcal{X}})}=\phi_{D}|_{\operatorname{Sk}({\mathcal{X}})}$ is affine on each face of $\Delta({\mathcal{X}})$ . Now pick $v\in\operatorname{Sk}(\psi)$ . By (5.1), we get

\kappa(v)=A_{\mathcal{X}}(v)+\phi_{D}(v)\geq\inf\phi_{D}=\min_{i}\phi_{D}(v_{i})=\min_{i}(A_{\mathcal{X}}+\phi_{D})(v_{i})\geq\inf_{X^{\mathrm{an}}}\kappa.

It follows that $A_{\mathcal{X}}(v)=0$ , and hence $v\in\operatorname{Sk}({\mathcal{X}})$ , by Proposition 5.6. ∎