Set $S:=\operatorname{Spec}k[\![{t}]\!]$. Following the convention of [MN15], we define a model of $X$ to be a normal scheme ${\mathcal{X}}$, flat and of finite type (but possibly non-proper) over $S$, together with an identification of the generic fiber of the structure morphism $\pi\colon{\mathcal{X}}\to S$ with $X$.

For any two models ${\mathcal{X}}$, ${\mathcal{X}}^{\prime}$, the identifications of the generic fibers with $X$ induces a unique birational map ${\mathcal{X}}^{\prime}\dashrightarrow{\mathcal{X}}$. We say that ${\mathcal{X}}^{\prime}$ dominates ${\mathcal{X}}$ if this map is a morphism. Any two models can be dominated by a third.

For any model ${\mathcal{X}}$ and every irreducible component $E$ of ${\mathcal{X}}_{0}$, we set $b_{E}:=\operatorname{ord}_{E}({t})$, and view the divisorial valuation

as an element of $X^{\operatorname{val}}\subset X^{\mathrm{an}}$. The set of such points is a dense subset $X^{\mathrm{div}}\subset X^{\mathrm{an}}$.

We usually denote by ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ the irreducible decomposition of the central fiber, and write $E_{J}:=\bigcap_{i\in J}E_{i}$ for $J\subset I$. We say that ${\mathcal{X}}$ is snc if (${\mathcal{X}}$ is regular and) ${\mathcal{X}}_{0,\mathrm{red}}$ has simple normal crossing support. Since $k$ has characteristic $0$, this means that each non-empty $E_{J}$ is smooth over $k$, of codimension $|J|$ in ${\mathcal{X}}$.

More generally, a model ${\mathcal{X}}$ is toroidal if ${\mathcal{X}}\setminus{\mathcal{X}}_{0}\subset{\mathcal{X}}$ is a strict toroidal embedding in the sense of [KKMS], i.e. is formally isomorphic, at each closed point of ${\mathcal{X}}_{0}$, to the inclusion of ${\mathbb{G}}_{m,k}^{n+1}$ in a toric $k$-variety, and such that each $E_{i}$ is normal (which then implies that each non-empty $E_{J}$ is normal).

Every model ${\mathcal{X}}$ contains a largest snc Zariski open subset ${\mathcal{X}}_{\mathrm{snc}}\subset{\mathcal{X}}$. By Temkin’s version of Hironaka’s theorem [Tem12], ${\mathcal{X}}$ is dominated by an snc model ${\mathcal{X}}^{\prime}$ such that the induced birational morphism $\mu\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is projective, and an isomorphism over ${\mathcal{X}}_{\mathrm{snc}}$.

If ${\mathcal{X}}$ is a model of $X$, the set ${\mathcal{X}}^{\mathrm{an}}\subset X^{\mathrm{an}}$ of semivaluations that admit a center (or reduction) on ${\mathcal{X}}_{0}$ is a closed subset; it can be viewed as the generic fiber of a suitable formal scheme [MN15, 2.2.2]. By the valuative criterion of properness, we have ${\mathcal{X}}^{\mathrm{an}}={\mathcal{X}}^{\prime\mathrm{an}}$ for each proper morphism of models ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, and ${\mathcal{X}}^{\mathrm{an}}=X^{\mathrm{an}}$ if ${\mathcal{X}}$ is proper (over $S$, that is). The reduction map $c_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to{\mathcal{X}}_{0}$, taking a semivaluation to its center, is anticontinuous.⁷⁷ 7 Anticontinuity means that the inverse image of an open set is closed.

The set ${\mathcal{X}}^{\mathrm{div}}:={\mathcal{X}}^{\mathrm{an}}\cap X^{\mathrm{div}}$ consists of all divisorial valuations $v$ on $F({\mathcal{X}})=F(X)$ that are centered on ${\mathcal{X}}_{0}$, trivial on $k$ and such that $v({t})=1$.

If $L$ is a line bundle on $X$, a model ${\mathcal{L}}$ of $L$ is a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on a proper model ${\mathcal{X}}$, together with an identification ${\mathcal{L}}|_{X}=L$. It defines a model metric $\phi_{\mathcal{L}}$ on the Berkovich analytification $L^{\mathrm{an}}$ of $L$. If ${\mathcal{L}}^{\prime}$ is another model of $L$, determined on a proper model ${\mathcal{X}}^{\prime}$ of $X$, then $\phi_{\mathcal{L}}=\phi_{{\mathcal{L}}^{\prime}}$ if and only if the pull-backs of ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ to some higher model ${\mathcal{X}}^{\prime\prime}$ coincide.

