5.6. From log discrepancies to Temkin’s metric [016T]

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5.6. From log discrepancies to Temkin’s metric

As noted in [FJ04, BFJ08, JM12] in increasing order of generality, log discrepancy functions extend in a natural way to Berkovich spaces. More precisely, let ${\mathcal{X}}$ be any model of $X$ such that $K^{\mathrm{log}}_{\mathcal{X}}$ is ${\mathbb{Q}}$ -Cartier, with log discrepancy function $A_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{div}}\to{\mathbb{Q}}$ . For each snc model ${\mathcal{X}}^{\prime}$ properly dominating ${\mathcal{X}}$ , a simple computation going back (at least) to [Kol97, Lemma 3.11] shows the following:

(i)

the restriction of $A_{\mathcal{X}}$ to $\operatorname{Sk}({\mathcal{X}}^{\prime})$ is ${\mathbb{Z}}$ -affine on each face of $\Delta({\mathcal{X}}^{\prime})$ ;
(ii)

we have $A_{\mathcal{X}}\geq A_{\mathcal{X}}\circ r_{{\mathcal{X}}^{\prime}}$ , the inequality being strict outside $\operatorname{Sk}({\mathcal{X}}^{\prime})$ .

We may thus extend $A_{\mathcal{X}}$ to an lsc function $A_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to[0,+\infty]$ by setting

A_{\mathcal{X}}(v):=\sup_{{\mathcal{X}}^{\prime}}A_{\mathcal{X}}(r_{{\mathcal{X}}^{\prime}}(v))

(5.3)

for any $v\in{\mathcal{X}}^{\mathrm{an}}$ . When ${\mathcal{X}}$ is dlt, the log discrepancy function $A_{\mathcal{X}}$ determines the skeleton as follows.

Proposition 5.6.

If ${\mathcal{X}}$ is dlt, then $\operatorname{Sk}({\mathcal{X}})=\left\{v\in{\mathcal{X}}^{\mathrm{an}}\mid A_{\mathcal{X}}(v)=0\right\}$ .

Lemma 5.7.

Assume that ${\mathcal{X}}$ is lc, and pick $v\in{\mathcal{X}}^{\mathrm{an}}$ with $A_{\mathcal{X}}(v)=0$ . Then $c_{\mathcal{X}}(v)$ is an lc center of ${\mathcal{X}}$ .

Proof.

We claim that, for every sufficiently high snc model ${\mathcal{X}}^{\prime}$ proper over ${\mathcal{X}}$ , $v^{\prime}:=r_{{\mathcal{X}}^{\prime}}(v)$ and $v$ have the same center on ${\mathcal{X}}$ . Indeed, the center of $v$ on ${\mathcal{X}}^{\prime}$ is a specialization of that of $r_{{\mathcal{X}}^{\prime}}(v)$ , and hence $c_{\mathcal{X}}(v)\in\overline{c_{\mathcal{X}}(r_{{\mathcal{X}}^{\prime}}(v))}$ . On the other hand, we have $\lim_{{\mathcal{X}}^{\prime}}r_{{\mathcal{X}}^{\prime}}(v)=v$ . Since $c_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to{\mathcal{X}}_{0}$ is anticontinuous, $c_{\mathcal{X}}^{-1}(\overline{\{c_{\mathcal{X}}(v)\}})$ is open, and hence contains $v^{\prime}:=r_{{\mathcal{X}}^{\prime}}(v)$ for some snc model ${\mathcal{X}}^{\prime}$ proper over ${\mathcal{X}}$ . As a result, $c_{\mathcal{X}}(v^{\prime})$ is a specialization of $c_{\mathcal{X}}(v)$ , and the claim follows.

By (5.3), we have $A_{\mathcal{X}}(v^{\prime})=0$ , and it is thus enough to prove the result for $v^{\prime}\in\operatorname{Sk}({\mathcal{X}}^{\prime})$ . If $\sigma$ is the unique face of $\Delta({\mathcal{X}}^{\prime})$ containing $v^{\prime}$ in its interior, then $A_{\mathcal{X}}\equiv 0$ on $\sigma$ , since $A_{\mathcal{X}}$ is non-negative and affine on $\sigma$ . For any divisorial point $w$ in the relative interior of $\sigma$ , we thus have $A_{\mathcal{X}}(w)=0$ and $c_{{\mathcal{X}}^{\prime}}(v^{\prime})=c_{{\mathcal{X}}^{\prime}}(w)$ , which shows that $c_{\mathcal{X}}(v^{\prime})=c_{\mathcal{X}}(w)$ is an lc center. ∎

Proof of Proposition 5.6.

When ${\mathcal{X}}$ is snc, the result is a direct consequence of (i) and (ii) above. When ${\mathcal{X}}$ is dlt, we have by definition

\operatorname{Sk}({\mathcal{X}})=\operatorname{Sk}({\mathcal{X}}_{\mathrm{snc}})\subset{\mathcal{X}}_{\mathrm{snc}}^{\mathrm{an}}\subset{\mathcal{X}}^{\mathrm{an}},

and $A_{\mathcal{X}}=A_{{\mathcal{X}}_{\mathrm{snc}}}$ on ${\mathcal{X}}_{\mathrm{snc}}^{\mathrm{an}}$ . It is thus enough to show that any $v\in{\mathcal{X}}^{\mathrm{an}}$ with $A_{\mathcal{X}}(v)=0$ belongs to ${\mathcal{X}}_{\mathrm{snc}}^{\mathrm{an}}$ , i.e. satisfies $c_{\mathcal{X}}(v)\in{\mathcal{X}}_{\mathrm{snc}}$ . But $c_{\mathcal{X}}(v)$ is an lc center by Lemma 5.7, and hence $c_{\mathcal{X}}(v)\in{\mathcal{X}}_{\mathrm{snc}}$ by definition of dlt singularities. ∎

Let ${\mathcal{X}}$ be a proper model with $K^{\mathrm{log}}_{{\mathcal{X}}/S}$ ${\mathbb{Q}}$ -Cartier. Viewed as a ${\mathbb{Q}}$ -line bundle, the latter is then a model of $K_{X}$ , and hence defines a model metric $\phi_{K^{\mathrm{log}}_{{\mathcal{X}}/S}}$ on $K_{X}^{\mathrm{an}}$ . Further, (5.2) shows that the lsc metric

A_{X}:=\phi_{K^{\mathrm{log}}_{{\mathcal{X}}/S}}+A_{\mathcal{X}}

(5.4)

on $K_{X}^{\mathrm{an}}$ is independent of ${\mathcal{X}}$ . This is a special case of Temkin’s canonical metrization of the canonical bundle [Tem14].⁸⁸ 8 That we obtain Temkin’s metric follows from [Tem14, Theorem 8.1.2]. Note that Temkin uses multiplicative terminology. The weight function of [MN15] associated to a pluricanonical form $\omega\in H^{0}(X,mK_{X})$ is the function $A_{X}-\frac{1}{m}\log|\omega|$ on $X^{\mathrm{an}}$ .