A model of ${\mathcal{O}}_{X}$ is given by a ${\mathbb{Q}}$-Cartier divisor $D$ supported on the central fiber of a proper model ${\mathcal{X}}$; the corresponding model metric will then be identified with the model function $\phi_{D}\colon X^{\mathrm{an}}\to{\mathbb{R}}$ defined by $\phi_{D}(v)=v(D)$. It satisfies

where $E$ runs over the irreducible components of ${\mathcal{X}}_{0}$.

If ${\mathcal{X}}$ is a regular model, $\pi\colon{\mathcal{X}}\to S$ is a locally complete intersection morphism, so the dualizing sheaf $\omega_{{\mathcal{X}}/S}$ is a well-defined line bundle (see [MN15, §4.1] for a more detailed discussion). For an arbitrary (normal) model, we may thus introduce the relative canonical divisor (class) $K_{{\mathcal{X}}/S}$ as the Weil divisor class on ${\mathcal{X}}$ such that ${\mathcal{O}}_{{\mathcal{X}}_{\mathrm{reg}}}(K_{{\mathcal{X}}/S})=\omega_{{\mathcal{X}}_{\mathrm{reg}}/S}$. We then define:

• (i)

  the canonical divisor $K_{\mathcal{X}}:=K_{{\mathcal{X}}/S}+\pi^{*}K_{S}$;

• (ii)

  the log canonical divisor $K_{\mathcal{X}}^{\mathrm{log}}:=K_{\mathcal{X}}+{\mathcal{X}}_{0,\mathrm{red}}$;

• (iii)

  the relative log canonical divisor

  $$K^{\mathrm{log}}_{{\mathcal{X}}/S}:=K^{\mathrm{log}}_{\mathcal{X}}-\pi^{*}K^{\mathrm{log}}_{S}=K_{{\mathcal{X}}/S}+{\mathcal{X}}_{0,\mathrm{red}}-{\mathcal{X}}_{0}.$$

Note that $K_{\mathcal{X}}^{\mathrm{log}}$ is ${\mathbb{Q}}$-Cartier if and only if $K_{{\mathcal{X}}/S}^{\mathrm{log}}$ is ${\mathbb{Q}}$-Cartier.

We refer to [dFKX12], [NX13, §2.2] and [KNX15] for more details and references on what follows.

Let ${\mathcal{X}}$ be a model with $K^{\mathrm{log}}_{\mathcal{X}}$ ${\mathbb{Q}}$-Cartier, and recall that ${\mathcal{X}}^{\mathrm{div}}$ denotes the set of divisorial valuations $v$ on ${\mathcal{X}}$ such that $v({t})=1$. We define the log discrepancy $A_{\mathcal{X}}(v)$ as the log discrepancy of $v$ with respect to the pair $({\mathcal{X}},{\mathcal{X}}_{0,\mathrm{red}})$, in the usual sense of the Minimal Model Program.

The log discrepancy function $A_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{div}}\to{\mathbb{Q}}$ is characterized by the following property: if ${\mathcal{X}}^{\prime}$ is a model over ${\mathcal{X}}$ with proper birational morphism $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$, then

with $E$ running over the irreducible components of ${\mathcal{X}}^{\prime}_{0}$.

We say that a model ${\mathcal{X}}$ is log canonical (lc for short), Kawamata log terminal (klt) or divisorially log terminal (dlt) if the pair $({\mathcal{X}},{\mathcal{X}}_{0,\mathrm{red}})$ has this property, in the sense of the Minimal Model Program.

Since the generic fiber $X$ is smooth, a model ${\mathcal{X}}$ is thus lc (resp. klt) if and only if $K^{\mathrm{log}}_{\mathcal{X}}$ is ${\mathbb{Q}}$-Cartier, with log discrepancy function $A_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{div}}\to{\mathbb{Q}}$ taking non-negative (resp. positive) values. If ${\mathcal{X}}$ is lc, then the center $c_{\mathcal{X}}(v)\in{\mathcal{X}}_{0}$ of a valuation $v\in{\mathcal{X}}^{\mathrm{div}}$ with $A_{\mathcal{X}}(v)=0$ is called an lc center of ${\mathcal{X}}$, and an lc model ${\mathcal{X}}$ is dlt if and only if ${\mathcal{X}}_{\mathrm{snc}}$ contains all lc centers. The irreducible components of each non-empty $E_{J}$ are then normal, with generic point contained in ${\mathcal{X}}_{\mathrm{snc}}$ [Kol13, 4.16].

The dual complex $\Delta({\mathcal{X}})$ of an snc model ${\mathcal{X}}$ is defined as the dual complex of the snc divisor ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$, as in §2.1. It is equipped with a natural integral affine structure, in which the face $\sigma$ corresponding to a component $Y$ of a non-empty $E_{J}$ is identified with the simplex

in such a way that $M_{\sigma}={\mathbb{Z}}^{J}$.

As explained in [BFJ16, §3] and [MN15, §3], there is a natural embedding

that takes a point $w\in\sigma$ to the corresponding monomial valuation. In particular, the vertex corresponding to $E_{i}$ is sent to the divisorial valuation $v_{E_{i}}=b_{i}^{-1}\operatorname{ord}_{E_{i}}$. The value group of a valuation $v=\operatorname{emb}_{\mathcal{X}}(w)$, $w\in\sigma$, is given by

Further, if $w\in\mathring{\sigma}$, then $Y_{\sigma}$ is the closure of the center of $\operatorname{emb}_{\mathcal{X}}(w)$.

The resulting subspace $\operatorname{Sk}({\mathcal{X}}):=\operatorname{emb}_{\mathcal{X}}(\Delta_{X})\subset{\mathcal{X}}^{\mathrm{an}}\subset X^{\mathrm{an}}$ is called the skeleton of ${\mathcal{X}}$. It is naturally a ${\mathbb{Z}}$-PA space, the ${\mathbb{Z}}$-PA functions on $\operatorname{Sk}({\mathcal{X}})$ being precisely the restrictions of model functions $\phi_{D}$ determined by a Cartier divisor $D$ on some proper modification ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$.

We further have a natural retraction $r_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to\operatorname{Sk}({\mathcal{X}})$, mapping a valuation $v$ centered on ${\mathcal{X}}_{0}$ to the monomial valuation $r_{\mathcal{X}}(v)$ taking the same values on the $E_{i}$’s. These retractions induce a homeomorphism

where ${\mathcal{X}}$ runs over all proper (or projective) snc models, compare (4.3).

If ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is a proper morphism of snc models, then, by [MN15, 3.1.7],

the first inclusion being ${\mathbb{Z}}$-PA. Further,

coincides with the set of (quasi)monomial, or Abhyankar, valuations.

For a dlt model ${\mathcal{X}}$, the dual complex $\Delta({\mathcal{X}})$ and skeleton $\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$ are simply defined as those of ${\mathcal{X}}_{\mathrm{snc}}$, cf. [NX13]. The retraction $r_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to\operatorname{Sk}({\mathcal{X}})$ can be defined as above when ${\mathcal{X}}$ is ${\mathbb{Q}}$-factorial, but its existence is otherwise unclear (at least to us!).

By [KKMS], any toroidal model ${\mathcal{X}}$ has a dual complex $\Delta({\mathcal{X}})$ endowed with a natural integral affine structure. This dual complex is canonically realized as a subspace $\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$, for instance by setting $\operatorname{Sk}({\mathcal{X}}):=\operatorname{Sk}({\mathcal{X}}^{\prime})$ for any toroidal modification ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ with ${\mathcal{X}}^{\prime}$ snc. Thus $\operatorname{Sk}({\mathcal{X}})$ is equipped with a ${\mathbb{Z}}$-PA structure.

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

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.

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

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 $X^{\mathrm{an}}$.

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].

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

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

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 $\Delta({\mathcal{L}})$.

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

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

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

is effective.

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

Now we study how skeleta of snc models and of metrics behave under base change.

For $m\in{\mathbb{Z}}_{>0}$ consider the Galois extension $K^{\prime}:=k(\!({t}^{1/m})\!)$ of $K=k(\!({t})\!)$, with Galois group $G={\mathbb{Z}}/m{\mathbb{Z}}$, and set $X^{\prime}=X_{K^{\prime}}$. Then $G$ acts on $X^{\prime\mathrm{an}}$ and the canonical map $p\colon X^{\prime\mathrm{an}}\to X^{\mathrm{an}}$ induces a homeomorphism

If ${\mathcal{X}}$ is a model of $X$, then its normalized base change yields a model ${\mathcal{X}}^{\prime}$ of $X^{\prime}$ with a finite morphism $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$. If $D$ is a ${\mathbb{Q}}$-divisor on ${\mathcal{X}}$ defining a model function $\phi_{D}$ on $X^{\mathrm{an}}$, then

When ${\mathcal{X}}$ is an snc model, ${\mathcal{X}}^{\prime}$ is toroidal, by [KKMS, pp.98–102]. The following rather detailed description will be useful later on.

Next we study skeleta of metrics. Generalizing [NX13, Lemma 4.1.9], we prove: