The essential skeleton of a product of degenerations

Abstract

We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration $\mathscr{X}_{R}$ changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to $X_{K}$. Using the Kato fan, we define a skeleton $\text{Sk}(\mathscr{X}_{R})$ when the model $\mathscr{X}_{R}$ is log-regular. We show that if $\mathscr{X}_{R}$ and $\mathscr{Y}_{R}$ are log-smooth, and at least one is semistable, then $\text{Sk}(\mathscr{X}_{R}\times _{R}\mathscr{Y}_{R})\simeq \text{Sk}(\mathscr{X}_{R})\times \text{Sk}(\mathscr{Y}_{R})$. The essential skeleton $\text{Sk}(X_{K})$, defined by Mustaţă and Nicaise, is a birational invariant of $X_{K}$ and is independent of the choice of $R$-model. We extend their definition to pairs, and show that if both $X_{K}$ and $Y_{K}$ admit semistable models, $\text{Sk}(X_{K}\times _{K}Y_{K})\simeq \text{Sk}(X_{K})\times \text{Sk}(Y_{K})$. As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the $2n$-dimensional degeneration is homeomorphic to a point, $n$-simplex, or $\mathbb{C}\mathbb{P}^{n}$, depending on the type of the degeneration.

1 Introduction

Let $R$ be a discrete valuation ring with quotient field $K$ and residue field $k$ , and let $X$ be a smooth proper variety over $K$ . While there may be no way to extend $X$ to a smooth proper variety over $R$ , in $\text{char}(k)=0$ resolution of singularities guarantees that we can always produce an $R$ -model $\mathscr{X}$ where the special fiber $\mathscr{X}_{k}$ is a strict normal crossings (snc) divisor. Given such a model, we associate the dual complex ${\mathcal{D}}(\mathscr{X}_{k})$ , which is the dual intersection complex of the components of the special fiber.

The dual complex of the special fiber of a such degeneration reflects the geometry of the generic fiber. If the generic fiber is rationally connected, then the dual complex of the special fiber is contractible [dKX17]. For Calabi–Yau varieties, degenerations are classified by the action of monodromy on the cohomology. The principle is that the degenerations with maximally unipotent actions have the richest combinatorial structure in the dual complex. In this case, the dual complex is always a $\mathbb{Q}$ -homology sphere, and Kollár and Xu [KX16] show that it is a sphere if $n\leqslant 3$ or $n\leqslant 4$ and the special fiber of a minimal divisorially log terminal (dlt) model is snc.

The goal of this work is to understand the dual complex of a model for the product of two smooth proper varieties over $K$ . We consider this problem from two perspectives.

1.1 Skeletons of Berkovich spaces

The first is via the theory of Berkovich spaces. In this setting we assume that $K$ is complete with respect to the valuation induced by $R$ , which gives rise to a non-archimedean norm on $K$ . In [Ber90], Berkovich develops a theory of analytic geometry over $K$ . He associates a $K$ -analytic space to $X$ ; each point corresponds to a real valuation on the residue field of a point of $X$ , extending the discrete valuation on $K$ . This space, denoted by $X^{\text{an}}$ , is called the Berkovich space associated to $X$ .

From any snc model $\mathscr{X}$ of $X$ one can construct a subspace of $X^{\text{an}}$ , called the Berkovich skeleton of $\mathscr{X}$ and denoted by $\text{Sk}(\mathscr{X})$ : it is homeomorphic to the dual intersection complex of the divisor $\mathscr{X}_{k}$ [MN15]. The Berkovich skeletons turn out to be relevant in the study of the topology of $X^{\text{an}}$ . They shape the Berkovich space, as $X^{\text{an}}$ is homeomorphic to the inverse limit $\underset{\longleftarrow }{\lim }\,\text{Sk}(\mathscr{X})$ where $\mathscr{X}$ runs through all snc models of $X$ . Also, the homotopy type of $X^{\text{an}}$ is determined by any snc model $\mathscr{X}$ : indeed, Berkovich and Thuillier prove that $\text{Sk}(\mathscr{X})$ is a strong deformation retract of $X^{\text{an}}$ [Ber90, Thu07].

1.2 The dual complex of a dlt model

The other approach to the study of the dual complexes comes from birational geometry. In this setting, we consider a pair $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ over the germ of a curve. In the log general type case, running the minimal model program (MMP) distinguishes a canonical model for the degeneration $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ [Ale96, KS88, HMX18], at the cost of possibly introducing worse singularities. If we are willing to tolerate some ambiguity in our choice of model, we can choose instead to produce a minimal dlt model. One advantage of dlt models is that they are expected to exist for all pairs admitting a log pluricanonical form. The singularities are mild enough that it is possible to define the dual complex as the dual intersection complex of divisors of coefficient $1$ , denoted ${\mathcal{D}}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ . In [dKX17], de Fernex, Kollár, and Xu investigate how the dual complex is affected by the operations of the MMP. They show, under mild hypotheses, that every step of the MMP induces a homotopy equivalence between dual complexes. Moreover, ${\mathcal{D}}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ is a piecewise linear (PL) invariant under log crepant birational maps.

1.3 The essential skeleton

Recently there has been much interest in a synthesis of the two approaches. Kontsevich and Soibelmann [KS06] define a version of the skeleton of a variety with trivial canonical bundle, which detects the locus of simple poles along the special fiber of the distinguished canonical form. Mustaţă and Nicaise [MN15] extend their definition to any variety with non-negative Kodaira dimension. The key technical tool is the definition, for a rational pluricanonical form, of a weight function on the Berkovich space. The essential skeleton $\text{Sk}(X)$ is the union over all regular pluricanonical forms of the minimality locus of the associated weight functions.

Thus the essential skeleton has the advantage of being intrinsic to the variety $X$ , with no dependence on a choice of model. As the weight function is closely related to the log discrepancy from birational geometry, it is natural to expect that the essential skeleton in some way encodes some of the minimal model theory of $X$ . Nicaise and Xu [NX16] show, when $X$ is a smooth projective variety with $K_{X}$ semiample, and $\mathscr{X}$ is a good minimal dlt model, that the dual complex of $\mathscr{X}_{k}$ can be identified with the essential skeleton of $X$ . While it is in general a difficult problem to produce good minimal dlt models, Kollár, Nicaise, and Xu [KNX17] show that for any smooth projective $X$ with $K_{X}$ semiample, $X$ extends to a good minimal dlt model over a finite extension of the valuation ring.

1.4 Skeletons for log-regular models

To produce nice models of the product, we work in the context of logarithmic geometry. To any log-regular scheme $\mathscr{X}^{+}$ , in [Kat94] Kato attaches a combinatorial structure $F_{\mathscr{X}}$ called a fan: if we denote by $D_{\mathscr{X}}$ the locus where the log structure is non-trivial, then the fan $F_{\mathscr{X}}$ consists of the set of the generic points of intersections of irreducible components of $D_{\mathscr{X}}$ , equipped with a sheaf of monoids. We define a logarithmic version of the Berkovich skeleton for a log-regular model $\mathscr{X}^{+}$ of $X$ over $R$ : it gives rise to a polyhedral complex in $X^{\text{an}}$ whose faces correspond to the points of $F_{\mathscr{X}}$ .

Given two log-smooth log schemes $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ over $R$ , their product $\mathscr{Z}^{+}$ in the category of fine and saturated log schemes is naturally log-regular, hence $\mathscr{Z}^{+}$ has an associated skeleton, and it is a model of the product $\mathscr{X}_{K}\times _{K}\mathscr{Y}_{K}$ of the generic fibers. If one of the two underlying schemes $\mathscr{X}$ or $\mathscr{Y}$ is semistable, which means it has reduced special fiber, then we show that the skeleton of the product $\mathscr{Z}^{+}$ is the product of the skeletons, with the projection maps given by restricting the valuation to the corresponding function fields (Proposition 3.4.3).

1.5 Skeletons for pairs

Working in the logarithmic setting, we may also allow a non-trivial log structure over the generic fiber. Geometrically this corresponds to adding horizontal divisors to the special fiber and yields the addition of some unbounded faces to the skeleton. In [GRW16] Gubler, Rabinoff, and Werner construct a skeleton for strictly semistable snc models with suitable horizontal divisors. Both constructions recover the Berkovich skeleton when there is no horizontal component and the special fiber is snc.

Pairs arise frequently in the MMP. We say that a pair $(X,\unicode[STIX]{x1D6E5}_{X})$ is fractional snc if $\unicode[STIX]{x1D6E5}_{X}$ is an effective $\mathbb{Q}$ -divisor such that $\unicode[STIX]{x1D6E5}_{X}=\sum a_{i}\unicode[STIX]{x1D6E5}_{X,i}$ with $0\leqslant a_{i}\leqslant 1$ and the round-up $(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ is snc. Taking advantage of a construction that admits horizontal components, the definition of the essential skeleton extends to the case of a fractional snc pair $(X,\unicode[STIX]{x1D6E5}_{X})$ over $K$ , and to pluricanonical forms of some positive index $r$ with divisor of poles no worse than $r\unicode[STIX]{x1D6E5}_{X}$ . We extend to pairs the result of Mustaţă and Nicaise [MN15] on the birational invariance of the essential skeleton (Proposition 5.1.5), as well as Nicaise and Xu’s result [NX16] that the essential skeleton is homeomorphic to the dual complex of a good minimal dlt model (Proposition 5.1.7). It follows from these results that we can define the notion of essential skeleton for a dlt pair.

1.6 Main result

Our main result establishes the behavior of essential skeletons under products.

Theorem 1.6.1. Assume that the residue field $k$ is algebraically closed. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ and $(Y,\unicode[STIX]{x1D6E5}_{Y})$ be fractional snc pairs. Suppose that both pairs have non-negative Kodaira–Iitaka dimension and admit semistable log-regular models $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ over $S^{+}$ . Then the PL homeomorphism of skeletons induces a PL homeomorphism of essential skeletons

$$\begin{eqnarray}\text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z})\xrightarrow[{}]{{\sim}}\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})\times \text{Sk}(Y,\unicode[STIX]{x1D6E5}_{Y}),\end{eqnarray}$$

where $Z$ and $\unicode[STIX]{x1D6E5}_{Z}$ are the respective products.

Semistability is a key assumption; without it the projection map might fail to be injective; see Example 3.4.4. As expected, we get a corresponding result for dual complexes of semistable good minimal dlt models (Theorem 5.3.5). Unfortunately, semistability is not well behaved under birational transformations so it seems possible that a degeneration admits a semistable good minimal dlt model but no semistable log-regular model.

1.7 Application to degenerations of hyper-Kähler varieties

As an application of Theorem 1.6.1, we study certain degenerations of hyper-Kähler varieties. One way to produce hyper-Kähler varieties is by taking the Hilbert scheme of points on a K3 surface. Another is to extend the Kummer construction to higher-dimensional abelian varieties. Aside from two other examples found by O’Grady in dimensions 6 [O’Gr03] and 10 [O’Gr99] there are no other known examples, up to deformation equivalence.

Just as for Calabi–Yau varieties, degenerations of hyper-Kähler varieties can be understood in terms of the monodromy operator on cohomology, with classification into types I, II, III. Type I is the case where the dual complex is just a single point, but types II and III have more interesting structure. Kollár, Laza, Saccà, and Voisin [KLSV17] show that in the type II case the dual complex has the rational homology type of a point, and in the type III case of a complex projective space. Gulbrandsen, Halle, Hulek, and Zhang [GHH15, GHH16, GHHZ18] use geometric invariant theory (GIT) to construct a model of the degeneration of $n$ th-order Hilbert schemes arising from some type II degenerations of K3 surfaces, and show that the dual complex is an $n$ -simplex. There are considerations from mirror symmetry that suggest that for a type III degeneration the dual complex should be homeomorphic to $\mathbb{C}\mathbb{P}^{n}$ [Hwa08, KS06].

Theorem 1.7.1. Assume that the residue field $k$ is algebraically closed. Let $S$ be a $K3$ surface over $K$ . If $S$ admits a semistable log-regular model or a semistable good minimal dlt model, then the essential skeleton of the Hilbert scheme of $n$ points on $S$ is isomorphic to the $n$ th symmetric product of the essential skeleton of $S$ ,

$$\begin{eqnarray}\text{Sk}(\text{Hilb}^{n}(S))\xrightarrow[{}]{{\sim}}\text{Sym}^{n}(\text{Sk}(S)).\end{eqnarray}$$

Computing these complexes gives a single point in the type I case, an $n$ -simplex in the type II case, and $\mathbb{C}\mathbb{P}^{n}$ in the type III case. The same types arise in the Kummer case.

Theorem 1.7.2. Assume the residue field $k$ is algebraically closed. Let $A$ be an abelian surface over $K$ . Suppose that $A$ admits a semistable log-regular model or a semistable good minimal dlt model. If the essential skeleton of $A$ is homeomorphic to a point, the circle $S^{1}$ or the torus $S^{1}\times S^{1}$ , then the essential skeleton of the $n$ th generalized Kummer variety $K_{n}(A)$ is isomorphic to a point, the standard $n$ -simplex or $\mathbb{C}\mathbb{P}^{n}$ , respectively.

Analysis of the weight function gives a powerful yet accessible approach to controlling the skeletons of these varieties. In both cases we use Theorem 1.6.1 to establish that the skeleton of the hyper-Kähler variety is a finite quotient of the $n$ -fold product of the skeleton of the original surface under the action of a symmetric group. In the case of Hilbert schemes we can get a complete description of the action using functoriality of the projection maps, but in the Kummer case we additionally need to understand the restriction of the multiplication map to the essential skeleton of an abelian surface [Ber90, HN17, Tem16].

To our knowledge these are the first examples of type III degenerations of hyper-Kähler varieties where the PL homeomorphism type of the dual complex is known.

1.8 Structure of the paper

In § 2 we recall the Kato fan, and develop necessary background for the rest of the paper. Section 3 contains the definition and properties of the skeleton of a log-regular scheme. These first two sections require no hypothesis on the characteristic of $K$ .

For the remainder of the paper, we assume that $K$ has equicharacteristic $0$ . In § 4 we introduce the weight function and essential skeleton from [MN15], extending their definition to the case of a pair. In § 5 we give connections to birational geometry, along with proofs of the main theorems. We discuss an application to degenerations of hyper-Kähler varieties in § 6.

1.9 Notation

1.9.1. Let $R$ be a complete discrete valuation ring with maximal ideal $\mathfrak{m}$ , residue field $k=R/\mathfrak{m}$ , and quotient field $K$ . We assume that the valuation $v_{K}$ is normalized, namely $v_{K}(\unicode[STIX]{x1D70B})=1$ for any uniformizer $\unicode[STIX]{x1D70B}$ of $R$ . We define by $|\cdot |_{K}=\exp (-v_{K}(\cdot ))$ the absolute value on $K$ corresponding to $v_{K}$ ; this turns $K$ into a non-archimedean complete valued field.

1.9.2. We write $S=\text{Spec}\,R$ and we denote by $s$ the closed point of $S$ . Let $\mathscr{X}$ be an $R$ -scheme of finite type. We will denote by $\mathscr{X}_{k}$ the special fiber of $\mathscr{X}$ and by $\mathscr{X}_{K}$ the generic fiber.

1.9.3. Let $X$ be a proper $K$ -scheme. A model for $X$ over $R$ is a flat separated $R$ -scheme $\mathscr{X}$ of finite type endowed with an isomorphism of $K$ -schemes $\mathscr{X}_{K}\rightarrow X$ . If $X$ is smooth over $K$ , we say that $\mathscr{X}$ is an snc model for $X$ if it is regular, proper over $R$ , and the special fiber $\mathscr{X}_{k}$ is an snc divisor on $\mathscr{X}$ . In equicharacteristic $0$ , such a model always exists, by Hironaka’s resolution of singularities.

We say that a model $\mathscr{X}$ over $R$ (not necessarily regular) is semistable if the special fiber $\mathscr{X}_{k}$ is reduced.

1.9.4. All log schemes in this paper are fine and saturated (fs) log schemes and defined with respect to the Zariski topology. We denote a log scheme by $\mathscr{X}^{+}=(\mathscr{X},{\mathcal{M}}_{\mathscr{X}})$ , where ${\mathcal{M}}_{\mathscr{X}}$ is the structural sheaf of monoids. The quotient sheaf

$$\begin{eqnarray}{\mathcal{C}}_{\mathscr{X}}={\mathcal{M}}_{\mathscr{X}}/{\mathcal{O}}_{\mathscr{X}}^{\times }\end{eqnarray}$$

is called the the characteristic sheaf of $\mathscr{X}^{+}$ . For every point $x$ of $\mathscr{X}$ , we denote by ${\mathcal{I}}_{\mathscr{X},x}$ the ideal in ${\mathcal{O}}_{\mathscr{X},x}$ generated by

$$\begin{eqnarray}{\mathcal{M}}_{\mathscr{X},x}\setminus {\mathcal{O}}_{\mathscr{X},x}^{\times }.\end{eqnarray}$$

If a log scheme $\mathscr{X}^{+}$ has divisorial log structure induced from a divisor $D$ , we denote it by $\mathscr{X}^{+}=(\mathscr{X},D)$ . In particular, we denote by $S^{+}$ the scheme $S$ endowed with the standard log structure (the divisorial log structure induced by $s$ ), namely $S^{+}=(S,s)$ .

1.9.5. A log scheme is log-regular at a point $x$ if the following two conditions are satisfied: ${\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}$ is a regular local ring; and $\dim {\mathcal{O}}_{\mathscr{X},x}=\dim {\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}+\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ . For example, a toric variety with its toric logarithmic structure is log-regular. More generally, working over perfect fields, log-regular varieties correspond to toroidal embeddings (without self-intersections).

If $\mathscr{X}^{+}$ is a log-regular log scheme over $S^{+}$ , then the locus where the log structure is non-trivial is a divisor that we will denote by $D_{\mathscr{X}}$ . Thus, the log structure on $\mathscr{X}^{+}$ is the divisorial log structure induced by $D_{\mathscr{X}}$ , by [Kat94, Theorem 11.6].

For a more extended dissertation of the theory of log schemes we refer to [Kat89, GR04, Kat94].

1.9.6. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ be a pair such that $X$ is proper over $K$ and the round-up $(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ is log-regular. A log-regular log scheme $\mathscr{X}^{+}$ over $S^{+}$ is a model for $(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ over $S^{+}$ if $\mathscr{X}$ is a model of $X$ over $R$ , the closure of any component of $\unicode[STIX]{x1D6E5}_{X}$ in $\mathscr{X}$ has non-empty intersection with $\mathscr{X}_{k}$ , and $D_{\mathscr{X}}=\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}$ .

1.9.7. We denote by $(\cdot )^{\text{an}}$ the analytification functor from the category of $K$ -schemes of finite type to Berkovich’s category of $K$ -analytic spaces. For every $K$ -scheme of finite type $X$ , as a set, $X^{\text{an}}$ consists of the pairs $x=(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})$ where $\unicode[STIX]{x1D709}_{x}$ is a point of $X$ and $|\cdot |_{x}$ is an absolute value on the residue field $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D709}_{x})$ of $X$ at $\unicode[STIX]{x1D709}_{x}$ extending the absolute value $|\cdot |_{K}$ on $K$ . We endow $X^{\text{an}}$ with the Berkovich topology, that is, the weakest one such that:

1. (i) the forgetful map $\unicode[STIX]{x1D719}:X^{\text{an}}\rightarrow X$ , defined as $(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})\mapsto \unicode[STIX]{x1D709}_{x}$ , is continuous;

2. (ii) for any Zariski open subset $U$ of $X$ and any regular function $f$ on $U$ the map $|f|:\unicode[STIX]{x1D719}^{-1}(U)\rightarrow \mathbb{R}$ defined by $|f|(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})=|f(\unicode[STIX]{x1D709}_{x})|$ is continuous.

The set $\text{Bir}(X)$ of birational points of $X^{\text{an}}$ is defined as the inverse image under $\unicode[STIX]{x1D719}$ of the generic point of $X$ . By definition, it is a birational invariant of $X$ .

1.9.8. Let $X$ be a smooth, proper $K$ -scheme of finite type. For every point $x\in X^{\text{an}}$ , we denote by $\mathscr{H}(x)$ the completion of the residue field $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D709}_{x})$ with respect to the absolute value $|\cdot |_{x}$ , and by $\mathscr{H}(x)^{\circ }$ the valuation ring of the valued field $\mathscr{H}(x)$ .

Given any snc model $\mathscr{X}$ of $X$ , by the valuative criterion of properness, the $K$ -morphism $\text{Spec}\,(\mathscr{H}(x))\rightarrow X$ extends to an $R$ -morphism $\text{Spec}\,(\mathscr{H}(x)^{\circ })\rightarrow \mathscr{X}$ . The image of the closed point via $\text{Spec}\,(\mathscr{H}(x)^{\circ })\rightarrow \mathscr{X}$ is called the center, or reduction, of $x$ on $\mathscr{X}$ . The map

$$\begin{eqnarray}\text{red}_{\mathscr{X}}:X^{\text{an}}\rightarrow \mathscr{X}_{k},\end{eqnarray}$$

which sends $x$ to its center on $\mathscr{X}$ , is called the reduction map with respect to $\mathscr{X}$ .

1.9.9. We recall a few notions from birational geometry; see [KM08]. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ be a pair such that $X$ is normal, $\unicode[STIX]{x1D6E5}_{X}$ is an effective divisor, and $K_{X}+\unicode[STIX]{x1D6E5}_{X}$ is $\mathbb{Q}$ -Cartier. Then we say that $(X,\unicode[STIX]{x1D6E5}_{X})$ is log canonical if for every log resolution $f:Z\rightarrow X$ , in the formula

$$\begin{eqnarray}K_{Z}+\unicode[STIX]{x1D6E5}_{Z}=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{X})+\sum a_{D}D,\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}_{Z}$ is the round-up of the strict transform of $\unicode[STIX]{x1D6E5}_{X}$ plus the reduced exceptional divisor, all the $a_{D}$ are non-negative. The sum ranges over all components of $\unicode[STIX]{x1D6E5}_{Z}$ . In fact the quantity $a_{D}$ , called the log discrepancy of $D$ with respect to $(X,\unicode[STIX]{x1D6E5}_{X})$ , depends only on the valuation corresponding to $D$ , and this condition need only be tested on a single log resolution.

We are most interested in the case where $(X,\unicode[STIX]{x1D6E5}_{X})$ satisfies the stronger condition of being dlt. A closed subset $Y\subset X$ is called a log canonical center if for some (equivalently any) log resolution, $Y$ is the image of a divisor $D$ with $a_{D}=0$ . The pair $(X,\unicode[STIX]{x1D6E5}_{X})$ is said to be dlt if for every log canonical center $Y$ , there is a neighborhood of the generic point of $Y$ inside which $(X,\unicode[STIX]{x1D6E5}_{X})$ is snc. Every fractional snc pair is dlt by [KM08, Corollary 2.39].

We say that a pair $(X,\unicode[STIX]{x1D6E5}_{X})$ is Kawamata log terminal (klt) if it is dlt and the coefficients of $\unicode[STIX]{x1D6E5}_{X}$ are all strictly less than $1$ .

1.9.10. Given a morphism $f:X\rightarrow B$ , and a dlt pair $(X,\unicode[STIX]{x1D6E5}_{X})$ , we say that $(X,\unicode[STIX]{x1D6E5}_{X})$ is a minimal dlt model over $B$ if $f$ is proper and $K_{X/B}+\unicode[STIX]{x1D6E5}_{X}$ is nef. If, in addition, $K_{X/B}+\unicode[STIX]{x1D6E5}_{X}$ is semiample, we say that $(X,\unicode[STIX]{x1D6E5}_{X})$ is a good minimal dlt model.

Given a pair $(X,\unicode[STIX]{x1D6E5}_{X})$ over $B$ , a birational map $\unicode[STIX]{x1D719}:X{\dashrightarrow}X^{\prime }$ is called a (good) minimal dlt model of $(X,\unicode[STIX]{x1D6E5}_{X})$ if $(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }})$ is a (good) minimal dlt model over $B$ , where $\unicode[STIX]{x1D6E5}_{X^{\prime }}$ is the sum of the strict transform of $\unicode[STIX]{x1D6E5}_{X}$ and the reduced exceptional divisor of $\unicode[STIX]{x1D719}^{-1}$ , the sum of all divisors of $X^{\prime }$ such that $\unicode[STIX]{x1D719}^{-1}$ does not define an isomorphism along the generic point. We often suppress the map $\unicode[STIX]{x1D719}$ . For example, if $(X,\unicode[STIX]{x1D6E5}_{X})$ is proper over $K$ , and $X^{\prime }$ is proper over $R$ extending $X$ , then $\unicode[STIX]{x1D6E5}_{X^{\prime }}$ is the sum of the strict transform of $\unicode[STIX]{x1D6E5}_{X}$ with the reduced special fiber of $X^{\prime }$ .

2 The Kato fan of a log-regular log scheme

2.1 Definition of Kato fans

2.1.1. According to [Kat94, Definition 9.1], a monoidal space $(T,{\mathcal{M}}_{T})$ is a topological space $T$ endowed with a sharp sheaf of monoids ${\mathcal{M}}_{T}$ , where sharp means that ${\mathcal{M}}_{T,t}^{\times }=\{1\}$ for every $t\in T$ . We often denote the monoidal space simply by $T$ .

A morphism of monoidal spaces is a pair $(f,\unicode[STIX]{x1D711}):(T,{\mathcal{M}}_{T})\rightarrow (T^{\prime },{\mathcal{M}}_{T^{\prime }})$ such that $f:T\rightarrow T^{\prime }$ is a continuous function of topological spaces and $\unicode[STIX]{x1D711}:f^{-1}({\mathcal{M}}_{T})\rightarrow {\mathcal{M}}_{T^{\prime }}$ is a sheaf homomorphism such that $\unicode[STIX]{x1D711}_{t}^{-1}(\{1\})=\{1\}$ for every $t\in T$ .

Example 2.1.2. If $\mathscr{X}^{+}$ is a log scheme then the Zariski topological space $\mathscr{X}$ is equipped with a sheaf of sharp monoids ${\mathcal{C}}_{\mathscr{X}}$ , namely the characteristic sheaf of $\mathscr{X}^{+}$ . Thus $(\mathscr{X},{\mathcal{C}}_{\mathscr{X}})$ is a monoidal space. Moreover, morphisms of log schemes induce morphisms of characteristic sheaves, hence morphisms of monoidal spaces. We therefore obtain a functor from the category of log schemes to the category of monoidal spaces.

Example 2.1.3. Given a monoid $P$ , we may associate to it a monoidal space called the spectrum of $P$ . As a set, $\text{Spec}\,P$ is the set of all prime ideals of $P$ . The topology is characterized by the basis open sets $D(f)=\{\mathfrak{p}\in \text{Spec}\,P|f\notin \mathfrak{p}\}$ for any $f\in P$ . The monoidal sheaf is defined by

$$\begin{eqnarray}{\mathcal{M}}_{\text{Spec}\,P}(D(f))=S^{-1}P/(S^{-1}P)^{\times }\end{eqnarray}$$

where $S=\{f^{n}|n\geqslant 0\}$ .

2.1.4. A monoidal space isomorphic to the monoidal space $\text{Spec}\,P$ for some monoid $P$ is called an affine Kato fan. A monoidal space is called a Kato fan if it has an open covering consisting of affine Kato fans. In particular, we call a Kato fan integral, saturated, of finite type or fs if it admits a cover by the spectra of monoids with the respective properties.

2.1.5. A morphism of fs Kato fans $F^{\prime }\rightarrow F$ is called a subdivision if it has finite fibers and the morphism

$$\begin{eqnarray}\text{Hom}(\text{Spec}\,\mathbb{N},F^{\prime })\rightarrow \text{Hom}(\text{Spec}\,\mathbb{N},F)\end{eqnarray}$$

is a bijection. By allowing subdivisions, a Kato fan might take the following shape.

Proposition 2.1.6 [Kat94, Proposition 9.8].

Let $F$ be an fs Kato fan. Then there is a subdivision $F^{\prime }\rightarrow F$ such that $F^{\prime }$ has an open cover $\{U_{i}^{\prime }\}$ by affine Kato fans with $U_{i}^{\prime }\simeq \text{Spec}\,\mathbb{N}^{r_{i}}$ .

The strategy of the proof of Proposition 2.1.6 goes back to [KKMS73] and relies on a sequence of particular subdivisions of the Kato fan, the so-called star and barycentric subdivisions [ACMUW16, Example 4.10].

2.2 Kato fans associated to log-regular log schemes

Theorem 2.2.1 [Kat94, Proposition 10.2].

Let $\mathscr{X}^{+}$ be a log-regular log scheme. Then there is an initial strict morphism $(\mathscr{X},{\mathcal{C}}_{\mathscr{X}})\rightarrow F$ to a Kato fan in the category of monoidal spaces. Explicitly, there exist a Kato fan $F$ and a morphism $\unicode[STIX]{x1D71A}:(\mathscr{X},{\mathcal{C}}_{\mathscr{X}})\rightarrow F$ such that $\unicode[STIX]{x1D71A}^{-1}({\mathcal{M}}_{F})\simeq {\mathcal{C}}_{\mathscr{X}}$ and any other morphism from $(\mathscr{X},{\mathcal{C}}_{\mathscr{X}})$ to a Kato fan factors through $\unicode[STIX]{x1D71A}$ .

The Kato fan $F$ in Theorem 2.2.1 is called the Kato fan associated to $\mathscr{X}^{+}$ . Concretely, it is the topological subspace of $\mathscr{X}$ consisting of the points $x$ such that the maximal ideal $\mathfrak{m}_{x}$ of ${\mathcal{O}}_{\mathscr{X},x}$ is equal to ${\mathcal{I}}_{\mathscr{X},x}$ , and ${\mathcal{M}}_{F}$ is the inverse image of ${\mathcal{C}}_{\mathscr{X}}$ on $F$ ; henceforth we write ${\mathcal{C}}_{F}$ for  ${\mathcal{M}}_{F}$ .

Example 2.2.2. Assume that $\mathscr{X}$ is regular, of finite type over $S$ , and $\mathscr{X}_{k}$ is a divisor with strict normal crossings. Then $\mathscr{X}^{+}$ is log-regular and $F$ is the set of generic points of intersections of irreducible components of $\mathscr{X}_{k}$ . For each point $x$ of $F$ , the stalk of ${\mathcal{C}}_{F}$ is isomorphic to $(\mathbb{N}^{r},+)$ , with $r$ the number of irreducible components of $\mathscr{X}_{k}$ that pass through $x$ .

This example admits the following partial generalization.

Lemma 2.2.3. Let $\mathscr{X}^{+}$ be a log-regular log scheme. Then the fan $F$ consists of the generic points of intersections of irreducible components of $D_{\mathscr{X}}$ .

Proof. First, we show that every such generic point is a point of $F$ . Let $E_{1},\ldots ,E_{r}$ be irreducible components of $D_{\mathscr{X}}$ and let $x$ be a generic point of the intersection $E_{1}\cap \cdots \cap E_{r}$ . We set $d=\dim {\mathcal{O}}_{\mathscr{X},x}$ . Since $\mathscr{X}^{+}$ is log-regular, we know that ${\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}$ is regular and that

(2.2.4) $$\begin{eqnarray}d=\dim {\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}+\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}.\end{eqnarray}$$

We denote by $V({\mathcal{I}}_{\mathscr{X},x})$ the vanishing locus of the ideal ${\mathcal{I}}_{\mathscr{X},x}$ in $\text{Spec}\,({\mathcal{O}}_{\mathscr{X},x})$ . We want to prove that ${\mathcal{I}}_{\mathscr{X},x}=\mathfrak{m}_{x}$ . We assume the contrary, hence that ${\mathcal{I}}_{\mathscr{X},x}\subsetneq \mathfrak{m}_{x}$ . This assumption implies that there exists $j$ such that $V({\mathcal{I}}_{\mathscr{X},x})\nsubseteq E_{j}$ : indeed, if the vanishing locus is contained in each irreducible component $E_{i}$ , that is,

$$\begin{eqnarray}V({\mathcal{I}}_{\mathscr{X},x})\subseteq E_{1}\cap \cdots \cap E_{r}\subseteq \overline{\{x\}},\end{eqnarray}$$

then ${\mathcal{I}}_{\mathscr{X},x}\supseteq \mathfrak{m}_{x}$ . From the assumption of log-regularity it follows that the vanishing locus $V({\mathcal{I}}_{\mathscr{X},x})$ is a regular subscheme, and, moreover, that $\mathscr{X}^{+}$ is Cohen–Macaulay by [Kat94, Theorem 4.1]. Thus, there exists a regular sequence $(f_{1},\ldots ,f_{l})$ in ${\mathcal{I}}_{\mathscr{X},x}$ , where $l$ is the codimension of $V({\mathcal{I}}_{\mathscr{X},x})$ , that is,

$$\begin{eqnarray}\dim {\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}=d-l.\end{eqnarray}$$

Moreover, by the equality (2.2.4), $\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}=l$ .

We claim that the residue classes of these elements $f_{i}$ in ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ are linearly independent. Assume the contrary. If $l=1$ the proposition holds because $f_{1}$ is not a unit. Otherwise, $l\geqslant 2$ and, up to renumbering the $f_{i}$ , there exist an integer $e$ with $1<e\leqslant l$ , non-negative integers $a_{1},\ldots ,a_{l}$ , not all zero, and a unit $u$ in ${\mathcal{O}}_{\mathscr{X},x}$ such that

$$\begin{eqnarray}f_{1}^{a_{1}}\cdot \ldots \cdot f_{e-1}^{a_{e-1}}=u\cdot f_{e}^{a_{e}}\cdot \ldots \cdot f_{l}^{a_{l}}.\end{eqnarray}$$

This contradicts the fact that $(f_{1},\ldots ,f_{l})$ is a regular sequence in ${\mathcal{I}}_{\mathscr{X},x}$ . Thus, the classes $\overline{f_{1}},\ldots ,\overline{f_{l}}$ are independent in ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ . As we also have the equality $\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}=l$ , it follows that these classes generate ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}\otimes _{\mathbb{Z}}\mathbb{Q}$ .

Let $g_{j}$ be a non-zero element of the ideal ${\mathcal{I}}_{\mathscr{X},x}$ that vanishes along $E_{j}$ : it necessarily exists as otherwise $E_{j}$ is not a component of the divisor $D_{\mathscr{X}}$ . Then $g_{j}$ satisfies

$$\begin{eqnarray}g_{j}^{N}=v\cdot f_{1}^{b_{1}}\cdot \ldots \cdot f_{l}^{b_{l}}\end{eqnarray}$$

with $b_{i}\in \mathbb{Z}$ , $v$ is a unit in ${\mathcal{O}}_{\mathscr{X},x}$ and $N$ is a positive integer. As $g_{j}$ vanishes along the irreducible component $E_{j}$ , at least one of the functions $f_{1},\ldots ,f_{l}$ has to vanish along $E_{j}$ : assume that it is  $f_{1}$ .

On the one hand, as $f_{1}$ is identically zero on $E_{j}$ , the trace of $E_{j}$ on $V({\mathcal{I}}_{\mathscr{X},x})$ has codimension at most $l-1$ in $E_{j}$ at the point $x$ . On the other hand, we assumed that $V({\mathcal{I}}_{\mathscr{X},x})$ is not contained in $E_{j}$ and it has codimension $l$ in ${\mathcal{O}}_{\mathscr{X},x}$ . Then the trace of $E_{j}$ on $V({\mathcal{I}}_{\mathscr{X},x})$ has codimension $l$ in $E_{j}$ at $x$ . This is a contradiction. We conclude that the ideal ${\mathcal{I}}_{\mathscr{X},x}$ is equal to the maximal ideal $\mathfrak{m}_{x}$ , therefore $x$ is a point of $F$ .

It remains to prove the converse implication: every point $x$ of the fan $F$ is a generic point of an intersection of irreducible components of $D_{\mathscr{X}}$ . Let $x$ be a point of $F$ : by construction of Kato fan $F$ , the maximal ideal of ${\mathcal{O}}_{\mathscr{X},x}$ is equal to ${\mathcal{I}}_{\mathscr{X},x}$ , thus it is generated by elements in ${\mathcal{M}}_{\mathscr{X},x}$ . The zero locus of such an element is contained in $D_{\mathscr{X}}$ by definition of the logarithmic structure on $\mathscr{X}^{+}$ . Therefore, the zero locus of a generator of $\mathfrak{m}_{x}$ in ${\mathcal{M}}_{\mathscr{X},x}$ is a union of irreducible components of the trace of $D_{\mathscr{X}}$ on $\text{Spec}\,{\mathcal{O}}_{\mathscr{X},x}$ and $x$ is a generic point of the intersection of all such irreducible components.◻

Remark 2.2.5. By convention, the generic point of the empty intersection of irreducible components is the generic point of $\mathscr{X}$ . By definition, this point is also included in the Kato fan $F$ . Thus, for example, the Kato fan associated to $S^{+}$ consists of two points: the generic point of $S$ that corresponds to the empty intersection, and the closed point $s$ corresponding to the unique irreducible component of the logarithmic divisorial structure.

Moreover, Example 2.2.2 also leads to the following characterization.

Proposition 2.2.6 [GR04, Corollary 12.5.35].

Let $\mathscr{X}^{+}$ be a log-regular log scheme over $S^{+}$ and $F$ its associated Kato fan. The following statements are equivalent:

1. (i) for every $x\in F$ , ${\mathcal{C}}_{F,x}\simeq \mathbb{N}^{r(x)}$ ,

2. (ii) the underlying scheme $\mathscr{X}$ is regular.

If this is the case, then the special fiber $\mathscr{X}_{k}$ is an snc divisor.

2.2.7. The construction of the Kato fan of a log scheme defines a functor from the category of log-regular log schemes to the category of Kato fans. Indeed, given a morphism of log schemes $\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}$ , we consider the embedding of the associated Kato fan $F_{\mathscr{X}}$ in $\mathscr{X}^{+}$ and the canonical morphism $\mathscr{Y}^{+}\rightarrow F_{\mathscr{Y}}$ : the composition

$$\begin{eqnarray}F_{\mathscr{X}}{\hookrightarrow}\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}\rightarrow F_{\mathscr{ Y}}\end{eqnarray}$$

functorially induces a map between associated Kato fans. Moreover, this association preserves strict morphisms [Uli17, Lemma 4.9].

2.3 Resolutions of log schemes via Kato fan subdivisions

Proposition 2.3.1 [Kat94, Proposition 9.9].

Let $\mathscr{X}^{+}$ be a log-regular log scheme and let $F$ be its associated Kato fan. Let $F^{\prime }\rightarrow F$ be a subdivision of fans. Then there exist a log scheme ${\mathscr{X}^{\prime }}^{+}$ , a morphism of log schemes ${\mathscr{X}^{\prime }}^{+}\rightarrow \mathscr{X}^{+}$ , and a commutative diagram

such that $p^{-1}({\mathcal{M}}_{F^{\prime }})\simeq {\mathcal{C}}_{\mathscr{X}^{\prime }}$ ; they define a final object in the category of such diagrams and the refinement $F^{\prime }\rightarrow F$ is functorially induced by the morphism of log-regular log schemes ${\mathscr{X}^{\prime }}^{+}\rightarrow \mathscr{X}^{+}$ .

2.3.2. It follows that given any subdivision $F^{\prime }\rightarrow F$ of the Kato fan F associated with a log-regular log scheme $\mathscr{X}^{+}$ , we can construct a log scheme over $\mathscr{X}^{+}$ with prescribed associated Kato fan $F^{\prime }$ . Combining this fact with Propositions 2.1.6 and 2.2.6 yields a construction of resolutions of log schemes in the following sense: for any log-regular log scheme over $S^{+}$ we can find a birational modification by a regular log scheme with snc special fiber. Moreover, the morphism of log schemes ${\mathscr{X}^{\prime }}^{+}\rightarrow \mathscr{X}^{+}$ is obtained by a log blow-up [Niz06, Theorem 5.8].

2.4 Fibred products and associated Kato fans

2.4.1. Given morphisms of fs log schemes $f_{1}:\mathscr{X}_{1}^{+}\rightarrow \mathscr{Y}^{+}$ and $f_{2}:\mathscr{X}_{2}^{+}\rightarrow \mathscr{Y}^{+}$ , their fibred product exists in the category of log schemes. It is obtained by endowing the usual fibred product of schemes

(2.4.2)

with the log structure associated to $p_{1}^{-1}{{\mathcal{M}}_{\mathscr{X}_{1}}\oplus }_{p_{\mathcal{Y}}^{-1}{\mathcal{M}}_{\mathcal{Y}}}p_{2}^{-1}{\mathcal{M}}_{\mathscr{X}_{2}}$ . If $u_{1}:P\rightarrow Q_{1}$ and $u_{2}:P\rightarrow Q_{2}$ are charts for the morphisms $f_{1}$ and $f_{2}$ respectively, then the induced morphism $\mathscr{X}_{1}\times _{\mathscr{Y}}\mathscr{X}_{2}\rightarrow \text{Spec}\,\mathbb{Z}[{Q_{1}\oplus }_{P}Q_{2}]$ is a chart for $\mathscr{X}_{1}^{+}\times _{\mathscr{Y}^{+}}\mathscr{X}_{2}^{+}$ .

2.4.3. In general, the fibred product is not fs, but the category of fs log schemes also admits fibred products. Keeping the same notation, the following is a local description of the fibred product in the category of fine and saturated log schemes

$$\begin{eqnarray}\mathscr{X}_{1}^{+}\times _{\mathscr{ Y}^{+}}^{\mathit{ fs}}\mathscr{X}_{2}^{+}=(\mathscr{X}_{1}^{+}\times _{\mathscr{ Y}^{+}}\mathscr{X}_{2}^{+})\times _{\mathbb{ Z}[{Q_{1}\oplus }_{P}Q_{2}]}\mathbb{Z}[({Q_{1}\oplus }_{P}Q_{2})^{\text{sat}}]\end{eqnarray}$$

in terms of the charts for $f_{1}$ and $f_{2}$ [Bul15, 3.6.16]. We remark that the two fibred products above may not only have different log structures, but also the underlying schemes may differ. Nevertheless, this obviously does not occur when the monoid ${Q_{1}\oplus }_{P}Q_{2}$ is saturated.

2.4.4. Log-smoothness is preserved under fs base change and composition [GR04, Proposition 12.3.24]. In particular, if $f_{1}:\mathscr{X}_{1}^{+}\rightarrow \mathscr{Y}^{+}$ is log-smooth and $\mathscr{X}_{2}^{+}$ is log-regular, then $\mathscr{X}_{1}^{+}\times _{\mathscr{Y}^{+}}^{\mathit{fs}}\mathscr{X}_{2}^{+}$ is log-regular, by [Kat94, Theorem 8.2].

Consider log-smooth morphisms of fs log schemes $\mathscr{X}_{1}^{+}\rightarrow \mathscr{Y}^{+}$ and $\mathscr{X}_{2}^{+}\rightarrow \mathscr{Y}^{+}$ . The sheaves of logarithmic differentials are related by the isomorphism

(2.4.5) $$\begin{eqnarray}p_{1}^{\ast }\unicode[STIX]{x1D6FA}_{\mathscr{X}_{1}^{+}/\mathscr{Y}^{+}}^{\log }\oplus p_{2}^{\ast }\unicode[STIX]{x1D6FA}_{\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }\simeq \unicode[STIX]{x1D6FA}_{\mathscr{X}_{1}^{+}\times _{\mathcal{Y}^{+}}^{\mathit{fs}}\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }\end{eqnarray}$$

by [GR04, Proposition 12.3.13]. Furthermore, by assumption of log-smoothness over $S^{+}$ the logarithmic differential sheaves are locally free of finite rank [Kat94, Proposition 3.10] and we can consider their determinants; they are called log canonical bundles and denoted by $\unicode[STIX]{x1D714}^{\log }$ . The following isomorphism is a direct consequence of (2.4.5):

(2.4.6) $$\begin{eqnarray}p_{1}^{\ast }\unicode[STIX]{x1D714}_{\mathscr{X}_{1}^{+}/\mathscr{Y}^{+}}^{\log }\otimes p_{2}^{\ast }\unicode[STIX]{x1D714}_{\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }\simeq \unicode[STIX]{x1D714}_{\mathscr{X}_{1}^{+}\times _{\mathcal{Y}^{+}}^{\mathit{fs}}\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }.\end{eqnarray}$$

2.4.7. Similarly to the construction of fibred products of fs log schemes, the category of fs Kato fans admits fibred products: on affine Kato fans $F=\text{Spec}\,P$ and $G=\text{Spec}\,Q$ over $H=\text{Spec}\,T$ , $F\times _{H}G$ is the spectrum of the amalgamated sum $(P\oplus _{T}Q)^{\text{sat}}$ in the category of fs monoids [Uli19, Proposition 2.4] and on the underlying topological spaces this coincides with the usual fibred product.

We seek to compare the Kato fan associated to the fibred product of log-regular log schemes with the fibred product of associated Kato fans.

Proposition 2.4.8 [Sai04, Lemma 2.8].

Given $\mathscr{T}^{+}$ a log-regular log scheme, let $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ be log-smooth log schemes over $\mathscr{T}^{+}$ . We denote by $\mathscr{Z}^{+}$ the fs fibred product $\mathscr{X}^{+}\times _{\mathscr{T}^{+}}^{\mathit{fs}}\mathscr{Y}^{+}$ . Then, the natural morphisms $F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}$ and $F_{\mathscr{Z}}\rightarrow F_{\mathscr{Y}}$ induce a morphism of Kato fans

(2.4.9) $$\begin{eqnarray}F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}\times _{F_{\mathscr{T}}}F_{\mathscr{Y}}\end{eqnarray}$$

that is locally an isomorphism.

2.4.10. For any pair of points $(x,y)$ in $F_{\mathscr{X}}\times _{F_{\mathscr{T}}}F_{\mathscr{Y}}$ , we denote by $n(x,y)$ the number of preimages of $(x,y)$ in the Kato fan of $\mathscr{Z}^{+}$ under the local isomorphism (2.4.9).

Lemma 2.4.11. If $x^{\prime }$ is in the closure of $x$ , and $y^{\prime }$ in the closure of $y$ , then $n(x^{\prime },y^{\prime })\geqslant n(x,y)$ .

Proof. Let $z^{\prime }$ be a preimage of the pair $(x^{\prime },y^{\prime })$ . By Proposition 2.4.8, there exists an open neighborhood $U_{z^{\prime }}$ of $z^{\prime }$ such that the restriction of $F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}\times _{F_{\mathscr{T}}}F_{\mathscr{Y}}$ to $U_{z^{\prime }}$ is an isomorphism onto its image. In particular, $(x,y)$ lies in this image. Thus, there exists a unique preimage of $(x,y)$ that is contained in $U_{z^{\prime }}$ . It follows that $n(x^{\prime },y^{\prime })\geqslant n(x,y)$ .◻

2.5 Semistability and Kato fans associated to the fibred products

2.5.1. We recall that a log-regular log scheme $\mathscr{X}^{+}$ is said to be semistable if the special fiber is reduced. We will see that semistability is a sufficient condition to establish injectivitiy of the local isomorphism (2.4.9).

2.5.2. Given a log-regular log scheme $\mathscr{X}^{+}$ over $S^{+}$ , the morphism $f:\mathscr{X}^{+}\rightarrow S^{+}$ is called saturated if for any $x\in \mathscr{X}$ the morphism ${\mathcal{C}}_{S,f(x)}\rightarrow {\mathcal{C}}_{\mathscr{X},x}$ on the stalks of the characteristic sheaves is a saturated morphism of monoids; that is, if for any $x$ and any morphism $u:{\mathcal{C}}_{S,f(x)}\rightarrow P$ of fs monoids, the amalgamated sum ${{\mathcal{C}}_{\mathscr{X},x}\oplus }_{{\mathcal{C}}_{S,f(x)}}P$ is still a saturated monoid.

Following the work by T. Tsuji in an unpublished 1997 preprint, Vidal in [Vid04] defines the notion of saturation indexes for morphisms of log schemes. In the case of a log-regular log scheme $\mathscr{X}^{+}$ over $S^{+}$ , the saturation index at a point $x\in \mathscr{X}_{k}$ is the least common multiple of the multiplicities in $\mathscr{X}_{k}$ of the prime components of $\mathscr{X}_{k}$ passing through $x$ . The following criterion holds.

Lemma 2.5.3 [Vid04, § 1.3].

The morphism $\mathscr{X}^{+}\rightarrow S^{+}$ is saturated if and only if the saturation index at any point of $\mathscr{X}_{k}$ is equal to $1$ .

Proposition 2.5.4. Assume that the residue field $k$ is algebraically closed. Let $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ be log-smooth log schemes over $S^{+}$ . Let $\mathscr{Z}^{+}$ be their fs fibred product. If $\mathscr{X}^{+}$ is semistable, then for any pair of points $(x,y)$ in $F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}$ whose closures intersect the special fibers $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ respectively, the morphism $F_{\mathscr{Z}}\xrightarrow[{}]{}F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}},$ induced by the projections $\mathscr{Z}^{+}\rightarrow \mathscr{X}^{+}$ and $\mathscr{Z}^{+}\rightarrow \mathscr{Y}^{+}$ , is a bijection above the pair $(x,y)$ , namely $n(x,y)=1$ .

Proof. Since $\mathscr{X}^{+}$ is a semistable log-regular log scheme over $S^{+}$ , the saturation index of $\mathscr{X}^{+}\rightarrow S^{+}$ at any point $x$ in $\mathscr{X}_{k}$ is $1$ . Thus, by Lemma 2.5.3 the morphism $\mathscr{X}^{+}\rightarrow S^{+}$ is saturated, and it follows that the fibred product in the category of log schemes coincides with the fibred product in the category of fs log schemes. This fact is crucial in the sequel of the proof to describe explicitly the divisor $D_{\mathscr{Z}}$ , and to characterize the points of $\mathscr{Z}_{k}$ .

We write $D_{\mathscr{X}}=\unicode[STIX]{x1D6E5}_{\mathscr{X}}+\mathscr{X}_{k}$ , where $\unicode[STIX]{x1D6E5}_{\mathscr{X}}=\sum _{i}\unicode[STIX]{x1D6E5}_{\mathscr{X},i}$ is the horizontal part and $\mathscr{X}_{k}=\sum _{i}E_{\mathscr{X},i}$ ; and $D_{\mathscr{Y}}=\unicode[STIX]{x1D6E5}_{\mathscr{Y}}+\mathscr{Y}_{k,\text{red}}$ , with $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}=\sum _{j}\unicode[STIX]{x1D6E5}_{\mathscr{Y},j}$ and $\mathscr{Y}_{k,\text{red}}=\sum _{j}E_{\mathscr{Y},j}$ . The divisor $D_{\mathscr{Z}}$ , associated to the log structure of the fs fibred product, is given by $D_{\mathscr{Z}}=\unicode[STIX]{x1D6E5}_{\mathscr{Z}}+\mathscr{Z}_{k,\text{red}}$ , where $\unicode[STIX]{x1D6E5}_{\mathscr{Z}}=\unicode[STIX]{x1D6E5}_{\mathscr{X}}\times \mathscr{Y}+\mathscr{X}\times \unicode[STIX]{x1D6E5}_{\mathscr{Y}}=\sum _{l}\unicode[STIX]{x1D6E5}_{\mathscr{Z},l}$ and $\mathscr{Z}_{k,\text{red}}=\sum _{h}E_{\mathscr{Z},h}.$ As $k$ is algebraically closed, the irreducible components of $\mathscr{Z}_{k}=\mathscr{X}_{k}\times _{k}\mathscr{Y}_{k}$ are given by the products of irreducible components of $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ ; that is, for any $h$ we have $E_{\mathscr{Z},h}=E_{\mathscr{X},i_{h}}\times _{k}E_{\mathscr{Y},j_{h}}$ .

The points in the special fiber $\mathscr{Z}_{k}$ are characterized as follows:

$$\begin{eqnarray}z=(x,y,s,\mathfrak{p})\quad \text{and}\quad {\mathcal{O}}_{\mathscr{Z},z}=({\mathcal{O}}_{\mathscr{X},x}\otimes _{R}{\mathcal{O}}_{\mathscr{Y},y})_{\mathfrak{p}},\end{eqnarray}$$

where $x$ and $y$ are points of $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ both mapped to $s$ , while $\mathfrak{p}$ is a prime ideal of the tensor product of residue fields $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ . We assume now that $z$ is a point in $F_{\mathscr{Z}}\cap \mathscr{Z}_{k}$ , and see the restrictions on $x$ , $y$ , and $\mathfrak{p}$ that follow from this assumption.

1. (i) We claim that the projections $x$ and $y$ are points of the respective Kato fans. By Lemma 2.2.3, $z$ is a generic point of an intersection of irreducible components of $D_{\mathscr{Z}}$ .

   If every irreducible component of $D_{\mathscr{Z}}$ containing $z$ is in the special fiber, then the intersection is given by

   $$\begin{eqnarray}\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ Z},h}=\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ X},i_{h}}\times E_{\mathscr{Y},j_{h}}=\bigg(\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ X},i_{h}}\bigg)\times \bigg(\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ Y},j_{h}}\bigg).\end{eqnarray}$$
   The projections $x$ and $y$ of $z$ are generic points of strata in $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ , hence lie in the respective Kato fans.

   Otherwise, let $\unicode[STIX]{x1D6E5}_{\mathscr{Z},1},\ldots ,\unicode[STIX]{x1D6E5}_{\mathscr{Z},r^{\prime }}$ be the horizontal components of $D_{\mathscr{Z}}$ passing through $z$ . We consider the subscheme $\mathscr{D}=\bigcap _{l=1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{Z},l}$ of $\mathscr{Z}$ , endowed with the divisorial log structure $D_{\mathscr{D}}=\mathscr{D}_{k,\text{red}}$ . Then, $\mathscr{D}^{+}=(\mathscr{D},D_{\mathscr{D}})$ is a log-regular log scheme and $z$ is a point in the Kato fan of $\mathscr{D}^{+}$ that corresponds to the generic point of an intersection of irreducible components of $D_{\mathscr{D}}$ , that are all contained in the special fiber. Since

   $$\begin{eqnarray}\mathscr{D}=\mathop{\bigcap }_{l=1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{Z},l}=\bigg(\mathop{\bigcap }_{l=1}^{r^{\prime \prime }}\unicode[STIX]{x1D6E5}_{\mathscr{X},l}\times \mathscr{Y}\bigg)\cap \bigg(\mathop{\bigcap }_{l=r^{\prime \prime }+1}^{r^{\prime }}\mathscr{X}\times \unicode[STIX]{x1D6E5}_{\mathscr{Y},l}\bigg)=\bigg(\mathop{\bigcap }_{l=1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{X},l}\bigg)\times \bigg(\mathop{\bigcap }_{l=r^{\prime \prime }+1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{Y},l}\bigg)\end{eqnarray}$$
   is itself a fibred product of log-smooth log schemes, we conclude by the previous case.

2. (ii) We claim that the ideal $\mathfrak{p}$ is minimal. Since $z$ lies in $F_{\mathscr{Z}}$ , we have that $\dim {\mathcal{O}}_{\mathscr{Z},z}=\text{rank}{\mathcal{C}}_{\mathscr{Z},z}^{\text{gp}}$ . At the level of characteristic sheaves

   $$\begin{eqnarray}\text{rank}{\mathcal{C}}_{\mathscr{Z},z}^{\text{gp}}=\text{rank}{\mathcal{C}}_{\mathscr{ X},x}^{\text{gp}}+\text{rank}{\mathcal{C}}_{\mathscr{ Y},y}^{\text{gp}}-1\end{eqnarray}$$
   holds, since the stalk at $z$ of the characteristic sheaf of $\mathscr{Z}^{+}$ is equal to ${\mathcal{C}}_{\mathscr{Z},z}={{\mathcal{C}}_{\mathscr{X},x}\oplus }_{\mathbb{N}}{\mathcal{C}}_{\mathscr{Y},y}$ and the morphism $\mathbb{N}\rightarrow {\mathcal{C}}_{\mathscr{X},x}$ is saturated. As $x$ and $y$ are both points in the associated Kato fans, the equality between the dimension of the local ring and the rank of the groupification of the characteristic sheaves holds for $x$ and $y$ as well, and we obtain
   $$\begin{eqnarray}\displaystyle \dim {\mathcal{O}}_{\mathscr{Z},z}=\text{rank}{\mathcal{C}}_{\mathscr{Z},z}^{\text{gp}}=\text{rank}{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}+\text{rank}{\mathcal{C}}_{\mathscr{Y},y}^{\text{gp}}-1=\dim {\mathcal{O}}_{\mathscr{X},x}+\dim {\mathcal{O}}_{\mathscr{Y},y}-1. & & \displaystyle \nonumber\end{eqnarray}$$
   By log-regularity of $\mathscr{Z}^{+}$ we have the inequality $\dim {\mathcal{O}}_{\mathscr{Z},z^{\prime }}\geqslant \text{rank}{\mathcal{C}}_{\mathscr{Z},z^{\prime }}^{\text{gp}}$ at any point $z^{\prime }$ , hence the equality that holds for $z$ necessarily implies that $\mathfrak{p}$ is a minimal prime ideal of $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ .

Let $(x,y)$ be a pair of points of $F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}$ that lie in the respective special fibers. In order to determine the number $n(x,y)$ of preimages of $(x,y)$ in $F_{\mathscr{Z}}$ , we need to study the number of minimal prime ideals of the tensor product $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ . Since the residue field $k$ is an algebraically closed field, the tensor product $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ is a domain; in particular, it has a unique minimal prime ideal, namely $0$ . We obtain that $n(x,y)=1$ .

Let $(x,y)$ be a pair of points in $F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}$ whose closures intersect the special fibers, namely there exist $x^{\prime }\in F_{\mathscr{X}}\cap \mathscr{X}_{k}$ and $y^{\prime }\in F_{\mathscr{Y}}\cap \mathscr{Y}_{k}$ such that $x^{\prime }$ is in the closure of $x$ and $y^{\prime }$ in the closure of $y$ . Then, by the previous part of the proof and by Lemma 2.4.11, we have $n(x,y)\leqslant n(x^{\prime },y^{\prime })=1$ .◻

3 The skeleton of a log-regular log scheme

3.1 Construction

3.1.1. Let $\mathscr{X}^{+}$ be a log-regular log scheme over $S^{+}$ . Let $x$ be a point of the associated Kato fan $F$ . Denote by $F(x)$ the set of points $y$ of $F$ such that $x$ lies in the closure of $\{y\}$ , and by ${\mathcal{C}}_{F(x)}$ the restriction of ${\mathcal{C}}_{F}$ to $F(x)$ . Denote by $\text{Spec}\,{\mathcal{C}}_{\mathscr{X},x}$ the spectrum of the monoid ${\mathcal{C}}_{\mathscr{X},x}={\mathcal{C}}_{F,x}$ . Then there exists a canonical isomorphism of monoid spaces

$$\begin{eqnarray}(F(x),{\mathcal{C}}_{F(x)})\rightarrow \text{Spec}\,{\mathcal{C}}_{\mathscr{X},x}:y\mapsto \{s\in {\mathcal{C}}_{\mathscr{X},x}\,|\,s(y)=0\},\end{eqnarray}$$

where the expression $s(y)=0$ means that $s^{\prime }(y)=0$ for any representative $s^{\prime }$ of $s$ in ${\mathcal{M}}_{\mathscr{X},x}$ . In particular, we obtain a bijective correspondence between the faces of the monoid ${\mathcal{C}}_{\mathscr{X},x}$ and the points of $F(x)$ , and for every point $y$ of $F(x)$ , a surjective cospecialization morphism of monoids

$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{x,y}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{C}}_{\mathscr{X},y}\end{eqnarray}$$

which induces an isomorphism of monoids

$$\begin{eqnarray}S^{-1}{\mathcal{C}}_{\mathscr{X},x}/(S^{-1}{\mathcal{C}}_{\mathscr{X},x})^{\times }\cong {\mathcal{C}}_{\mathscr{ X},x}/S\xrightarrow[{}]{{\sim}}{\mathcal{C}}_{\mathscr{X},y},\end{eqnarray}$$

where $S$ denotes the monoid of elements $s$ in ${\mathcal{C}}_{\mathscr{X},x}$ such that $s(y)\neq 0$ .

3.1.2. For each point $x$ in $F$ , we denote by $\unicode[STIX]{x1D70E}_{x}$ the set of morphisms of monoids

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow (\mathbb{R}_{{\geqslant}0},+)\end{eqnarray}$$

such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=1$ for every uniformizer $\unicode[STIX]{x1D70B}$ in $R$ . We endow $\unicode[STIX]{x1D70E}_{x}$ with the topology of pointwise convergence, where $\mathbb{R}_{{\geqslant}0}$ carries the usual Euclidean topology. Note that $\unicode[STIX]{x1D70E}_{x}$ is a polyhedron, possibly unbounded, in the real affine space

$$\begin{eqnarray}\{\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}\rightarrow (\mathbb{R},+)\,|\,\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=1\text{ for every uniformizer }\unicode[STIX]{x1D70B}\text{ in }R\}.\end{eqnarray}$$

If $y$ is a point of $F(x)$ , then the surjective cospecialization morphism $\unicode[STIX]{x1D70F}_{x,y}$ induces a topological embedding $\unicode[STIX]{x1D70E}_{y}\rightarrow \unicode[STIX]{x1D70E}_{x}$ that identifies $\unicode[STIX]{x1D70E}_{y}$ with a face of $\unicode[STIX]{x1D70E}_{x}$ .

3.1.3. We denote by $T$ the disjoint union of the topological spaces $\unicode[STIX]{x1D70E}_{x}$ with $x$ in $F$ . On the topological space $T$ , we consider the equivalence relation ${\sim}$ generated by couples of the form $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}\circ \unicode[STIX]{x1D70F}_{x,y})$ where $x$ and $y$ are points in $F$ such that $x$ lies in the closure of $\{y\}$ and $\unicode[STIX]{x1D6FC}$ is a point of $\unicode[STIX]{x1D70E}_{y}$ .

The skeleton of $\mathscr{X}^{+}$ is defined as the quotient of the topological space $T$ by the equivalence relation ${\sim}$ . We denote this skeleton by $\text{Sk}(\mathscr{X}^{+})$ . It is clear that $\text{Sk}(\mathscr{X}^{+})$ has the structure of a polyhedral complex with cells $\{\unicode[STIX]{x1D70E}_{x},\,x\in F\}$ , so it comes equipped with a PL structure, and that the faces of a cell $\unicode[STIX]{x1D70E}_{x}$ are precisely the cells $\unicode[STIX]{x1D70E}_{y}$ with $y$ in $F(x)$ .

3.1.4. We note that $\unicode[STIX]{x1D70E}_{x}$ is empty for any point $x$ that does not lie in the special fiber $\mathscr{X}_{k}$ : indeed, outside the special fiber any uniformizer is an invertible element, so it is trivial in ${\mathcal{C}}_{\mathscr{X},x}$ and is mapped to $0$ by any morphism of monoids. Therefore, the construction of the skeleton associated to $\mathscr{X}^{+}$ only concerns the points in the Kato fan $F$ that lie in the special fiber. Moreover, given a generic point $x\in \mathscr{X}_{k}$ of an intersection of components of $D_{\mathscr{X}}$ , where at least one component is not in the special fiber, the corresponding face $\unicode[STIX]{x1D70E}_{x}$ is unbounded.

In other words, the skeleton associated to a log-regular scheme $\mathscr{X}^{+}$ , where $D_{\mathscr{X}}$ allows horizontal components, generalizes Berkovich’s skeletons by admitting unbounded faces in the direction of the horizontal components as well as by allowing singularities in the special fiber. It also generalizes the construction performed by Gubler, Rabinoff, and Werner in [GRW16] of a skeleton associated to a strictly semistable snc pair.

3.2 Embedding the skeleton in the non-archimedean generic fiber

3.2.1. Let $\mathscr{X}^{+}$ be a log-regular log scheme over $S^{+}$ . Let $x$ be a point of the associated Kato fan $F$ . As the log structure on $\mathscr{X}^{+}$ is of finite type, the characteristic monoid ${\mathcal{C}}_{\mathscr{X},x}$ is of finite type too, and thus ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ is a free abelian group of finite rank. Hence there exists a section

$$\begin{eqnarray}\unicode[STIX]{x1D701}:{\mathcal{M}}_{\mathscr{X},x}^{\text{gp}}/{\mathcal{M}}_{\mathscr{ X},x}^{\times }\rightarrow {\mathcal{M}}_{\mathscr{X},x}^{\text{gp}}.\end{eqnarray}$$

The section $\unicode[STIX]{x1D701}$ restricts to ${\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{M}}_{\mathscr{X},x}$ ; indeed, if $x\in {\mathcal{M}}_{\mathscr{X},x}$ then $\unicode[STIX]{x1D701}(\overline{x})-x\in {\mathcal{M}}_{\mathscr{X},x}^{\times }$ . Therefore we may choose a section

(3.2.2) $$\begin{eqnarray}{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{M}}_{\mathscr{X},x}\end{eqnarray}$$

of the projection homomorphism

$$\begin{eqnarray}{\mathcal{M}}_{\mathscr{X},x}\rightarrow {\mathcal{C}}_{\mathscr{X},x}\end{eqnarray}$$

and use this section to view ${\mathcal{C}}_{\mathscr{X},x}$ as a submonoid of ${\mathcal{M}}_{\mathscr{X},x}$ . Note that ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ generates the ideal ${\mathcal{I}}_{\mathscr{X},x}$ of ${\mathcal{O}}_{\mathscr{X},x}$ .

We propose a generalization of [MN15, Lemma 2.4.4].

Lemma 3.2.3. Let $A$ be a Noetherian ring, let $I$ be an ideal of $A$ , and let $(y_{1},\ldots ,y_{m})$ be a system of generators for $I$ . We denote by $\hat{A}$ the $I$ -adic completion of $A$ . Let $B$ be a subring of $A$ such that the elements $y_{1},\ldots ,y_{m}$ belong to $B$ and generate the ideal $B\cap I$ in $B$ . Then, in the ring $\hat{A}$ , every element $f$ of $B$ can be written as

(3.2.4) $$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \mathbb{Z}_{{\geqslant}0}^{m}}c_{\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}},\end{eqnarray}$$

where the coefficients $c_{\unicode[STIX]{x1D6FD}}$ belong to $((A\setminus I)\cap B)\cup \{0\}$ .

Proof. Let $f$ be an element of $B$ ; we construct an expansion for $f$ of the form (3.2.4) by induction. If $f$ belongs to the complement of $I$ , the conclusion trivially holds. Otherwise, $f$ belongs to $I$ and we can write $f$ as a linear combination of the elements $y_{1},\ldots ,y_{m}$ with coefficients in $B$ :

$$\begin{eqnarray}f=\mathop{\sum }_{j=1}^{m}b_{j}y_{j},\quad b_{j}\in B.\end{eqnarray}$$

By the induction hypothesis, we suppose that $i$ is a positive integer and that we can write every $f$ in $B$ as a sum of an element $f_{i}$ of the form (3.2.4) and a linear combination of degree- $i$ monomials in the elements $y_{1},\ldots ,y_{m}$ with coefficients in $B$ . We apply this assumption to the coefficients $b_{j}$ , hence

$$\begin{eqnarray}b_{j}=b_{j,i}+\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FD}\in \mathbb{Z}_{{\geqslant}0}^{m} \\ |\unicode[STIX]{x1D6FD}|=i}}b_{j,\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}},\quad b_{j,\unicode[STIX]{x1D6FD}}\in B.\end{eqnarray}$$

Then we can write $f$ as a sum of an element $f_{i+1}$ of the form (3.2.4) and a linear combination of degree-( $i+1$ ) monomials in the elements $y_{1},\ldots ,y_{m}$ with coefficients in $B$ :

$$\begin{eqnarray}f=\underbrace{\mathop{\sum }_{j=1}^{m}b_{j,i}y_{j}}_{f_{i+1}}+\mathop{\sum }_{j=1}^{m}\Bigg(\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FD}\in \mathbb{Z}_{{\geqslant}0}^{m} \\ |\unicode[STIX]{x1D6FD}|=i}}b_{j,\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}}\Bigg)y_{j}\end{eqnarray}$$

such that $f_{i}$ and $f_{i+1}$ have the same coefficients in degree less than or equal to $i$ . Iterating this construction, we finally find an expansion of $f$ of the required form.◻

3.2.5. Let $f$ be an element of ${\mathcal{O}}_{\mathscr{X},x}$ . Considering $A=B={\mathcal{O}}_{\mathscr{X},x}$ , $I=\mathfrak{m}_{x}$ , and a system of generators for $\mathfrak{m}_{x}$ in ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ , by Lemma 3.2.3 we can write $f$ as a formal power series

(3.2.6) $$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FE}\end{eqnarray}$$

in $\widehat{{\mathcal{O}}}_{\mathscr{X},x}$ , where each coefficient $c_{\unicode[STIX]{x1D6FE}}$ is either zero or a unit in ${\mathcal{O}}_{\mathscr{X},x}$ . We call this formal series an admissible expansion of $f$ . We set

(3.2.7) $$\begin{eqnarray}S=\{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}\,|\,c_{\unicode[STIX]{x1D6FE}}\neq 0\}\end{eqnarray}$$

and we denote by $\unicode[STIX]{x1D6E4}_{x}(f)$ the set of elements of $S$ that lie on a compact face of the convex hull of $S+{\mathcal{C}}_{\mathscr{X},x}$ in ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}\otimes _{\mathbb{Z}}\mathbb{R}$ . We call $\unicode[STIX]{x1D6E4}_{x}(f)$ the initial support of $f$ at $x$ , notation which is justified by the next proposition.

Proposition 3.2.8.

1. (i) The element

   $$\begin{eqnarray}f_{x}=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{x}(f)}c_{\unicode[STIX]{x1D6FE}}(x)\unicode[STIX]{x1D6FE}\in k(x)[{\mathcal{C}}_{\mathscr{X},x}]\end{eqnarray}$$
   depends on the choice of the section (3.2.2), but not on the expansion (3.2.6).

2. (ii) The subset $\unicode[STIX]{x1D6E4}_{x}(f)$ of ${\mathcal{C}}_{\mathscr{X},x}$ only depends on $f$ and $x$ , and not on the choice of the section (3.2.2) or the expansion (3.2.6).

Proof. If we denote by $I$ the ideal of $k(x)[{\mathcal{C}}_{\mathscr{X},x}]$ generated by ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ , then it follows from [Kat94] that there exists an isomorphism of $k(x)$ -algebras

(3.2.9) $$\begin{eqnarray}\text{gr}_{I}\,k(x)[{\mathcal{C}}_{\mathscr{X},x}]\rightarrow \text{gr}_{\mathfrak{m}_{x}}\,{\mathcal{O}}_{\mathscr{X},x}.\end{eqnarray}$$

Using this result and following the argument of [MN15, Proposition 2.4.4], we show now that $f_{x}$ does not depend on the expansion of $f$ . Let

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE}}^{\prime }\unicode[STIX]{x1D6FE}\end{eqnarray}$$

be another admissible expansion of $f$ with associated set $\unicode[STIX]{x1D6E4}_{x}(f)^{\prime }$ and element $f_{x}^{\prime }$ . Then

$$\begin{eqnarray}0=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}(c_{\unicode[STIX]{x1D6FE}}-c_{\unicode[STIX]{x1D6FE}}^{\prime })\unicode[STIX]{x1D6FE}=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}d_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FE}\end{eqnarray}$$

where the right-hand side is an admissible expansion obtained by choosing admissible expansions for the elements $c_{\unicode[STIX]{x1D6FE}}-c_{\unicode[STIX]{x1D6FE}}^{\prime }$ that do not lie in ${\mathcal{O}}_{\mathscr{X},x}^{\times }\cup \{0\}$ . In particular, $d_{\unicode[STIX]{x1D6FE}}(x)=c_{\unicode[STIX]{x1D6FE}}(x)-c_{\unicode[STIX]{x1D6FE}}^{\prime }(x)$ for any $\unicode[STIX]{x1D6FE}$ in $\unicode[STIX]{x1D6E4}_{x}(f)\cup \unicode[STIX]{x1D6E4}_{x}(f)^{\prime }$ . The isomorphism of graded algebras in (3.2.9) implies that the elements $d_{\unicode[STIX]{x1D6FE}}$ must all vanish, hence $\unicode[STIX]{x1D6E4}_{x}(f)=\unicode[STIX]{x1D6E4}_{x}(f)^{\prime }$ and $f_{x}=f_{x}^{\prime }$ .

Point (ii) follows from the fact that the coefficients $c_{\unicode[STIX]{x1D6FE}}$ of $f_{x}$ are independent of the chosen section up to multiplication by a unit in ${\mathcal{O}}_{\mathscr{X},x}$ , so that the support $\unicode[STIX]{x1D6E4}_{x}(f)$ of $f_{x}$ only depends on $f$ and $x$ .◻

Proposition 3.2.10. Let $x$ be a point of $F$ and let

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow (\mathbb{R}_{{\geqslant}0},+)\end{eqnarray}$$

be an element of $\unicode[STIX]{x1D70E}_{x}$ .Then there exists a unique minimal real valuation

$$\begin{eqnarray}v:{\mathcal{O}}_{\mathscr{X},x}\setminus \{0\}\rightarrow \mathbb{R}_{{\geqslant}0}\end{eqnarray}$$

such that $v(m)=\unicode[STIX]{x1D6FC}(\overline{m})$ for each element $m$ of ${\mathcal{M}}_{\mathscr{X},x}$ .

Proof. We will prove that the map

(3.2.11) $$\begin{eqnarray}v:{\mathcal{O}}_{\mathscr{X},x}\setminus \{0\}\rightarrow \mathbb{R}:f\mapsto \min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})\,|\,\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{x}(f)\}\end{eqnarray}$$

satisfies the requirements in the statement. We fix a section

$$\begin{eqnarray}{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{M}}_{\mathscr{X},x}.\end{eqnarray}$$

It is straightforward to check that $(f\cdot g)_{x}=f_{x}\cdot g_{x}$ for all $f$ and $g$ in ${\mathcal{O}}_{\mathscr{X},x}$ . This implies that $v$ is a valuation. It is obvious that $v(m)=\unicode[STIX]{x1D6FC}(\overline{m})$ for all $m$ in ${\mathcal{M}}_{\mathscr{X},x}$ , since we can write $m$ as the product of an element of ${\mathcal{C}}_{\mathscr{X},x}$ and a unit in ${\mathcal{O}}_{\mathscr{X},x}$ .

Now we prove minimality. Consider any real valuation

$$\begin{eqnarray}w:{\mathcal{O}}_{\mathscr{X},x}\rightarrow \mathbb{R}\end{eqnarray}$$

such that $w(m)=\unicode[STIX]{x1D6FC}(\overline{m})$ for each element $m$ of ${\mathcal{M}}_{\mathscr{X},x}$ , and let $f$ be an element of ${\mathcal{O}}_{\mathscr{X},x}$ . We must show that $w(f)\geqslant v(f)$ .

We set

$$\begin{eqnarray}C_{\unicode[STIX]{x1D6FC}}={\mathcal{C}}_{\mathscr{X},x}\setminus \unicode[STIX]{x1D6FC}^{-1}(0).\end{eqnarray}$$

We denote by $I$ the ideal in ${\mathcal{O}}_{\mathscr{X},x}$ generated by $C_{\unicode[STIX]{x1D6FC}}$ and by $A$ the $I$ -adic completion of ${\mathcal{O}}_{\mathscr{X},x}$ . By Lemma 3.2.3, we see that we can write $f$ in $A$ as

(3.2.12) $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in C_{\unicode[STIX]{x1D6FC}}\cup \{1\}}d_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FD},\end{eqnarray}$$

where $d_{\unicode[STIX]{x1D6FD}}$ is either zero or contained in the complement of $I$ in ${\mathcal{O}}_{\mathscr{X},x}$ .

Since $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})>0$ for every $\unicode[STIX]{x1D6FD}\in C_{\unicode[STIX]{x1D6FC}}$ , we can find an integer $N>0$ such that $w(g)>w(f)$ for every element $g$ in $I^{N}$ . So we have

$$\begin{eqnarray}w(f)\geqslant \min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})\,|\,d_{\unicode[STIX]{x1D6FD}}\neq 0\},\end{eqnarray}$$

recalling that $w(\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}$ in ${\mathcal{C}}_{\mathscr{X},x}$ .

We consider the coefficients in the expansion (3.2.12) of $f$ . Applying Lemma 3.2.3 as in paragraph (3.2.5), we can write admissible expansions of these coefficients in $\widehat{{\mathcal{O}}}_{\mathscr{X},x}$ as

$$\begin{eqnarray}d_{\unicode[STIX]{x1D6FD}}=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FE},\quad c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\in {\mathcal{O}}_{\mathscr{X},x}^{\times }\cup \{0\},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})=0$ in the expansions of $d_{\unicode[STIX]{x1D6FD}}$ that belong to $\mathfrak{m}_{x}\setminus I$ .

Therefore we obtain an admissible expansion of $f$

$$\begin{eqnarray}f=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FD}\in C_{\unicode[STIX]{x1D6FC}}\cup \{1\}\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}}c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\,\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}\end{eqnarray}$$

and we have $v(f)=\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD})\,|\,c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\neq 0\}=\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})\,|\,d_{\unicode[STIX]{x1D6FD}}\neq 0\}\leqslant w(f)$ .◻

Remark 3.2.13. In the definition (3.2.11) of the valuation $v$ , we compute the minimum over the terms in the initial support of $f$ : these elements are a finite number and they only depend on $x$ and $f$ by Proposition 3.2.8. Therefore, this minimum provides a well-defined function on ${\mathcal{O}}_{\mathscr{X},x}\setminus \{0\}$ . Nevertheless, it is equivalent to consider the minimum over all the terms of an admissible expansion of $f$ , that is, for any admissible expansion $f=\sum _{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FE}$ ,

$$\begin{eqnarray}\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})\,|\,\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{x}(f)\}=\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})\,|\,\unicode[STIX]{x1D6FE}\in S\},\end{eqnarray}$$

where $S=\{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}\,|\,c_{\unicode[STIX]{x1D6FE}}\neq 0\}$ as in (3.2.7). Indeed, any element that belongs to $S$ can be written as a sum of an element of the initial support of $f$ and an element of ${\mathcal{C}}_{\mathscr{X},x}$ . Since the morphism $\unicode[STIX]{x1D6FC}$ is additive and takes positive real values, the minimum is necessarily attained by the elements in the initial support.

3.2.14. We will denote the valuation $v$ from Proposition 3.2.10 by $v_{x,\unicode[STIX]{x1D6FC}}$ . Since $v_{x,\unicode[STIX]{x1D6FC}}$ induces a real valuation on the function field of $\mathscr{X}_{K}$ that extends the discrete valuation $v_{K}$ on $K$ , it defines a point of the $K$ -analytic space $\mathscr{X}_{K}^{\text{an}}$ , which we will denote by the same symbol $v_{x,\unicode[STIX]{x1D6FC}}$ . We now show that the characterization of $v_{x,\unicode[STIX]{x1D6FC}}$ in Proposition 3.2.10 implies that

$$\begin{eqnarray}v_{y,\unicode[STIX]{x1D6FC}^{\prime }}=v_{x,\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}}\end{eqnarray}$$

for every $y$ in $F(x)$ and every $\unicode[STIX]{x1D6FC}^{\prime }$ in $\unicode[STIX]{x1D70E}_{y}$ .

First we note that ${\mathcal{O}}_{\mathscr{X},y}$ is the localization of ${\mathcal{O}}_{\mathscr{X},x}$ with respect to the elements $m\in {\mathcal{M}}_{\mathscr{X},x}$ in the kernel of $\unicode[STIX]{x1D70F}_{x,y}$ . Indeed, by construction of $\unicode[STIX]{x1D70F}_{x,y}$ , the kernel is given by

$$\begin{eqnarray}\ker (\unicode[STIX]{x1D70F}_{x,y})=\{s\in {\mathcal{C}}_{\mathscr{X},x}\mid s(y)\neq 0\}.\end{eqnarray}$$

To obtain ${\mathcal{O}}_{\mathscr{X},y}$ from ${\mathcal{O}}_{\mathscr{X},x}$ , we localize by

$$\begin{eqnarray}S=\{a\in {\mathcal{O}}_{\mathscr{X},x}\mid a(y)\neq 0\}.\end{eqnarray}$$

Therefore we can identify the set of elements of ${\mathcal{M}}_{\mathscr{X},x}$ whose reduction is in $\ker (\unicode[STIX]{x1D70F}_{x,y})$ with the set $S$ , recalling that, for points in the Kato fan, ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ generates the maximal ideal of ${\mathcal{O}}_{\mathscr{X},x}$ . Therefore we are dealing with these two morphisms:

$$\begin{eqnarray}\displaystyle {\mathcal{O}}_{\mathscr{X},x}{\hookrightarrow}S^{-1}{\mathcal{O}}_{\mathscr{X},x} & = & \displaystyle {\mathcal{O}}_{\mathscr{X},y},\nonumber\\ \displaystyle {\mathcal{C}}_{\mathscr{X},x}{\twoheadrightarrow}{\mathcal{C}}_{\mathscr{X},x}/S & = & \displaystyle {\mathcal{C}}_{\mathscr{X},y}.\nonumber\end{eqnarray}$$

Let $f$ be an element of ${\mathcal{O}}_{\mathscr{X},x}$ . In the notation of Lemma 3.2.3, we apply the lemma to $A={\mathcal{O}}_{\mathscr{X},y}$ and $B={\mathcal{O}}_{\mathscr{X},x}$ , choosing a system of generators of $\mathfrak{m}_{y}$ in ${\mathcal{C}}_{\mathscr{X},x}$ : we can find an admissible expansion of $f$ of the form

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FF}\in {\mathcal{C}}_{\mathscr{X},y}}d_{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FF}\quad \text{with }d_{\unicode[STIX]{x1D6FF}}\in ({\mathcal{O}}_{\mathscr{X},x}\cap {\mathcal{O}}_{\mathscr{X},y}^{\times })\cup \{0\}.\end{eqnarray}$$

Admissible expansions of coefficients $d_{\unicode[STIX]{x1D6FF}}$ induce an admissible expansion for $f$ by

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FF}\in {\mathcal{C}}_{\mathscr{X},y}}\bigg(\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in S}c_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FE}\bigg)\unicode[STIX]{x1D6FF}\quad \text{with }c_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}_{\mathscr{X},x}^{\times }\cup \{0\},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ runs through the set $S$ since $d_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}_{\mathscr{X},y}^{\times }$ . Thus we have

$$\begin{eqnarray}\displaystyle v_{y,\unicode[STIX]{x1D6FC}^{\prime }}(f) & = & \displaystyle \min \{\unicode[STIX]{x1D6FC}^{\prime }(\unicode[STIX]{x1D6FF})\,|\,\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E4}_{y}(f)\}\nonumber\\ \displaystyle & = & \displaystyle \min \{\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF})\,|\,\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E4}_{y}(f),\unicode[STIX]{x1D6FE}\in S\}\nonumber\\ \displaystyle & = & \displaystyle \min \{\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF})\,|\,\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E4}_{x}(f)\}\nonumber\\ \displaystyle & = & \displaystyle v_{x,\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}}(f).\nonumber\end{eqnarray}$$

Hence, we obtain a well-defined map

$$\begin{eqnarray}\unicode[STIX]{x1D704}:\text{Sk}(\mathscr{X}^{+})\rightarrow \mathscr{X}_{K}^{\text{an}}\end{eqnarray}$$

by sending $\unicode[STIX]{x1D6FC}$ to $v_{x,\unicode[STIX]{x1D6FC}}$ for every point $x$ of $F$ and every $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D70E}_{x}$ .

Proposition 3.2.15. The map

$$\begin{eqnarray}\unicode[STIX]{x1D704}:\text{Sk}(\mathscr{X}^{+})\rightarrow \mathscr{X}_{K}^{\text{an}}\end{eqnarray}$$

is a topological embedding.

Proof. First, we show that $\unicode[STIX]{x1D704}$ is injective. Let $x$ be a point of $F$ and $\unicode[STIX]{x1D6FC}$ an element of $\unicode[STIX]{x1D70E}_{x}$ . Let $y$ be the point of $F(x)$ corresponding to the face ${\mathcal{C}}_{\mathscr{X},x}\setminus \unicode[STIX]{x1D6FC}^{-1}(0)$ of ${\mathcal{C}}_{\mathscr{X},x}$ . Then $\unicode[STIX]{x1D6FC}$ factors through an element

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\prime }:{\mathcal{C}}_{\mathscr{ X},y}\rightarrow \mathbb{R}_{{\geqslant}0}\end{eqnarray}$$

of $\unicode[STIX]{x1D70E}_{y}$ . Note that $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}^{\prime }$ in $\text{Sk}(\mathscr{X}^{+})$ because $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}$ . Moreover, since $(\unicode[STIX]{x1D6FC}^{\prime })^{-1}(0)=\{1\}$ , the center of the valuation $v_{y,\unicode[STIX]{x1D6FC}^{\prime }}$ is the point $y$ , so that $\text{red}_{\mathscr{X}}(v_{y,\unicode[STIX]{x1D6FC}^{\prime }})=y$ . Thus we can recover $y$ from $v_{y,\unicode[STIX]{x1D6FC}^{\prime }}$ . Then we can also reconstruct $\unicode[STIX]{x1D6FC}^{\prime }$ by looking at the values of $v_{y,\unicode[STIX]{x1D6FC}^{\prime }}$ at the elements of ${\mathcal{M}}_{\mathscr{X},y}$ . We conclude that $\unicode[STIX]{x1D704}$ is injective.

Now, we show that $\unicode[STIX]{x1D704}$ is a homeomorphism onto its image. For any valuation $v$ in $\text{Sk}(\mathscr{X}^{+})$ and any small open neighborhood $U$ of $\unicode[STIX]{x1D704}(v)$ in $\mathscr{X}_{K}^{\text{an}}$ , there exists a closed subset $C$ in $\text{Sk}(\mathscr{X}^{+})$ such that $U\cap \unicode[STIX]{x1D704}(\text{Sk}(\mathscr{X}^{+}))\subseteq \unicode[STIX]{x1D704}(C)$ and, up to subdivisions, we can assume that the $C$ is a closed cell of $\text{Sk}(\mathscr{X}^{+})$ . Therefore, it suffices to prove that the restriction of $\unicode[STIX]{x1D704}$ to any closed cell $\unicode[STIX]{x1D70E}_{x}$ of $\text{Sk}(\mathscr{X}^{+})$ is a homeomorphism. The restriction $\unicode[STIX]{x1D704}_{|\unicode[STIX]{x1D70E}_{x}}$ is an injective map from a compact set to the Hausdorff space $\mathscr{X}_{K}^{\text{an}}$ , so we reduce to showing that $\unicode[STIX]{x1D704}_{|\unicode[STIX]{x1D70E}_{x}}$ is continuous, to conclude that $\unicode[STIX]{x1D704}_{|\unicode[STIX]{x1D70E}_{x}}$ is a homeomorphism. By definition of the Berkovich topology, it is enough to prove that the map

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{x}\rightarrow \mathbb{R}:\unicode[STIX]{x1D6FC}\mapsto v_{x,\unicode[STIX]{x1D6FC}}(f)\end{eqnarray}$$

is continuous for every $f$ in ${\mathcal{O}}_{\mathscr{X},x}$ . This is obvious from the formula (3.2.11).◻

3.2.16. From now on, we will view $\text{Sk}(\mathscr{X}^{+})$ as a topological subspace of $\mathscr{X}_{K}^{\text{an}}$ by means of the embedding $\unicode[STIX]{x1D704}$ in Proposition 3.2.15. If $\mathscr{X}$ is regular over $R$ and $\mathscr{X}_{k}$ is a divisor with strict normal crossings, the skeleton $\text{Sk}(\mathscr{X}^{+})$ was described in [MN15, §3.1].

3.3 Contracting the generic fiber to the skeleton

3.3.1. We denote by $D_{\mathscr{X},\text{hor}}$ the component of $D_{\mathscr{X}}$ not contained in the special fiber $\mathscr{X}_{k}$ . The inclusion $\unicode[STIX]{x1D704}:\text{Sk}(\mathscr{X}^{+})\rightarrow \mathscr{X}_{K}^{\text{an}}$ is actually an inclusion in $(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}$ and it admits a continuous retraction

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\mathscr{X}}:(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}\rightarrow \text{Sk}(\mathscr{X}^{+})\end{eqnarray}$$

constructed as follows. Let $x$ be a point of $(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}$ and consider the reduction map

$$\begin{eqnarray}\text{red}_{\mathscr{X}}:(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}\rightarrow \mathscr{X}_{k}.\end{eqnarray}$$

Let $E_{1},\ldots ,E_{r}$ be the irreducible components of $D_{\mathscr{X}}$ passing through the point $\text{red}_{\mathscr{X}}(x)$ . We denote by $\unicode[STIX]{x1D709}$ the generic point of the connected component of $E_{1}\cap \cdots \cap E_{r}$ that contains $\text{red}_{\mathscr{X}}(x)$ . By Lemma 2.2.3, $\unicode[STIX]{x1D709}$ is a point in the associated Kato fan $F$ . We set $\unicode[STIX]{x1D6FC}$ to be the morphism of monoids

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},\unicode[STIX]{x1D709}}\rightarrow \mathbb{R}_{{\geqslant}0}\end{eqnarray}$$

such that $\unicode[STIX]{x1D6FC}(\overline{m})=v_{x}(m)$ for any element $m$ of ${\mathcal{M}}_{\mathscr{X},\unicode[STIX]{x1D709}}$ . In particular, $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=v_{x}(\unicode[STIX]{x1D70B})=1$ as we assumed the normalization of all valuations in the Berkovich space. Then $\unicode[STIX]{x1D70C}_{\mathscr{X}}(x)$ is the point of $\text{Sk}(\mathscr{X}^{+})$ corresponding to the couple $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D6FC})$ . By construction $\unicode[STIX]{x1D70C}_{\mathscr{X}}$ is continuous and right inverse to the inclusion $\unicode[STIX]{x1D704}$ .

3.3.2. Given a dominant morphism $f:\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}$ of integral flat separated log-regular log schemes over $S$ , it induces a map between the set of birational points $\text{Bir}(\mathscr{X}_{K})\rightarrow \text{Bir}(\mathscr{Y}_{K})$ . As $\text{Bir}(\mathscr{X}_{K})\subseteq (\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}$ , we can employ the retraction $\unicode[STIX]{x1D70C}$ to define a map of skeletons as follows.

This association makes the skeleton construction $\text{Sk}(\mathscr{X}^{+})$ functorial in $\mathscr{X}^{+}$ with respect to dominant morphisms.

3.4 Skeleton of an fs fibred product

3.4.1. Let $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ be log-smooth log schemes over $S^{+}$ , and let $\mathscr{Z}^{+}$ be their fs fibred product. Let

$$\begin{eqnarray}\text{Sk}(\mathscr{Z}^{+})\rightarrow \text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})\end{eqnarray}$$

be the continuous map of skeletons functorially associated to the projections $\text{pr}_{\mathscr{X}}:\mathscr{Z}^{+}\rightarrow \mathscr{X}^{+}$ and $\text{pr}_{\mathscr{Y}}:\mathscr{Z}^{+}\rightarrow \mathscr{Y}^{+}$ . We denote this map by $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ and we recall that it is constructed considering the following diagram.

(3.4.2)

Proposition 3.4.3. Assume that the residue field $k$ is algebraically closed. If $\mathscr{X}^{+}$ is semistable, then the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is a PL homeomorphism.

Proof. We first provide an explicit description of the map $\text{pr}_{\text{Sk}(\mathscr{X})}$ (respectively, of $\text{pr}_{\text{Sk}(\mathscr{Y})}$ ) in diagram (3.4.2). Then we show that the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is injective and surjective.

Let $v_{z,\unicode[STIX]{x1D700}}$ be the valuation in $\text{Sk}(\mathscr{Z}^{+})$ corresponding to a couple $(z,\unicode[STIX]{x1D700})$ with $z\in F_{\mathscr{Z}}\cap \mathscr{Z}_{k}$ and $\unicode[STIX]{x1D700}\in \unicode[STIX]{x1D70E}_{z}$ . We consider the morphism of associated Kato fans

$$\begin{eqnarray}F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}\end{eqnarray}$$

as established in Proposition 2.4.8. We denote respectively by $\text{pr}_{F_{\mathscr{X}}}$ and $\text{pr}_{F_{\mathcal{Y}}}$ the projection to the first and second factor. Then $\text{pr}_{F_{\mathscr{X}}}(z)$ is a point in the associated Kato fan $F_{\mathscr{X}}$ , which we denote by $x$ . We consider the morphism of monoids

$$\begin{eqnarray}i_{x}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{C}}_{\mathscr{Z},z}\end{eqnarray}$$

and the composition

It trivially satisfies $\unicode[STIX]{x1D700}\circ i_{x}(\unicode[STIX]{x1D70B})=1$ . In order to conclude that it correctly defines a point in the skeleton $\text{Sk}(\mathscr{X}^{+})$ , we need to check the compatibility with respect to the equivalence relation ${\sim}$ . Indeed, suppose that $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}^{\prime }\circ \unicode[STIX]{x1D70F}_{z,z^{\prime }}$ for some $z\in \overline{\{z^{\prime }\}}$ . We denote by $x^{\prime }$ the projection of $z^{\prime }$ under the local isomorphism of associated Kato fans. The diagram

is commutative as it is made up of a commutative square and a commutative triangle of arrows. Therefore, by commutativity

$$\begin{eqnarray}\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})=\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700}^{\prime })\circ \unicode[STIX]{x1D70F}_{x,x^{\prime }}\end{eqnarray}$$

and this implies that $\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})$ defines a well-defined point $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ of $\text{Sk}(\mathscr{X}^{+})$ .

We claim that $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ is indeed the image of $v_{z,\unicode[STIX]{x1D700}}$ under the map $\text{pr}_{\text{Sk}(\mathscr{X})}$ . We recall that the projection $\widehat{\text{pr}_{\mathscr{X}}}$ , in diagram (3.4.2), is such that a valuation $v$ on the function field $K(\mathscr{Z}_{K})$ maps to the composition $v\circ i$ where $i:K(\mathscr{X}_{K}){\hookrightarrow}K(\mathscr{Z}_{K})$ . Thus, we need to prove that the equality in the following inner diagram holds.

We denote $\unicode[STIX]{x1D70C}_{\mathscr{X}}(v_{z,\unicode[STIX]{x1D700}}\circ i)$ by $(x,\unicode[STIX]{x1D6FC})$ as a point of $\text{Sk}(\mathscr{X}^{+})$ . By definition of the retraction $\unicode[STIX]{x1D70C}_{\mathscr{X}}$ , the morphism $\unicode[STIX]{x1D6FC}$ is characterized by the fact that $\unicode[STIX]{x1D6FC}(\overline{m})=(v_{z,\unicode[STIX]{x1D700}}\circ i)(m)$ for any $m$ in ${\mathcal{M}}_{\mathscr{X},x}$ and then we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(\overline{m})=(v_{z,\unicode[STIX]{x1D700}}\circ i)(m)=v_{z,\unicode[STIX]{x1D700}}(m)=\unicode[STIX]{x1D700}(\overline{m}).\end{eqnarray}$$

On the other hand, for any $m$ in ${\mathcal{M}}_{\mathscr{X},x}$ ,

$$\begin{eqnarray}v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}(m)=\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})(\overline{m})=\unicode[STIX]{x1D700}(\overline{m})\end{eqnarray}$$

hence we obtain that $\unicode[STIX]{x1D6FC}$ coincides with the morphism $\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})$ . This means that their associated points $\unicode[STIX]{x1D70C}_{\mathscr{X}}(v_{z,\unicode[STIX]{x1D700}}\circ i)$ and $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ coincide in $\text{Sk}(\mathscr{X}^{+})$ .

We now prove injectivity and surjectivity of the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ . Given a pair of points in $\text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})$ , they are of the form $(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}}),$ with $x\in F_{\mathscr{X}}\cap \mathscr{X}_{k}$ , $y\in F_{\mathscr{Y}}\cap \mathscr{Y}_{k}$ , $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D70E}_{x}$ , and $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D70E}_{y}$ . The assumptions of semistability of $\mathscr{X}^{+}$ and algebraic closedness of $k$ guarantee that there is a unique $z$ in $F_{\mathscr{Z}}$ in the fiber of $x$ and $y$ , by Proposition 2.5.4 and Remark 3.1.4. We claim that we can construct a unique $\unicode[STIX]{x1D700}\in \unicode[STIX]{x1D70E}_{z}$ such that $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})(v_{z,\unicode[STIX]{x1D700}})=(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}}).$ We set

this is well defined as $\unicode[STIX]{x1D700}([\unicode[STIX]{x1D70B},1])=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=1=\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70B})=\unicode[STIX]{x1D700}([1,\unicode[STIX]{x1D70B}])$ . By construction, the image of $v_{z,\unicode[STIX]{x1D700}}$ is $(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}})$ , so the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is surjective. Moreover, we can uniquely reconstruct $\unicode[STIX]{x1D700}$ by looking at the values of $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ at the elements of ${\mathcal{M}}_{\mathscr{X},x}$ , and of $v_{y,\text{pr}_{\mathcal{Y}}(\unicode[STIX]{x1D700})}$ at the elements of ${\mathcal{M}}_{\mathscr{Y},y}$ . Thus $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is injective.◻

The assumption of semistability is crucial in the result of Proposition 3.4.3. To see this, it is helpful to consider an example.

Example 3.4.4. Let $q$ be the equation of a generic quartic curve in $\mathbb{P}_{\mathbb{C}((t))}^{2}$ . Then $\mathscr{X}:tq+x^{2}y^{2}=0$ gives the equation of a family of genus- $3$ curves, degenerating to two double lines. The dual complex ${\mathcal{D}}(\mathscr{X}_{\mathbb{C}})$ of the special fiber $\mathscr{X}_{\mathbb{C}}$ is a line segment and $\mathscr{X}$ has four singularities of type $A_{1}$ in each component of the special fiber, corresponding to the base points of the family. In this case taking a semistable model of $\mathscr{X}_{\mathbb{C}((t))}$ requires an order- $2$ base change $R^{\prime }$ of $R=\mathbb{C}[[t]]$ , which induces coverings branched at each of these singular points (see [HM98, p. 133] for details). Let $\mathscr{Y}$ be such a semistable reduction. Thus the special fiber of $\mathscr{Y}$ consists of two elliptic curves, call them $E_{1}$ and $E_{2}$ , which intersect in two points, $p_{A}$ and $p_{B}$ , which are the preimages of the point $(0:0:1)$ . The dual complex ${\mathcal{D}}(\mathscr{Y}_{\mathbb{C}})$ of the special fiber $\mathscr{Y}_{\mathbb{C}}$ is isomorphic to $S^{1}$ .

We will compare the dual complex of $(\mathscr{X}\times _{R}\mathscr{X})_{\mathbb{C}}$ with that of $(\mathscr{Y}\times _{R^{\prime }}\mathscr{Y})_{\mathbb{C}}$ . The models $\mathscr{X}$ and $\mathscr{Y}$ are not log-regular at every point, but from our perspective it is enough that they are log-regular at the generic point of each stratum. For the product with a semistable model, the dual complex is the product of the dual complexes, and ${\mathcal{D}}((\mathscr{Y}\times _{R^{\prime }}\mathscr{Y})_{\mathbb{C}})$ is therefore a real $2$ -torus $S^{1}\times S^{1}$ .

On the other hand, the dual complex of the product $(\mathscr{X}\times _{R}\mathscr{X})_{\mathbb{C}}$ is given by a quotient of $S^{1}\times S^{1}$ by the action of $\mathbb{Z}/2\mathbb{Z}$ . ${\mathcal{D}}((\mathscr{Y}\times _{R^{\prime }}\mathscr{Y})_{\mathbb{C}})$ has the structure of a cell complex, whose cells correspond to ordered pairs of strata in ${\mathcal{D}}(\mathscr{Y}_{\mathbb{C}})$ , so

$$\begin{eqnarray}\displaystyle \text{zero-dimensional strata: } & & \displaystyle \quad (E_{1},E_{2}),(E_{1},E_{1}),(E_{2},E_{1}),(E_{2},E_{2})\nonumber\\ \displaystyle \text{one-dimensional strata: } & & \displaystyle \quad (E_{1},p_{A}),(E_{1},p_{B}),(E_{2},p_{A}),(E_{2},p_{B})\nonumber\\ \displaystyle & & \displaystyle \quad (p_{A},E_{1}),(p_{B},E_{1}),(p_{A},E_{2}),(p_{B},E_{2})\nonumber\\ \displaystyle \text{two-dimensional strata: } & & \displaystyle \quad (p_{A},p_{A}),(p_{A},p_{B}),(p_{B},p_{A}),(p_{B},p_{B}).\nonumber\end{eqnarray}$$

The action of $\mathbb{Z}/2\mathbb{Z}$ fixes $E_{1}$ and $E_{2}$ , while switching $p_{A}$ and $p_{B}$ . Therefore it fixes exactly the zero-dimensional strata while acting freely on the other points. The quotient, the complex ${\mathcal{D}}((\mathscr{X}\times _{R}\mathscr{X})_{\mathbb{C}})$ , is piecewise linearly homeomorphic to the sphere $S^{2}$ . In particular, it is not isomorphic to the product of two line segments.

4 The weight function

From now on we assume $\text{char}(K)=\text{char}(k)=0$ .

4.1 Weight function associated to a logarithmic pluricanonical form

4.1.1. Let $X$ be a connected, smooth and proper $K$ -variety of dimension $n$ . We introduce the following notation: for any log-regular model $\mathscr{X}^{+}$ of $X$ , for any point $x=(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})\in X^{\text{an}}$ , and for any $\mathbb{Q}$ -Cartier divisor $D$ on $\mathscr{X}^{+}$ whose support does not contain $\unicode[STIX]{x1D709}_{x}$ , we set

$$\begin{eqnarray}v_{x}(D)=-\ln |f(x)|^{1/m},\end{eqnarray}$$

where $f$ is any element of $K(X)^{\times }$ such that $mD=\text{div}(f)$ locally at $\text{red}_{\mathscr{X}}(x)$ , and $m\in \mathbb{Z}_{{>}0}$ is such that $mD$ is Cartier.

4.1.2. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ be a fractional snc pair as in § 1.5, so that $\unicode[STIX]{x1D6E5}_{X}$ is an effective $\mathbb{Q}$ -divisor such that $\unicode[STIX]{x1D6E5}_{X}=\sum a_{i}\unicode[STIX]{x1D6E5}_{X,i}$ has $0\leqslant a_{i}\leqslant 1$ , and the round-up $X^{+}=(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ is snc. Let $\unicode[STIX]{x1D714}$ be a regular $m$ -pluricanonical form on $X^{+}$ with poles of order at most $ma_{i}$ along $\unicode[STIX]{x1D6E5}_{X,i}$ , for some $m$ such that $ma_{i}\in \mathbb{N}$ for any $i$ . Thus, such a form is a section of ${\mathcal{O}}_{X}(m(K_{X}+\unicode[STIX]{x1D6E5}_{X}))$ , and we call it a $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical form.

Given a log-regular model $\mathscr{X}^{+}$ of $X^{+}$ , where $D_{\mathscr{X}}=\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}$ , we can view any $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical forms as rational sections of the logarithmic $m$ -pluricanonical bundle $(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })^{\otimes m}$ , that is, of

$$\begin{eqnarray}(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })^{\otimes m}=\unicode[STIX]{x1D714}_{\mathscr{ X}/S}^{\otimes m}\otimes {\mathcal{O}}_{\mathscr{ X}}(m(\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}-\text{div}(\unicode[STIX]{x1D70B}))).\end{eqnarray}$$

The form $\unicode[STIX]{x1D714}$ , viewed as a rational section of $(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })^{\otimes m}$ , defines a divisor $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})$ on $\mathscr{X}^{+}$ . Note that the multiplicity in $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})$ of the closure $\overline{\unicode[STIX]{x1D6E5}_{X,i}}$ in $\mathscr{X}$ of $\unicode[STIX]{x1D6E5}_{X,i}$ is at least $m(1-a_{i})$ .

4.1.3. Given a $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical form $\unicode[STIX]{x1D714}$ , we can consider it as a rational section of $\unicode[STIX]{x1D714}_{X/K}^{\otimes m}$ . Hence, we can associate to $\unicode[STIX]{x1D714}$ the weight function $\text{wt}_{\unicode[STIX]{x1D714}}$ as in [MN15]. The following lemma gives an interpretation of the weight function associated to $\unicode[STIX]{x1D714}$ in terms of logarithmic differentials, which we will use in the sequel.

Lemma 4.1.4. Let $\mathscr{X}^{+}$ be a log-regular model of $X^{+}$ . Then for every point $x$ of $\text{Sk}(\mathscr{X}^{+})$ ,

$$\begin{eqnarray}\text{wt}_{\unicode[STIX]{x1D714}}(x)=v_{x}(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))+m.\end{eqnarray}$$

Proof. It suffices to prove the equality for the divisorial points in $\text{Sk}(\mathscr{X}^{+})$ , since they are dense in the skeleton, and both $\text{wt}_{\unicode[STIX]{x1D714}}(\cdot )$ and $v_{\cdot }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))$ are continuous functions on the skeleton $\text{Sk}(\mathscr{X}^{+})$ .

Let $x$ be a divisorial point in $\text{Sk}(\mathscr{X}^{+})$ . If $x$ corresponds to a component of the special fiber $\mathscr{X}_{k}$ , then the center $\text{red}_{\mathscr{X}}(x)$ of $x$ does not contain the closure of any components of $\unicode[STIX]{x1D6E5}_{X}$ . Thus, locally around $\text{red}_{\mathscr{X}}(x)$ the log schemes $\mathscr{X}^{+}$ and $(\mathscr{X},\mathscr{X}_{k,\text{red}})$ are isomorphic and, in particular, $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})=\text{div}_{(\mathscr{X},\mathscr{X}_{k,\text{red}})}(\unicode[STIX]{x1D714})$ . The computation in [NX16, §3.2.2] shows that $\text{wt}_{\unicode[STIX]{x1D714}}(x)=v_{x}(\text{div}_{(\mathscr{X},\mathscr{X}_{k,\text{red}})}(\unicode[STIX]{x1D714}))+m$ , so we obtain the required equality.

Otherwise, we consider the blow-up $h:{\mathscr{X}^{\prime }}^{+}\rightarrow \mathscr{X}^{+}$ at the closure of the center $\text{red}_{\mathscr{X}}(x)$ of $x$ in $\mathscr{X}^{+}$ . By [KM08, Lemma 2.45], ¹ iterating this procedure a finite number of times, we obtain a log-regular model $\mathscr{Y}^{+}$ such that the $x$ corresponds to a component $E$ of the special fiber $\mathscr{Y}_{k}$ . By reduction to the previous case, it is enough to check that the value $v_{x}(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))$ does not change under such a blow-up $h$ .

The morphism $h$ is the blow-up of a stratum of the log boundary $\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}$ , hence $h$ is a log étale morphism. By [Kat89, Proposition 3.12], we have $h^{\ast }(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })=\unicode[STIX]{x1D714}_{{\mathscr{ X}^{\prime }}^{+}/S^{+}}^{\log }$ . It follows that $\text{div}_{{\mathscr{X}^{\prime }}^{+}}(\unicode[STIX]{x1D714})=h^{\ast }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))$ and this concludes the proof.◻

4.1.5. We recall from [MN15, § 4.7] that the Kontsevich–Soibelman skeleton $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ is the closure in $\text{Bir}(X)$ of the set of divisorial points of $X^{\text{an}}$ where the weight function $\text{wt}_{\unicode[STIX]{x1D714}}$ reaches its minimal weight, namely

$$\begin{eqnarray}\text{wt}_{\unicode[STIX]{x1D714}}(X,\unicode[STIX]{x1D6E5}_{X})=\inf \{\text{wt}_{\unicode[STIX]{x1D714}}(x)\,|\,x\in \text{Div}(X)\}\in \mathbb{R}\cup \{-\infty \}.\end{eqnarray}$$

A priori the weight function associated to a rational pluricanonical form may have minimal weight $-\infty$ , hence the corresponding Kontsevich–Soibelman skeleton would be empty. We prove that this does not occur for $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic pluricanonical forms.

Proposition 4.1.6. Given a $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic  $m$ -pluricanonical form $\unicode[STIX]{x1D714}$ , for any log-regular model $\mathscr{X}^{+}$ of $X^{+}$ the inclusion $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})\subseteq \text{Sk}(\mathscr{X}^{+})$ holds.

Proof. Let $\mathscr{X}^{+}$ be a log-regular model of $X^{+}$ and let $y$ be a divisorial point of $X^{\text{an}}$ . It suffices to prove that

$$\begin{eqnarray}\text{wt}_{\unicode[STIX]{x1D714}}(y)\geqslant \text{wt}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D70C}_{\mathscr{X}}(y))\end{eqnarray}$$

and that the equality holds if and only if $y$ is in $\text{Sk}(\mathscr{X}^{+})$ . As in the proof of Lemma 4.1.4, we consider the blow-up of $\mathscr{X}^{+}$ at the closure of $\text{red}_{\mathscr{X}}(y)$ : iterating this procedure a finite number of times, we obtain a log-regular model $\mathscr{Y}^{+}$ such that $y\in \text{Sk}(\mathscr{Y}^{+})$ .

Let $h:\mathscr{Z}^{+}\rightarrow \mathscr{W}^{+}$ be a morphism of this sequence. If $\overline{\{\text{red}_{\mathscr{W}}(y)\}}$ is a stratum of $D_{\mathscr{W}}$ , then the morphism $h$ induces a subdivision of the skeleton $\text{Sk}(\mathscr{W}^{+})$ , so $\unicode[STIX]{x1D70C}_{\mathscr{Z}}(y)=\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)$ .

Otherwise, $\overline{\{\text{red}_{\mathscr{W}}(y)\}}$ is strictly contained in a stratum $V$ of ${\mathcal{D}}_{\mathscr{W}}$ . Let $E$ be the exceptional divisor of $h$ , $r=r_{h}+r_{v}$ be the codimension of $V$ in $\mathscr{W}$ , where $r_{h}$ and $r_{v}$ are the number of irreducible components of $\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }$ and respectively of the special fiber $\mathscr{W}_{k}$ , containing $V$ . Let $r+j$ be the codimension of $\overline{\{\text{red}_{\mathscr{W}}(y)\}}$ , where $j\geqslant 1$ . We denote the projections onto $S^{+}$ by $s_{\mathscr{W}}:\mathscr{W}^{+}\rightarrow S^{+}$ and $s_{\mathscr{Z}}:\mathscr{Z}^{+}\rightarrow S^{+}$ and by $\unicode[STIX]{x1D70B}$ a uniformizer in $R$ . Then we have that

$$\begin{eqnarray}\displaystyle h^{\ast }(\unicode[STIX]{x1D714}_{\mathscr{W}^{+}/S^{+}}^{\log }) & = & \displaystyle h^{\ast }(\unicode[STIX]{x1D714}_{\mathscr{ W}/R}\otimes {\mathcal{O}}_{\mathscr{W}}(\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }_{,\text{red}}+\mathscr{W}_{k,\text{red}}-s_{\mathscr{W}}^{\ast }\,\text{div}(\unicode[STIX]{x1D70B})))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D714}_{\mathscr{Z}/R}\otimes {\mathcal{O}}_{\mathscr{Z}}((1-r-j)E+\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }_{,\text{red}}+r_{h}E+\mathscr{Z}_{k,\text{red}}+(r_{v}-1)E-s_{\mathscr{Z}}^{\ast }\,\text{div}(\unicode[STIX]{x1D70B}))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D714}_{\mathscr{Z}^{+}/S^{+}}^{\log }\otimes {\mathcal{O}}_{\mathscr{ Z}}(-jE).\nonumber\end{eqnarray}$$

It follows that $\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D714})=h^{\ast }(\text{div}_{\mathscr{W}^{+}}(\unicode[STIX]{x1D714}))+mjE$ , so

$$\begin{eqnarray}\displaystyle v_{\unicode[STIX]{x1D70C}_{\mathscr{Z}}(y)}(\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D714})) & = & \displaystyle v_{\unicode[STIX]{x1D70C}_{\mathscr{Z}}(y)}(h^{\ast }(\text{div}_{\mathscr{W}^{+}}(\unicode[STIX]{x1D714}))+mjE)\nonumber\\ \displaystyle & {\geqslant} & \displaystyle v_{\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)}(\text{div}_{\mathscr{W}^{+}}(\unicode[STIX]{x1D714}))+mjv_{\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)}(E)\nonumber\\ \displaystyle & {>} & \displaystyle v_{\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)}(\text{div}_{\mathscr{W}^{+}}(\unicode[STIX]{x1D714})),\nonumber\end{eqnarray}$$

where for the first inequality we apply [MN15, Proposition 3.1.6], while the second strict inequality holds as $j>0$ and $v_{\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)}(E)>0$ since the center of the valuation $\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)$ is contained in $E$ . Therefore, for any such morphism $h$ , the weight is strictly increasing, namely $\text{wt}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D70C}_{\mathscr{Z}}(y))>\text{wt}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D70C}_{\mathscr{W}}(y))$ . This concludes the proof.◻

Only the components of $\unicode[STIX]{x1D6E5}_{X}$ with coefficient $a_{i}=1$ determine strata that are contained in the Kontsevich–Soibelman skeletons. The introduction of $\unicode[STIX]{x1D6E5}_{X}$ -log pluricanonical forms allows us to construct non-empty Kontsevich–Soibelman skeletons even for varieties with Kodaira dimension $-\infty$ , as in the following examples.

Example 4.1.7. Let $X$ be the projective line $\mathbb{P}_{K}^{1}$ with affine coordinates $x$ and $y$ , and $\unicode[STIX]{x1D6E5}_{X}=(0:1)+(1:0)$ . Then $a_{i}=1$ for any $i$ and there exist $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic canonical forms. For example, we consider

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{dx}{x}=-\frac{dy}{y}.\end{eqnarray}$$

Let $\mathscr{X}=\mathbb{P}_{R}^{1}$ and $D_{\mathscr{X}}=(0:1)+(1:0)+\mathbb{P}_{k}^{1}$ . The log scheme $\mathscr{X}^{+}=(\mathscr{X},D_{\mathscr{X}})$ is a log-regular model of $X^{+}=(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ and the associated skeleton $\text{Sk}(\mathscr{X}^{+})$ looks like this:

Since $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})=0$ , the weight associated to $\unicode[STIX]{x1D714}$ is minimal at any point of the skeleton $\text{Sk}(\mathscr{X}^{+})$ . Thus $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})=\text{Sk}(\mathscr{X}^{+})\simeq \mathbb{R}$ .

Example 4.1.8. Let $X=\mathbb{P}_{K}^{1}$ and $\unicode[STIX]{x1D6E5}_{X}=\frac{2}{3}(0:1)+\frac{2}{3}(1:0)+\frac{2}{3}(1:1)$ . So $a_{i}=\frac{2}{3}$ for any $i$ and there exist $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $3$ -pluricanonical forms. We set

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{1}{(x-1)^{2}}\cdot \frac{1}{x^{2}}(dx)^{3}=-\frac{1}{(1-y)^{2}}\cdot \frac{1}{y^{2}}(dy)^{3}.\end{eqnarray}$$

We consider $\mathscr{X}=\mathbb{P}_{R}^{1}$ and $D_{\mathscr{X}}=(0:1)+(1:0)+(1:1)+\mathbb{P}_{k}^{1}$ ; then $\mathscr{X}^{+}=(\mathscr{X},D_{\mathscr{X}})$ is a log-regular model of $X^{+}=(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ and $\text{Sk}(\mathscr{X}^{+})$ is

Since $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})=(0:1)+(1:0)+(1:1)$ , the weight associated to $\unicode[STIX]{x1D714}$ is minimal at the divisorial point $v_{\mathbb{P}_{k}^{1}}$ corresponding to $\mathbb{P}_{k}^{1}$ and is strictly increasing with slope $1$ along the unbounded edges, when we move away from the point $v_{\mathbb{P}_{k}^{1}}$ . Therefore, $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})=\{v_{\mathbb{P}_{k}^{1}}\}$ .

4.2 Weight function and Kontsevich–Soibelman skeleton for products

4.2.1. Let $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ be log-regular models over $S^{+}$ of $X^{+}=(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ and $Y^{+}=(Y,\lceil \unicode[STIX]{x1D6E5}_{Y}\rceil )$ respectively. Then the fs fibred product $\mathscr{Z}^{+}=\mathscr{X}^{+}\times _{S^{+}}^{\mathit{fs}}\mathscr{Y}^{+}$ is a log-regular model of $Z^{+}:=X^{+}\times _{K}^{\mathit{fs}}Y^{+}$ . Therefore, given $\unicode[STIX]{x1D714}_{X^{+}}$ and $\unicode[STIX]{x1D714}_{Y^{+}}$ $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic and $\unicode[STIX]{x1D6E5}_{Y}$ -logarithmic $m$ -pluricanonical forms on $(X,\unicode[STIX]{x1D6E5}_{X})$ and $(Y,\unicode[STIX]{x1D6E5}_{Y})$ respectively, the form

$$\begin{eqnarray}\unicode[STIX]{x1D71B}=\text{pr}_{X^{+}}^{\ast }\,\unicode[STIX]{x1D714}_{X^{+}}\otimes \text{pr}_{Y^{+}}^{\ast }\,\unicode[STIX]{x1D714}_{Y^{+}}\end{eqnarray}$$

is a $\unicode[STIX]{x1D6E5}_{Z}$ -logarithmic $m$ -pluricanonical form on $(Z,\unicode[STIX]{x1D6E5}_{Z})$ , where $\unicode[STIX]{x1D6E5}_{Z}=X\times _{K}\unicode[STIX]{x1D6E5}_{Y}+\unicode[STIX]{x1D6E5}_{X}\times _{K}Y$ . Viewing these forms as rational sections of logarithmic $m$ -pluricanonical bundles, we see that $\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D71B})=\text{pr}_{\mathscr{X}^{+}}^{\ast }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{X^{+}}))+\text{pr}_{\mathscr{Y}^{+}}^{\ast }(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{Y^{+}}))$ according to (2.4.6).

4.2.2. Let $z$ be a point of $F_{\mathscr{Z}}\cap \mathscr{Z}_{k}$ ; as before, we denote by $x$ and $y$ the images of $z$ under the local isomorphism $F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}$ . Any morphism $\unicode[STIX]{x1D700}\in \unicode[STIX]{x1D70E}_{z}$ defines a point $v_{z,\unicode[STIX]{x1D700}}$ in $\text{Sk}(\mathscr{Z}^{+})$ . For the sake of convenience, we simply denote the valuations by the corresponding morphism and we denote $\unicode[STIX]{x1D6FC}=\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})$ and $\unicode[STIX]{x1D6FD}=\text{pr}_{\mathscr{Y}}(\unicode[STIX]{x1D700})$ . We aim to relate the valuation $v_{\unicode[STIX]{x1D700}}(\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D71B}))$ to the values

$$\begin{eqnarray}v_{\unicode[STIX]{x1D6FC}}(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{X^{+}})),\quad v_{\unicode[STIX]{x1D6FD}}(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{Y^{+}})).\end{eqnarray}$$

4.2.3. Let $f_{x}\in {\mathcal{O}}_{\mathscr{X},x}$ be a local equation of $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{X^{+}})$ around $x$ . In order to evaluate $v_{x,\unicode[STIX]{x1D6FC}}$ on $f_{x}$ , we consider an admissible expansion of $f_{x}$ as in (3.2.6),

$$\begin{eqnarray}f_{x}=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FE}.\end{eqnarray}$$

Furthermore, this expansion also induces an expansion of $\text{pr}_{\mathscr{X}}^{\ast }(f_{x})$ by

$$\begin{eqnarray}\text{pr}_{\mathscr{X}}^{\ast }(f_{x})=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}\text{pr}_{\mathscr{X}}^{\ast }(c_{\unicode[STIX]{x1D6FE}})\unicode[STIX]{x1D6FE}\end{eqnarray}$$

as formal power series in $\widehat{{\mathcal{O}}}_{\mathscr{Z},z}$ , since the morphism of characteristic sheaves ${\mathcal{C}}_{\mathscr{X},x}{\hookrightarrow}{\mathcal{C}}_{\mathscr{Z},z}$ is injective. Following the same procedure for a local equation $f_{y}\in {\mathcal{O}}_{\mathscr{Y},y}$ of $\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{Y^{+}})$ around $y$ , we get an expansion of $f_{y}$ that extends to $\text{pr}_{\mathscr{Y}}^{\ast }(f_{y})$ :

$$\begin{eqnarray}f_{y}=\mathop{\sum }_{\unicode[STIX]{x1D6FF}\in {\mathcal{C}}_{\mathcal{Y},y}}d_{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FF}.\end{eqnarray}$$

4.2.4. A local equation of $\unicode[STIX]{x1D71B}$ around $z$ is determined by $\text{pr}_{\mathscr{X}}^{\ast }(f_{x})\,\,\text{pr}_{\mathscr{Y}}^{\ast }(f_{y})$ . Thus

$$\begin{eqnarray}v_{\unicode[STIX]{x1D700}}(\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D71B}))=v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{X}}^{\ast }(f_{x})\,\text{pr}_{\mathscr{Y}}^{\ast }(f_{y})),\end{eqnarray}$$

and by multiplicativity of the valuation $v_{\unicode[STIX]{x1D700}}$ ,

$$\begin{eqnarray}v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{X}}^{\ast }(f_{x})\,\text{pr}_{\mathscr{Y}}^{\ast }(f_{y}))=v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{X}}^{\ast }(f_{x}))+v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{Y}}^{\ast }(f_{y})).\end{eqnarray}$$

Recalling Remark 3.2.13, the valuation can be computed as follows:

$$\begin{eqnarray}v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{X}}^{\ast }(f_{x}))=\min \{\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FE})\,|\,c_{\unicode[STIX]{x1D6FE}}\neq 0\};\end{eqnarray}$$

as the elements $\unicode[STIX]{x1D6FE}$ belong to ${\mathcal{C}}_{\mathscr{X},x}$ and $\unicode[STIX]{x1D6FC}$ is defined to be $\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})$ , we have

$$\begin{eqnarray}\min \{\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FE})\,|\,c_{\unicode[STIX]{x1D6FE}}\neq 0\}=\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})\,|\,c_{\unicode[STIX]{x1D6FE}}\neq 0\}=v_{x,\unicode[STIX]{x1D6FC}}(f_{x}).\end{eqnarray}$$

Hence, we conclude that

(4.2.5) $$\begin{eqnarray}\displaystyle v_{\unicode[STIX]{x1D700}}(\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D71B})) & = & \displaystyle v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{X}}^{\ast }(f_{x}))+v_{\unicode[STIX]{x1D700}}(\text{pr}_{\mathscr{Y}}^{\ast }(f_{y}))\nonumber\\ \displaystyle & = & \displaystyle v_{\unicode[STIX]{x1D6FC}}(f_{x})+v_{\unicode[STIX]{x1D6FD}}(f_{y})\nonumber\\ \displaystyle & = & \displaystyle v_{\unicode[STIX]{x1D6FC}}(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{X^{+}}))+v_{\unicode[STIX]{x1D6FD}}(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{Y^{+}})).\end{eqnarray}$$

4.2.6. This result turns out to be advantageous to compute the weight function $\text{wt}_{\unicode[STIX]{x1D71B}}$ on divisorial points of $\text{Sk}(\mathscr{Z}^{+})$ :

(4.2.7) $$\begin{eqnarray}\displaystyle \text{wt}_{\unicode[STIX]{x1D71B}}(\unicode[STIX]{x1D700}) & = & \displaystyle v_{\unicode[STIX]{x1D700}}(\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D71B}))+m\nonumber\\ \displaystyle & = & \displaystyle v_{\unicode[STIX]{x1D6FC}}(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{X^{+}}))+v_{\unicode[STIX]{x1D6FD}}(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{Y^{+}}))+m\nonumber\\ \displaystyle & = & \displaystyle \text{wt}_{\unicode[STIX]{x1D714}_{X^{+}}}(\unicode[STIX]{x1D6FC})+\text{wt}_{\unicode[STIX]{x1D714}_{Y^{+}}}(\unicode[STIX]{x1D6FD})-m.\end{eqnarray}$$

4.2.8. In the notation of the previous paragraphs, our computations lead to the following result.

Theorem 4.2.9. Suppose that the residue field $k$ is algebraically closed and that $\mathscr{X}^{+}$ is semistable. Then, the PL homeomorphism of skeletons

$$\begin{eqnarray}\text{Sk}(\mathscr{Z}^{+})\xrightarrow[{}]{{\sim}}\text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})\end{eqnarray}$$

given in Proposition 3.4.3 restricts to a PL homeomorphism of Kontsevich–Soibelman skeletons

$$\begin{eqnarray}\text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z},\unicode[STIX]{x1D71B})\xrightarrow[{}]{{\sim}}\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714}_{X^{+}})\times \text{Sk}(Y,\unicode[STIX]{x1D6E5}_{Y},\unicode[STIX]{x1D714}_{Y^{+}}).\end{eqnarray}$$

Proof. This follows immediately from equality (4.2.7), which shows that a point in $\text{Sk}(\mathscr{Z}^{+})$ has minimal value $\text{wt}_{\unicode[STIX]{x1D71B}}(Z,\unicode[STIX]{x1D6E5}_{Z})$ if and only if its projections have minimal value $\text{wt}_{\unicode[STIX]{x1D714}_{X^{+}}}(X,\unicode[STIX]{x1D6E5}_{X})$ and $\text{wt}_{\unicode[STIX]{x1D714}_{Y^{+}}}(Y,\unicode[STIX]{x1D6E5}_{Y})$ .◻

5 The essential skeleton of a product

For this section, we assume the residue field $k$ is algebraically closed.

5.1 Essential skeleton of a pair

5.1.1. Let $X$ be a proper variety over $K$ and $(X,\unicode[STIX]{x1D6E5}_{X})$ a fractional snc pair as in § 1.5. Let $\mathscr{X}^{+}$ be a log-regular model of $(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ over $S^{+}$ . Let $\unicode[STIX]{x1D714}$ be a non-zero regular $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical form on $(X,\unicode[STIX]{x1D6E5}_{X})$ . There exist minimal positive integers $d$ and $n$ such that the divisor $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}^{\otimes d}\unicode[STIX]{x1D70B}^{-n})$ is effective and the multiplicity of some component of the special fiber is zero: we denote this divisor by $D_{\min }(\mathscr{X},\unicode[STIX]{x1D714})$ . It follows from the properties of the weight function (see [MN15, Proposition 4.5.5]) that for any $x\in \text{Sk}(\mathscr{X}^{+})$ ,

$$\begin{eqnarray}v_{x}(D_{\min }(\mathscr{X},\unicode[STIX]{x1D714}))+dm=\text{wt}_{\unicode[STIX]{x1D714}^{\otimes d}\unicode[STIX]{x1D70B}^{-n}}(x)=d\cdot \text{wt}_{\unicode[STIX]{x1D714}}(x)+v_{x}(\unicode[STIX]{x1D70B}^{-n})=d\cdot \text{wt}_{\unicode[STIX]{x1D714}}(x)-n.\end{eqnarray}$$

Lemma 5.1.2. Let $v_{x,\unicode[STIX]{x1D6FC}}$ be a divisorial point in $\text{Sk}(\mathscr{X}^{+})$ . Then $v_{x,\unicode[STIX]{x1D6FC}}\in \text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ if and only if $D_{\min }(\mathscr{X},\unicode[STIX]{x1D714})$ does not contain $x$ .

Proof. We denote $v_{x,\unicode[STIX]{x1D6FC}}$ simply by $\unicode[STIX]{x1D6FC}$ . By the above series of equalities, the weight function $\text{wt}_{\unicode[STIX]{x1D714}}$ reaches its minimum at $\unicode[STIX]{x1D6FC}$ if and only if $v_{\unicode[STIX]{x1D6FC}}(D_{\min }(\mathscr{X},\unicode[STIX]{x1D714}))$ is minimal, hence, in particular, equal to zero.

Let $h:\mathscr{Y}^{+}\rightarrow \mathscr{X}^{+}$ be a sequence of blow-up morphisms of strata of $D_{\mathscr{X}}$ such that $\unicode[STIX]{x1D6FC}$ corresponds to an irreducible component $E$ of $D_{\mathscr{Y}}$ . As in the proof of Proposition 4.1.6,

$$\begin{eqnarray}h^{\ast }(D_{ min}(\mathscr{X},\unicode[STIX]{x1D714}))=h^{\ast }(\text{div}_{\mathscr{ X}^{+}}(\unicode[STIX]{x1D714}^{\otimes d}\unicode[STIX]{x1D70B}^{-n}))=\text{div}_{\mathscr{ Y}^{+}}(\unicode[STIX]{x1D714}^{\otimes d}\unicode[STIX]{x1D70B}^{-n}).\end{eqnarray}$$

Therefore we have that $v_{\unicode[STIX]{x1D6FC}}(D_{\min }(\mathscr{X},\unicode[STIX]{x1D714}))>0$ if and only if $E\subseteq \text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}^{\otimes d}\unicode[STIX]{x1D70B}^{-n})$ , and this holds if and only if $x\in D_{\min }(\mathscr{X},\unicode[STIX]{x1D714})$ .◻

5.1.3. We define the essential skeleton $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})$ of a fractional snc pair $(X,\unicode[STIX]{x1D6E5}_{X})$ as the union of all Kontsevich–Soibelman skeletons $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ , where $\unicode[STIX]{x1D714}$ ranges over all regular $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic pluricanonical forms. In the case of an empty boundary, this recovers the notions introduced in [MN15].

The further reason to define the essential skeleton this way is that it behaves nicely under birational morphisms. Let $f:X^{\prime }\rightarrow X$ be a log resolution. Then there is a $\mathbb{Q}$ -divisor $\unicode[STIX]{x1D6E4}^{\prime }$ with snc support, and no coefficient exceeding $1$ , such that $K_{X^{\prime }}+\unicode[STIX]{x1D6E4}^{\prime }=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{X})$ . Take $\unicode[STIX]{x1D6E5}_{X^{\prime }}$ to be the positive part of $\unicode[STIX]{x1D6E4}^{\prime }$ and write

$$\begin{eqnarray}K_{X^{\prime }}+\unicode[STIX]{x1D6E5}_{X^{\prime }}=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{X})+N.\end{eqnarray}$$

For any $m$ , pullback along with multiplication by the divisor of discrepancies $N$ induces an isomorphism of vector spaces

(5.1.4) $$\begin{eqnarray}H^{0}(X,mK_{X}+m\unicode[STIX]{x1D6E5}_{X})\cong H^{0}(X^{\prime },mK_{X^{\prime }}+m\unicode[STIX]{x1D6E5}_{X^{\prime }}).\end{eqnarray}$$

Let $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D714}^{\prime }$ be corresponding forms via this isomorphism.

Proposition 5.1.5. Under the identification of the birational points of $X$ with those of $X^{\prime }$ , $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ is identified with $\text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }},\unicode[STIX]{x1D714}^{\prime })$ .

Proof. The Kontsevich–Soibelman skeleton $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ is contained in the skeleton associated to any log-regular model by Proposition 4.1.6. Thus, we choose log-regular models $\mathscr{X}^{+}$ and ${\mathscr{X}^{\prime }}^{+}$ so that $f$ extends to a log resolution $f_{R}:\mathscr{X}^{\prime }\rightarrow \mathscr{X}$ and the pair $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ is fractional snc, where $\unicode[STIX]{x1D6E5}_{\mathscr{X}}=\overline{\unicode[STIX]{x1D6E5}_{X}}+\mathscr{X}_{k,\text{red}}$ . Likewise we denote $\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }}=\overline{\unicode[STIX]{x1D6E5}_{X^{\prime }}}+\mathscr{X}_{k,\text{red}}^{\prime }$ , and $(\mathscr{X}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }})$ is fractional snc as $f_{R}$ is a log resolution. It suffices to check the proposition for divisorial valuations. By Lemma 5.1.2 a divisorial valuation $v$ of $\text{Sk}(\mathscr{X}^{+})$ is in $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ if and only if it is a log canonical center of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and the divisor $D_{\min }(\mathscr{X},\unicode[STIX]{x1D714})$ does not contain the center of $v$ .

Suppose $v$ is a divisorial point in $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ . Without loss of generality, we can assume that the divisor $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})$ in $\mathscr{X}^{+}$ does not contain the center of $v$ . As $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ is a fractional snc pair, it is dlt (1.9.9) and we have that

$$\begin{eqnarray}K_{\mathscr{X}^{\prime }}+\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }}=f_{R}^{\ast }(K_{\mathscr{ X}}+\unicode[STIX]{x1D6E5}_{\mathscr{X}})+M,\end{eqnarray}$$

where $M$ is effective, thus $\text{div}_{{\mathscr{X}^{\prime }}^{+}}(\unicode[STIX]{x1D714}^{\prime })=f_{R}^{\ast }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))+mM$ . As $v$ is a log canonical center of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ , $M$ does not vanish along $v$ , so neither does $\text{div}_{{\mathscr{X}^{\prime }}^{+}}(\unicode[STIX]{x1D714}^{\prime })$ . Likewise $v$ is a log canonical center of $(\mathscr{X}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }})$ . It follows that $v\in \text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }},\unicode[STIX]{x1D714}^{\prime })$ .

Conversely, if $v$ is a divisorial point in $\text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }},\unicode[STIX]{x1D714}^{\prime })$ , it is a log canonical center of $(\mathscr{X}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }})$ and the divisor $D_{\min }(\mathscr{X}^{\prime },\unicode[STIX]{x1D714}^{\prime })$ does not contain the center of $v$ . As a result, $v$ is also a log canonical center of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ , and the divisor $D_{\min }(\mathscr{X},\unicode[STIX]{x1D714})$ does not contain the center of $v$ since its pullback does not.◻

We define the Kontsevich–Soibelman skeleton $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ of a dlt pair $(X,\unicode[STIX]{x1D6E5}_{X})$ as the Kontsevich–Soibelman skeleton $\text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }},\unicode[STIX]{x1D714}^{\prime })$ where $(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }})$ is any log resolution of $(X,\unicode[STIX]{x1D6E5}_{X})$ , and $\unicode[STIX]{x1D714}^{\prime }$ is the form corresponding to $\unicode[STIX]{x1D714}$ under the isomorphism (5.1.4): Proposition 5.1.5 guarantees that this is well defined. It follows that we can define the essential skeleton $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})$ of a dlt pair $(X,\unicode[STIX]{x1D6E5}_{X})$ as the essential skeleton of any log resolution of $(X,\unicode[STIX]{x1D6E5}_{X})$ .

Moreover, notice that our construction works more generally for log canonical pairs, hence the notions of Kontsevich–Soibelman skeleton and essential skeleton generalize to such pairs.

5.1.6. Suppose that $(X,\unicode[STIX]{x1D6E5}_{X})$ is a proper dlt pair over $K$ , such that $K_{X}+\unicode[STIX]{x1D6E5}_{X}$ is semiample. Suppose also that $\mathscr{X}$ is a good minimal dlt model of $(X,\unicode[STIX]{x1D6E5}_{X})$ over $R$ and let $\unicode[STIX]{x1D6E5}_{\mathscr{X}}=\overline{\unicode[STIX]{x1D6E5}_{X}}+\mathscr{X}_{k,\text{red}}$ . We denote by ${\mathcal{D}}_{0}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ the dual complex of the strata of the coefficient- $1$ part of $\unicode[STIX]{x1D6E5}_{\mathscr{X}}$ that lie in the special fiber.

We consider a log resolution $f:X^{\prime }\rightarrow X$ that extends to a log resolution of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ . We write $K_{X^{\prime }}+\unicode[STIX]{x1D6E4}^{\prime }=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{X})$ . Let $\unicode[STIX]{x1D6E5}_{X^{\prime }}$ be the positive part of $\unicode[STIX]{x1D6E4}^{\prime }$ . We may embed the open dual complex ${\mathcal{D}}_{0}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ into the birational points of $X$ .

Proposition 5.1.7. This embedding identifies ${\mathcal{D}}_{0}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ with $\text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }})$ .

Proof. Choose a regular $R$ -model $\mathscr{X}^{\prime }$ for $X^{\prime }$ which is a log resolution of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ extending $f$ and let $\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }}=\overline{\unicode[STIX]{x1D6E5}_{X^{\prime }}}+\mathscr{X}_{k,\text{red}}^{\prime }$ , namely we have the log resolutions

$$\begin{eqnarray}\displaystyle & (X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }})\rightarrow (X,\unicode[STIX]{x1D6E5}_{X})\quad \text{where }(X,\unicode[STIX]{x1D6E5}_{X})\text{ dlt and }\unicode[STIX]{x1D6E5}_{X^{\prime }}\text{is the positive part of }\unicode[STIX]{x1D6E4}^{\prime }, & \displaystyle \nonumber\\ \displaystyle & (\mathscr{X}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }})\rightarrow (\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})\quad \text{where }\unicode[STIX]{x1D6E5}_{\mathscr{X}}=\overline{\unicode[STIX]{x1D6E5}_{X}}+\mathscr{X}_{k,\text{red}}\text{ and }\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }}=\overline{\unicode[STIX]{x1D6E5}_{X^{\prime }}}+\mathscr{X}_{k,\text{red}}^{\prime }. & \displaystyle \nonumber\end{eqnarray}$$

As in the previous proof, it suffices to check the proposition for divisorial valuations. Let $v$ be a divisorial valuation, and suppose $v\in {\mathcal{D}}_{0}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ . Then $v$ is a log canonical center for $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ , so $v$ is also a log canonical center of $(\mathscr{X}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }})$ . For a sufficiently divisible index, we may find a $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic pluricanonical form on $(X,\unicode[STIX]{x1D6E5}_{X})$ whose associated divisor in $\mathscr{X}$ has vanishing locus $C$ such that $C$ is a divisor not containing the center of $v$ . After pullback, we get a $\unicode[STIX]{x1D6E5}_{X^{\prime }}$ -logarithmic pluricanonical form $\unicode[STIX]{x1D714}^{\prime }$ whose associated divisor in $\mathscr{X}^{\prime }$ is supported on the strict transform of $C$ and the exceptional divisors of positive log discrepancy. But none of these contain $v$ . Thus, $v\in \text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }},\unicode[STIX]{x1D714}^{\prime })$ .

Conversely, if $v$ is a divisorial point in $\text{Sk}(X^{\prime },\unicode[STIX]{x1D6E5}_{X^{\prime }})$ , then $v$ is a log canonical center of $(\mathscr{X}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{X}^{\prime }})$ , so $v$ is a log canonical center of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ , hence an element of the open dual complex ${\mathcal{D}}_{0}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ .◻

Remark 5.1.8. Proposition 5.1.7 compares the essential skeleton of $(X,\unicode[STIX]{x1D6E5}_{X})$ to the skeleton of a good minimal dlt model of $(X,\unicode[STIX]{x1D6E5}_{X})$ . Thus, the result can be restated as follows: if $(X,\unicode[STIX]{x1D6E5}_{X})$ is a dlt pair with $K_{X}+\unicode[STIX]{x1D6E5}_{X}$ semiample and $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ is a good minimal dlt model of $(X,\unicode[STIX]{x1D6E5}_{X})$ over $R$ , then ${\mathcal{D}}_{0}^{=1}(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})=\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})$ . This generalizes [NX16, Theorem 3.3.3] to dlt pairs.

5.2 Essential skeletons and products of log-regular models

5.2.1. We say that a proper dlt pair $(X,\unicode[STIX]{x1D6E5}_{X})$ has non-negative Kodaira–Iitaka dimension if some multiple of the line bundle $K_{X}+\unicode[STIX]{x1D6E5}_{X}$ has a regular section. We also define products for pairs: if $(X,\unicode[STIX]{x1D6E5}_{X})$ and $(Y,\unicode[STIX]{x1D6E5}_{Y})$ are pairs over $K$ , then we define their product to be $(Z,\unicode[STIX]{x1D6E5}_{Z})$ , where $Z=X\times _{K}Y$ and $\unicode[STIX]{x1D6E5}_{Z}=\unicode[STIX]{x1D6E5}_{X}\times _{K}Y+X\times _{K}\unicode[STIX]{x1D6E5}_{Y}$ . If $(X,\unicode[STIX]{x1D6E5}_{X})$ and $(Y,\unicode[STIX]{x1D6E5}_{Y})$ have semistable models $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ over $R$ , then their product is $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ , where $\mathscr{Z}=\mathscr{X}\times _{R}\mathscr{Y}$ and $\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ is the sum of the strict transform of $\unicode[STIX]{x1D6E5}_{Z}$ with the special fiber $\mathscr{Z}_{k}$ , which is reduced.

Theorem 5.2.2. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ and $(Y,\unicode[STIX]{x1D6E5}_{Y})$ be proper dlt pairs such that $X^{+}=(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ and $Y^{+}=(Y,\lceil \unicode[STIX]{x1D6E5}_{Y}\rceil )$ are log-regular log schemes over $K$ . Suppose that both pairs have non-negative Kodaira–Iitaka dimension and both admit semistable log-regular models $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ over $S^{+}$ . Then the PL homeomorphism of skeletons

$$\begin{eqnarray}\text{Sk}(\mathscr{Z}^{+})\xrightarrow[{}]{{\sim}}\text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})\end{eqnarray}$$

of Proposition 3.4.3 induces a PL homeomorphism of essential skeletons

$$\begin{eqnarray}\text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z})\xrightarrow[{}]{{\sim}}\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})\times \text{Sk}(Y,\unicode[STIX]{x1D6E5}_{Y}),\end{eqnarray}$$

where $\mathscr{Z}^{+}$ , $Z$ and $\unicode[STIX]{x1D6E5}_{Z}$ are the respective products.

Proof. It follows immediately from Theorem 4.2.9 that we have the inclusion $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})\times \text{Sk}(Y,\unicode[STIX]{x1D6E5}_{Y})\subseteq \text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z})$ . Thus, we reduce to proving the following statement. Let $v_{z,\unicode[STIX]{x1D700}}$ be a divisorial point in $\text{Sk}(\mathscr{Z}^{+})$ and $(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}})$ be the corresponding pair in $\text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})$ under the isomorphism of Proposition 3.4.3; if $v_{z,\unicode[STIX]{x1D700}}$ lies in the essential skeleton $\text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z})$ , then $v_{x,\unicode[STIX]{x1D6FC}}$ lies in $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X})$ .

Assume that $v_{z,\unicode[STIX]{x1D700}}$ lies in the essential skeleton $\text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z})$ . Then there exists a non-zero regular $\unicode[STIX]{x1D6E5}_{Z}$ -logarithmic $m$ -pluricanonical form $\unicode[STIX]{x1D714}$ on $Z^{+}$ , such that $v_{z,\unicode[STIX]{x1D700}}\in \text{Sk}(Z,\unicode[STIX]{x1D6E5}_{Z},\unicode[STIX]{x1D714})$ . By Lemma 5.1.2, $D_{\min }(\mathscr{Z},\unicode[STIX]{x1D714})$ does not contain $z$ .

Let $E$ be an irreducible component of $\mathscr{Y}_{k}$ containing $y$ and denote by $\unicode[STIX]{x1D709}_{E}$ the generic point of $E$ . Then the point in the Kato fan of $\mathscr{Z}^{+}$ corresponding to $(x,\unicode[STIX]{x1D709}_{E})$ is not contained in $D_{\min }(\mathscr{Z},\unicode[STIX]{x1D714})$ , as otherwise $z$ would be contained in it.

As $k$ is algebraically closed, we can choose a $k$ -rational point $p$ in $E$ such that $p$ is contained in no other components of $D_{\mathscr{Y}}$ and $D_{\min }(\mathscr{Z},\unicode[STIX]{x1D714})$ does not contain the locus $\overline{\{x\}}\times _{R}\{p\}$ . By Hensel’s lemma and the assumption of semistability, $p$ can be lifted to an $R$ -rational point of $\mathscr{Y}$ . The pullback of $\mathscr{Z}^{+}$ along this $R$ -rational point is an embedding $i:\mathscr{X}^{+}\rightarrow \mathscr{Z}^{+}$ , so we have the following diagram.

Since $S$ has trivial normal bundle in $\mathscr{Y}$ , we have that

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{X^{+}/K}^{\log }=i^{\ast }(\unicode[STIX]{x1D714}_{Z^{+}/K}^{\log }),\end{eqnarray}$$

so $i^{\ast }(\unicode[STIX]{x1D714})$ is a non-zero pluricanonical form on $X$ and, in particular, is a regular $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical form. Moreover, $D_{\min }(\mathscr{X},i^{\ast }(\unicode[STIX]{x1D714}))=i^{\ast }(D_{\min }(\mathscr{Z},\unicode[STIX]{x1D714}))$ . Finally, $x$ is not contained in $D_{\min }(\mathscr{X},i^{\ast }(\unicode[STIX]{x1D714}))$ , as otherwise $i(x)=\{x\}\times _{R}\{p\}$ would be contained in $D_{\min }(\mathscr{Z},\unicode[STIX]{x1D714})$ . By Lemma 5.1.2, $x$ is a point of $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},i^{\ast }(\unicode[STIX]{x1D714}))$ and this concludes the proof.◻

Remark 5.2.3. Consider the case where the line bundles $K_{X}+\unicode[STIX]{x1D6E5}_{X}$ and $K_{Y}+\unicode[STIX]{x1D6E5}_{Y}$ are semiample, that is, some multiple of them is base point free. It follows from the arguments of [NX16, Theorem 3.3.3] that the essential skeleton of $(Z,\unicode[STIX]{x1D6E5}_{Z})$ is a finite union of Kontsevich–Soibelman skeletons where the union runs through a generating set of global sections of a sufficiently large multiple of $K_{Z}+\unicode[STIX]{x1D6E5}_{Z}$ . We can construct such a set from generating sets of global sections of multiples of $K_{X}+\unicode[STIX]{x1D6E5}_{X}$ and $K_{Y}+\unicode[STIX]{x1D6E5}_{Y}$ respectively, via tensor product. Then, in this case, the result of Theorem 5.2.2 follows directly from Theorem 4.2.9.

5.3 Essential skeletons and products of dlt models

5.3.1. We need a combinatorial lemma to understand the formally local behavior of products of semistable dlt models.

Lemma 5.3.2. Let $M$ be the monoid generated by $r_{1}\ldots r_{n_{1}}$ , $s_{1}\ldots s_{n_{2}}$ with the single relation $\sum _{i=1}^{n_{1}}r_{i}=\sum _{j=1}^{n_{2}}s_{j}$ . Then any small $\mathbb{Q}$ -factorialization of the affine toric variety $W=\text{Spec}\,(k[M])$ associated to $M$ is a log resolution.

Proof. We calculate the fan of $W$ . Let $N$ be the dual lattice of $M$ . The fan associated to $W$ is the cone of elements of $N\,\otimes \,\mathbb{R}$ which are non-negative on $M$ . We consider these as linear functions $l$ on the vector space spanned by the $r_{i}$ and $s_{i}$ , subject to the restriction that $l(\sum _{i=1}^{n_{1}}r_{i})=l(\sum _{j=1}^{n_{2}}s_{j})$ . Let $x_{ij}$ be the function which is $1$ on $r_{i}$ and $s_{j}$ and $0$ on all others. Then the fan of $W$ is given by the single cone $C_{W}$ spanned by the $x_{ij}$ .

Any $\mathbb{Q}$ -factorialization $\widetilde{W}$ corresponds to a simplicial subdivision of the cone $C_{W}$ (see [Ful93, p. 65]). We now check that every choice of $\widetilde{W}$ is non-singular.

A maximal cone of $\widetilde{W}$ is spanned by $n=n_{1}+n_{2}-1$ independent rays of $C_{W}$ . Each ray of $C_{W}$ corresponds to a choice of $x_{ij}$ , and we can index these by edges of the complete bipartite graph $B$ on the $r_{i}$ and $s_{j}$ . These $x_{ij}$ are independent if and only if the corresponding edges form a spanning tree. Let $w_{1}\ldots w_{n}$ span a maximal cone of $\widetilde{W}$ . On this affine chart, $\widetilde{W}$ is smooth if and only if the $w_{i}$ generate $N$ as a lattice. We have shown already that the $x_{ij}$ generate $N$ . But every $x_{ij}$ is either one of the $w_{i}$ , or it completes a cycle in $B$ , so that it is a $\mathbb{Z}$ -linear combination of the $w_{i}$ .◻

Proposition 5.3.3. Let $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ be semistable projective good minimal dlt pairs over the germ of a pointed curve $\mathscr{C}$ , such that both $\unicode[STIX]{x1D6E5}_{\mathscr{X}}$ and $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}$ contain their respective special fibers. The product $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is a log canonical pair, $K_{\mathscr{Z}}+\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ is semiample, and the log canonical centers of $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ are strata of the coefficient- $1$ part of $\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ .

Proof. We first show the product $\mathscr{Z}$ is normal. As $\mathscr{X}$ and $\mathscr{Y}$ are semistable, the special fibers $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ are reduced. The dlt condition guarantees that both $\mathscr{X}$ and $\mathscr{Y}$ are Cohen–Macaulay, hence the Cartier divisors $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ are Cohen–Macaulay. The special fiber $\mathscr{Z}_{k}$ is the product $\mathscr{X}_{k}\times _{k}\mathscr{Y}_{k}$ . Since $k$ is algebraically closed, $\mathscr{Z}_{k}$ is reduced, so $\mathscr{Z}$ is regular in codimension $1$ . The special fiber $\mathscr{Z}_{k}$ is the product of Cohen–Macaulay varieties over a field, so it is Cohen–Macaulay by [BK02]. As $\mathscr{Z}_{k}$ is a Cartier divisor in $\mathscr{Z}$ , $\mathscr{Z}$ must be Cohen–Macaulay and, in particular, it is $S_{2}$ .

The divisor $K_{\mathscr{Z}}+\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ is semiample by pullback of semiample divisors. Let $(\widetilde{\mathscr{X}},\unicode[STIX]{x1D6E5}_{\widetilde{\mathscr{X}}})$ and $(\widetilde{\mathscr{Y}},\unicode[STIX]{x1D6E5}_{\widetilde{\mathscr{Y}}})$ be log resolutions of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}\mathscr{Y})$ , respectively. Then we have

$$\begin{eqnarray}\displaystyle & K_{\widetilde{\mathscr{X}}}+\unicode[STIX]{x1D6E5}_{\widetilde{\mathscr{X}}}=f_{\mathscr{X}}^{\ast }(K_{\mathscr{X}}+\unicode[STIX]{x1D6E5}_{\mathscr{X}})+\sum a_{i}E_{\mathscr{X},i}, & \displaystyle \nonumber\\ \displaystyle & K_{\widetilde{\mathscr{Y}}}+\unicode[STIX]{x1D6E5}_{\widetilde{\mathscr{Y}}}=f_{\mathscr{Y}}^{\ast }(K_{\mathscr{Y}}+\unicode[STIX]{x1D6E5}_{\mathscr{Y}})+\sum b_{j}E_{\mathscr{Y},j}, & \displaystyle \nonumber\end{eqnarray}$$

where the coefficients $a_{i}$ and $b_{j}$ are non-negative. Let $\widetilde{\mathscr{Z}}$ be a toroidal log resolution of the fs product $\widetilde{\mathscr{X}}\times _{\mathscr{C}}^{\mathit{fs}}\widetilde{\mathscr{Y}}$ . In particular, $\widetilde{\mathscr{Z}}$ is a log resolution of $\mathscr{Z}$ and we can write

$$\begin{eqnarray}K_{\widetilde{\mathscr{Z}}}+\unicode[STIX]{x1D6E5}_{\widetilde{\mathscr{Z}}}=f_{\mathscr{Z}}^{\ast }(K_{\mathscr{ Z}}+\unicode[STIX]{x1D6E5}_{\mathscr{Z}})+\sum c_{h}E_{\mathscr{Z},h},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}_{\widetilde{\mathscr{Z}}}$ is effective. Over the generic fiber, $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is dlt, so we need only compute discrepancies over the special fiber, namely study the positivity of the coefficients $c_{h}$ .

Let $\unicode[STIX]{x1D6E4}$ be a divisor of $\widetilde{\mathscr{Z}}$ over the special fiber, denote by $v_{\unicode[STIX]{x1D6E4}}$ the corresponding divisorial valuation in $\text{Sk}(\widetilde{\mathscr{Z}}^{+})$ , and by $\unicode[STIX]{x1D6E4}_{\mathscr{X}}$ , $\unicode[STIX]{x1D6E4}_{\mathscr{Y}}$ , and $\unicode[STIX]{x1D6E4}_{\mathscr{Z}}$ its images in $\mathscr{X}$ , $\mathscr{Y}$ , and $\mathscr{Z}$ . The projections of $v_{\unicode[STIX]{x1D6E4}}$ in $\text{Sk}(\widetilde{\mathscr{X}}^{+})$ and $\text{Sk}(\widetilde{\mathscr{Y}}^{+})$ are divisorial valuations. Up to subdivisions of the skeletons, we can assume without loss of generality that the projections correspond to divisors $\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{X}}}$ and $\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{Y}}}$ .

Choose $\unicode[STIX]{x1D6E5}_{\mathscr{X}}$ -logarithmic and $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}$ -logarithmic pluricanonical forms $\unicode[STIX]{x1D714}_{\mathscr{X}}$ on $\mathscr{X}$ and $\unicode[STIX]{x1D714}_{\mathscr{Y}}$ on $\mathscr{Y}$ respectively, such that the divisors $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{X}})$ and $\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{Y}})$ do not contain $\unicode[STIX]{x1D6E4}_{\mathscr{X}}$ and $\unicode[STIX]{x1D6E4}_{\mathscr{Y}}$ respectively, where $\mathscr{X}^{+}=(\mathscr{X},\lceil \unicode[STIX]{x1D6E5}_{\mathscr{X}}\rceil )$ and $\mathscr{Y}^{+}=(\mathscr{Y},\lceil \unicode[STIX]{x1D6E5}_{\mathscr{Y}}\rceil )$ . Then, the divisor $\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{Z}})$ , associated to the wedge product $\unicode[STIX]{x1D714}_{\mathscr{Z}}$ of the pullbacks $\unicode[STIX]{x1D714}_{\mathscr{X}}$ and $\unicode[STIX]{x1D714}_{\mathscr{Y}}$ to $\mathscr{Z}$ , does not contain $\unicode[STIX]{x1D6E4}_{\mathscr{Z}}$ . Denote by $\unicode[STIX]{x1D714}_{\widetilde{\mathscr{X}}}$ , $\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Y}}}$ , and $\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Z}}}$ the pullback of the respective forms to $\widetilde{\mathscr{X}}$ , $\widetilde{\mathscr{Y}}$ , and $\widetilde{\mathscr{Z}}$ . Then we have

$$\begin{eqnarray}\displaystyle & \text{div}_{\widetilde{\mathscr{X}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{X}}})=f_{\mathscr{X}}^{\ast }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{X}}))+\sum a_{i}E_{\mathscr{X},i}, & \displaystyle \nonumber\\ \displaystyle & \text{div}_{\widetilde{\mathscr{Y}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Y}}})=f_{\mathscr{Y}}^{\ast }(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{Y}}))+\sum b_{j}E_{\mathscr{Y},j}, & \displaystyle \nonumber\\ \displaystyle & \text{div}_{\widetilde{\mathscr{Z}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Z}}})=f_{\mathscr{Z}}^{\ast }(\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{Z}}))+\sum c_{h}E_{\mathscr{Z},h}. & \displaystyle \nonumber\end{eqnarray}$$

As $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{X}})$ , $\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{Y}})$ , and $\text{div}_{\mathscr{Z}^{+}}(\unicode[STIX]{x1D714}_{\mathscr{Z}})$ do not contain $\unicode[STIX]{x1D6E4}_{\mathscr{X}}$ , $\unicode[STIX]{x1D6E4}_{\mathscr{Y}}$ , and $\unicode[STIX]{x1D6E4}_{\mathscr{Z}}$ respectively, we have

$$\begin{eqnarray}\displaystyle v_{\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{X}}}}(\text{div}_{\widetilde{\mathscr{X}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{X}}})) & = & \displaystyle \sum a_{i}v_{\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{X}}}}(E_{\mathscr{X},i})\geqslant 0,\nonumber\\ \displaystyle v_{\unicode[STIX]{x1D6E4}_{\widetilde{\mathcal{Y}}}}(\text{div}_{\widetilde{\mathscr{Y}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Y}}})) & = & \displaystyle \sum b_{j}v_{\unicode[STIX]{x1D6E4}_{\widetilde{\mathcal{Y}}}}(E_{\mathscr{Y},j})\geqslant 0,\nonumber\\ \displaystyle v_{\unicode[STIX]{x1D6E4}}(\text{div}_{\widetilde{\mathscr{Z}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Z}}})) & = & \displaystyle \sum c_{h}v_{\unicode[STIX]{x1D6E4}}(E_{\mathscr{Z},h}).\nonumber\end{eqnarray}$$

From formula (4.2.5), $v_{\unicode[STIX]{x1D6E4}}(\text{div}_{\widetilde{\mathscr{Z}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Z}}}))=v_{\unicode[STIX]{x1D6E4}_{\widetilde{\mathcal{Y}}}}(\text{div}_{\widetilde{\mathscr{Y}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{Y}}}))+v_{\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{X}}}}(\text{div}_{\widetilde{\mathscr{X}}^{+}}(\unicode[STIX]{x1D714}_{\widetilde{\mathscr{X}}}))$ , hence we obtain that the log discrepancy of $\unicode[STIX]{x1D6E4}$ with respect to the pair $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is non-negative. Moreover, it is zero if and only if the log discrepancies of $\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{X}}}$ and $\unicode[STIX]{x1D6E4}_{\widetilde{\mathscr{Y}}}$ are both zero, namely if and only if $\unicode[STIX]{x1D6E4}_{\mathscr{X}}$ and $\unicode[STIX]{x1D6E4}_{\mathscr{Y}}$ are log canonical centers of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ , respectively. Since for dlt pairs the log canonical centers are the strata of the coefficient- $1$ part of the boundary, it follows that any log canonical center of $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is a product of such strata, hence a stratum of the coefficient- $1$ part of $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ .◻

5.3.4. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ and $(Y,\unicode[STIX]{x1D6E5}_{Y})$ be dlt pairs over the germ of a punctured curve $C$ , and $(Z,\unicode[STIX]{x1D6E5}_{Z})$ their product, where $\unicode[STIX]{x1D6E5}_{Z}=X\times _{C}\unicode[STIX]{x1D6E5}_{Y}+\unicode[STIX]{x1D6E5}_{X}\times _{C}Y$ . Let $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ be semistable projective good minimal dlt models over the pointed curve.

Theorem 5.3.5. The product $(Z,\unicode[STIX]{x1D6E5}_{Z})$ has a semistable projective good minimal dlt model $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D6E5}_{\mathscr{Z}^{\prime }})$ and ${\mathcal{D}}_{0}^{=1}(\unicode[STIX]{x1D6E5}_{\mathscr{Z}^{\prime }})\simeq {\mathcal{D}}_{0}^{=1}(\unicode[STIX]{x1D6E5}_{\mathscr{X}})\times {\mathcal{D}}_{0}^{=1}(\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ .

Proof. Let $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ be the product of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ as in Proposition 5.3.3. Thus $K_{\mathscr{Z}}+\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ is semiample and log canonical, so any minimal dlt model over $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ will also have semiample log canonical divisor.

Let $\unicode[STIX]{x1D717}:\mathscr{W}\rightarrow \mathscr{Z}$ be a log resolution given by iterated blow-ups at centers of codimension at least $2$ . Then there exists an effective divisor $D$ supported on all of the exceptional divisors, such that $-D$ is $\unicode[STIX]{x1D717}$ -ample.

Claim. There exist $B_{\mathscr{X}}$ and $B_{\mathscr{Y}}$ effective divisors on $\mathscr{X}$ and $\mathscr{Y}$ whose respective supports contain no log canonical centers of $\unicode[STIX]{x1D6E5}_{\mathscr{X}}$ and $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}$ , and such that $B_{\mathscr{X}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ and $B_{\mathscr{Y}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{\mathscr{Y}}^{=1}$ are ample, for $\unicode[STIX]{x1D716}$ small and rational.

Choose $\unicode[STIX]{x1D716}$ small and rational. Then $\unicode[STIX]{x1D6E4}_{\mathscr{X}}=\unicode[STIX]{x1D6E5}_{\mathscr{X}}+B_{\mathscr{X}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ and $\unicode[STIX]{x1D6E4}_{\mathscr{Y}}=\unicode[STIX]{x1D6E5}_{\mathscr{Y}}+B_{\mathscr{Y}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5}_{\mathscr{Y}}^{=1}$ are effective, and $(\mathscr{X},\unicode[STIX]{x1D6E4}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E4}_{\mathscr{Y}})$ are klt. Let $\unicode[STIX]{x1D6E4}_{\mathscr{W}}^{\prime }$ be the log pullback to $\mathscr{W}$ of the product $\unicode[STIX]{x1D6E4}_{\mathscr{Z}}$ of $\unicode[STIX]{x1D6E4}_{\mathscr{X}}$ and $\unicode[STIX]{x1D6E4}_{\mathscr{Y}}$ , namely $K_{\mathscr{W}}+\unicode[STIX]{x1D6E4}_{\mathscr{W}}^{\prime }=\unicode[STIX]{x1D717}^{\ast }(K_{\mathscr{Z}}+\unicode[STIX]{x1D6E4}_{\mathscr{Z}})$ ; we denote by $\unicode[STIX]{x1D6E4}_{\mathscr{W}}$ the positive part of $\unicode[STIX]{x1D6E4}_{\mathscr{W}}^{\prime }$ . Since $(\mathscr{Z},\unicode[STIX]{x1D6E4}_{\mathscr{Z}})$ was klt, so is $(\mathscr{W},\unicode[STIX]{x1D6E4}_{\mathscr{W}})$ . For sufficiently small $\unicode[STIX]{x1D6FF}$ , $(\mathscr{W},\unicode[STIX]{x1D6E4}_{\mathscr{W}}+\unicode[STIX]{x1D6FF}D)$ is still klt.

Let $\unicode[STIX]{x1D6FC}_{i}$ be arbitrary small rational coefficients, one for each divisor $\unicode[STIX]{x1D6E5}_{\mathscr{Z},i}^{=1}$ of $\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ with coefficient $1$ and in the special fiber. We will recover the requested dlt model by running an MMP with scaling on the pair $(\mathscr{W},\unicode[STIX]{x1D6E4}_{\mathscr{W}}+\unicode[STIX]{x1D6FF}D-\sum \unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6E5}_{\mathscr{Z},i}^{=1})$ , scaling with respect to an ample divisor $A$ equivalent to $-D$ . By [BCHM10] this MMP terminates in a log terminal model $\unicode[STIX]{x1D719}:\mathscr{W}{\dashrightarrow}\mathscr{Z}^{\prime }$ . Moreover, as long as the $\unicode[STIX]{x1D6FC}_{i}$ are small relative to $\unicode[STIX]{x1D6FF}$ , when the MMP terminates it must be the case that every exceptional divisor is contracted. As running MMP induces birational contractions, the morphism $\unicode[STIX]{x1D713}:\mathscr{Z}^{\prime }\rightarrow \mathscr{Z}$ is small. Hence $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is log canonical, and every log canonical center dominates a stratum of the coefficient- $1$ part of $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ .

By construction of $\mathscr{Z}^{\prime }$ , the divisor $-\sum \,\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6E5}_{\mathscr{Z},i}^{=1}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D719}_{\ast }A$ is $\unicode[STIX]{x1D713}$ -ample, where $\unicode[STIX]{x1D707}$ is arbitrarily small. But for $\unicode[STIX]{x1D707}$ small enough, we can absorb $\unicode[STIX]{x1D707}\unicode[STIX]{x1D719}_{\ast }A$ into the term $\unicode[STIX]{x1D6FF}D$ . Thus in fact $-\sum \,\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6E5}_{\mathscr{Z},i}^{=1}$ is $\unicode[STIX]{x1D713}$ -ample. As a result we can represent

$$\begin{eqnarray}\mathscr{Z}^{\prime }={\text{Proj}_{\mathscr{ Z}}\bigoplus }_{m\geqslant 0}{\mathcal{O}}\bigg(-m\bigg(\sum \unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6E5}_{\mathscr{Z},i}^{=1}\bigg)\bigg).\end{eqnarray}$$

At this point we may take an arbitrarily large Veronese subring and assume the $\unicode[STIX]{x1D6FC}_{i}$ are all integers.

Now we show that $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is a dlt pair by looking at a formal toric model. Indeed, by Proposition 5.3.3,  $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is formally locally toric at the log canonical centers, and the condition of being dlt is a formally local property. Moreover, we reduce to checking this property at the log canonical centers of $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ that lie in the special fiber, as the generic fibers of $\mathscr{Z}^{\prime }$ and $\mathscr{Z}$ are isomorphic and the latter is dlt.

Let $z$ be the generic point of the image in $\mathscr{Z}$ of a log canonical center of $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ , hence $z$ is the generic point of a log canonical center of $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ and by Proposition 5.3.3 it is a stratum of the coefficient- $1$ part of $\unicode[STIX]{x1D6E5}_{\mathscr{Z}}$ . Let $x$ and $y$ be the generic points of the corresponding strata of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ . Let $E_{x}$ and $E_{y}$ be the monoids of effective Cartier divisors supported on the strata near $x$ and $y$ , respectively. Then the corresponding monoid for $z$ is $\langle E_{x}\oplus E_{y}\rangle /(t_{x}=t_{y})$ , where $t_{x}$ and $t_{y}$ are the respective sums of local equations of strata in the special fibers. This monoid has the form $M\oplus \mathbb{N}^{l}$ , where $l$ is the number of horizontal divisors containing $z$ , and $M$ is a monoid of the type considered in Lemma 5.3.2. The toric variety $T_{M,k}=\text{Spec}\,k[M\oplus \mathbb{N}^{l}]$ is a formal local model for $\mathscr{Z}$ near $z$ , so it suffices to consider

$$\begin{eqnarray}T^{\prime }={\text{Proj}_{T_{M,k}}\bigoplus }_{m\geqslant 0}{\mathcal{O}}\bigg(-m\bigg(\sum \unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6E5}_{T,i}\bigg)\bigg),\end{eqnarray}$$

where the divisors $\unicode[STIX]{x1D6E5}_{T,i}$ range over the torus invariant Weil divisors corresponding to the $\unicode[STIX]{x1D6E5}_{\mathscr{Z},i}^{=1}$ .

The divisors corresponding to the $l$ generators of $\mathbb{N}^{l}$ are Cartier, so their contribution to $T$ is trivial and we can reduce to the case $l=0$ . For sufficiently general choices for the $\unicode[STIX]{x1D6FC}_{i}$ , the toric variety $T^{\prime }$ is simplicial, its fan being induced by the simplicial subdivision of the fan of $T_{M,k}$ such that the $\unicode[STIX]{x1D6FC}_{i}$ induce a strictly convex PL function. Thus, $T^{\prime }$ is a small $\mathbb{Q}$ -factorialization of $T_{M,k}$ . By Lemma 5.3.2, for any such model, $T^{\prime }$ is a smooth toric variety, hence its invariant divisors are snc. Thus $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is dlt.

Finally, we compute the dual complex of the coefficient- $1$ part of $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ by looking at formal toric models again. Locally at the generic points of the log canonical centers, $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ are snc, hence log-regular, and are also semistable. By Proposition 2.5.4 there is a bijective correspondence between pairs of points in the Kato fans of $\mathscr{X}$ and $\mathscr{Y}$ locally around the log canonical centers and the points in the Kato fan of their product. This induces, by Proposition 3.4.3, a PL homeomorphism between the product of skeletons around the log canonical centers and the skeleton of their product, namely a bijective correspondence between pairs of points in the dual complexes $\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ and $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}^{=1}$ , and points in the skeleton of $(\mathscr{Z},\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ whose projections map to $\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ and $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}^{=1}$ . A face $\unicode[STIX]{x1D70E}_{z}$ of the skeleton of $\mathscr{Z}$ , where $z$ is a log canonical center corresponding to $(x,y)$ , corresponds to a prime ideal of the monoid $E_{z}=\langle E_{x}\oplus E_{y}\rangle /(t_{x}=t_{y})$ . Thus, we may reduce to considering such faces in the formal toric model $T_{M,k}$ , and to studying the dual complex of the coefficient- $1$ part of $(\mathscr{Z}^{\prime },\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ in the formal model $T^{\prime }$ . But $T^{\prime }$ is obtained by a simplicial subdivision of $T$ . Thus, we conclude that the dual complex ${\mathcal{D}}_{0}^{=1}(\unicode[STIX]{x1D713}^{\ast }\unicode[STIX]{x1D6E5}_{\mathscr{Z}})$ is identified with the product of the dual complexes of the coefficient- $1$ part of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ and $(\mathscr{Y},\unicode[STIX]{x1D6E5}_{\mathscr{Y}})$ .◻

Proof of Claim.

Let $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ be a good projective minimal dlt model of $(X,\unicode[STIX]{x1D6E5}_{X})$ and let $A_{\mathscr{X}}$ be an ample divisor on $\mathscr{X}$ . Let ${\mathcal{J}}$ be the ideal sheaf of $\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ and let

$$\begin{eqnarray}\unicode[STIX]{x1D712}:\mathscr{X}^{\prime }=\text{Bl}_{{\mathcal{J}}}\mathscr{X}\rightarrow \mathscr{X}\end{eqnarray}$$

be the blow-up of $\mathscr{X}$ with respect to ${\mathcal{J}}$ . Then the transform $-((\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1})^{\prime }+E)$ is a $\unicode[STIX]{x1D712}$ -ample Cartier divisor, where $(\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1})^{\prime }$ denotes the strict transform of $\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ and $E$ an effective divisor supported on the exceptional divisor of $\unicode[STIX]{x1D712}$ . For small positive rational $\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x1D712}^{\ast }(A_{\mathscr{X}})-\unicode[STIX]{x1D700}((\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1})^{\prime }+E)$ is ample. We choose such an $\unicode[STIX]{x1D700}$ . For a sufficiently large integer $n$ , we can find

$$\begin{eqnarray}G_{\mathscr{X}^{\prime }}\sim n\unicode[STIX]{x1D712}^{\ast }(A_{\mathscr{ X}})-n\unicode[STIX]{x1D700}((\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1})^{\prime }+E)\end{eqnarray}$$

such that $G_{\mathscr{X}^{\prime }}$ is effective and contains no log canonical centers. Then the pushforward $G_{\mathscr{X}}\sim nA_{\mathscr{X}}-n\unicode[STIX]{x1D700}\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}$ of $G_{\mathscr{X}^{\prime }}$ is effective and contains no log canonical centers of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ . In particular, $G_{\mathscr{X}}$ is Cartier at the log canonical centers of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ .

We can run the same construction for the ideal sheaf of the divisor $G_{\mathscr{X}}$ . Notice that the blow-up with respect to $G_{\mathscr{X}}$ is an isomorphism at the log canonical centers of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ as there $G_{\mathscr{X}}$ is Cartier. We obtain that, for sufficiently small positive rationals $\unicode[STIX]{x1D6FF}$ and then sufficiently large integers $m$ , we can find $H_{\mathscr{X}}\sim mA_{\mathscr{X}}-m\unicode[STIX]{x1D6FF}G_{\mathscr{X}}$ such that it is effective and does not contain any log canonical center of $(\mathscr{X},\unicode[STIX]{x1D6E5}_{\mathscr{X}})$ . Then

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D6FF}}A_{\mathscr{X}}-\frac{1}{m\unicode[STIX]{x1D6FF}}H_{\mathscr{X}}\sim G_{\mathscr{X}}\sim nA_{\mathscr{X}}-n\unicode[STIX]{x1D700}\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}\end{eqnarray}$$

implies that

$$\begin{eqnarray}\bigg(\frac{1}{\unicode[STIX]{x1D6FF}}-n\bigg)A_{\mathscr{X}}\sim \frac{1}{m\unicode[STIX]{x1D6FF}}H_{\mathscr{X}}-n\unicode[STIX]{x1D700}\unicode[STIX]{x1D6E5}_{\mathscr{X}}^{=1}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FF}$ sufficiently small the term $(1/\unicode[STIX]{x1D6FF}-n)$ is positive, and then for $m$ sufficiently large $B_{\mathscr{X}}=(1/m\unicode[STIX]{x1D6FF})H_{\mathscr{X}}$ , which concludes the proof.◻

6 Applications

6.1 Weight functions and skeletons for finite quotients

6.1.1. Let $X$ be a connected, smooth and proper $K$ -variety and let $G$ be a finite group acting on $X$ . Let $X^{\text{an}}$ be the analytification of $X$ . We recall that any point of $X^{\text{an}}$ is a pair $x=(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})$ with $\unicode[STIX]{x1D709}_{x}\in X$ and $|\cdot |_{x}$ an absolute value on the residue field $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D709}_{x})$ that extends the absolute value on $K$ . For any point $\unicode[STIX]{x1D709}_{x}$ of $X$ , an element $g$ of the group $G$ induces an isomorphism between the residue fields $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D709}_{x})$ and $\unicode[STIX]{x1D705}(g.\unicode[STIX]{x1D709}_{x})$ , which we still denote by $g$ . Then the action of $G$ extends to $X^{\text{an}}$ in the following way:

$$\begin{eqnarray}g.(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})=(g.\unicode[STIX]{x1D709}_{x},|\cdot |_{x}\circ g^{-1}).\end{eqnarray}$$

In particular, the action preserves the sets of divisorial and birational points of $X$ .

Let $f:X\rightarrow Y=X/G$ be the quotient map of $K$ -schemes, let $f^{\text{an}}:X^{\text{an}}\rightarrow Y^{\text{an}}$ be the map of Berkovich spaces induced by functoriality and let $\tilde{f}:X^{\text{an}}\rightarrow X^{\text{an}}/G$ be the quotient map of topological spaces.

Proposition 6.1.2 [Ber95, Corollary 5].

In the above notation, there is a canonical homeomorphism between $X^{\text{an}}/G$ and $Y^{\text{an}}$ such that $\tilde{f}$ and $f^{\text{an}}$ are identified.

Lemma 6.1.3. Let $\unicode[STIX]{x1D714}$ be an $m$ -pluricanonical rational form on $X$ . If $\unicode[STIX]{x1D714}$ is $G$ -invariant, then the weight function associated to $\unicode[STIX]{x1D714}$ on the set of birational points is stable under the action of $G$ .

Proof. Let $x$ be a birational point of $X$ and $g$ an element of $G$ . There exist snc models $\mathscr{X}$ and $\mathscr{X}^{\prime }$ over $R$ such that $x\in \text{Sk}(\mathscr{X})$ and $g.x\in \text{Sk}(\mathscr{X}^{\prime })$ . By replacing them by an snc model $\mathscr{Y}$ that dominates both $\mathscr{X}$ and $\mathscr{X}^{\prime }$ , we can assume that both points lies in $\text{Sk}(\mathscr{Y})$ . The weights of $\unicode[STIX]{x1D714}$ at $x$ and $g.x$ are such that

$$\begin{eqnarray}\displaystyle \text{wt}_{\unicode[STIX]{x1D714}}(g.x) & = & \displaystyle v_{g.x}(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}))+m=v_{x}((g^{-1})^{\ast }\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}))+m\nonumber\\ \displaystyle & = & \displaystyle v_{x}(\text{div}_{\mathscr{Y}^{+}}(\unicode[STIX]{x1D714}))+m=\text{wt}_{\unicode[STIX]{x1D714}}(x)\nonumber\end{eqnarray}$$

as $\unicode[STIX]{x1D714}$ is a $G$ -invariant form. Thus we see that birational points in the same $G$ -orbit have the same weight with respect to $\unicode[STIX]{x1D714}$ .◻

Corollary 6.1.4. Let $\unicode[STIX]{x1D714}$ be a $G$ -invariant pluricanonical rational form on $X$ . Then the Kontsevich–Soibelman skeleton $\text{Sk}(X,\unicode[STIX]{x1D714})$ is stable under the action of $G$ .

Proof. This follows immediately from Lemma 6.1.3.◻

6.1.5. Let $y$ be a divisorial point of $Y^{\text{an}}$ and consider a normal $R$ -model $\mathscr{Y}$ of $Y$ adapted to $y$ ; this means that $y$ is the divisorial valuation associated to $(\mathscr{Y},E)$ for some irreducible component $E$ of $\mathscr{Y}_{k}$ , and the model $\mathscr{Y}$ is regular in a neighborhood of the generic point of $E$ . We denote by $\mathscr{X}$ the normalization of $\mathscr{Y}$ inside $K(X)$ , where $K(\mathscr{Y})=K(Y)=K(X)^{G}{\hookrightarrow}K(X)$ . As $X$ is normal and the quotient map $f:X\rightarrow Y$ is finite, we obtain that $\mathscr{X}$ is an $R$ -model of $X$ . Moreover, by normality $\mathscr{X}$ is regular at the generic points of the special fiber $\mathscr{X}_{k}$ .

6.1.6. We assume $\text{char}(k)=0$ . We denote respectively by $R$ and by $B$ the codimension- $1$ components of the ramification locus and of the branch locus of $f:X\rightarrow Y$ . We set $X^{+}=(X,R_{\text{red}})$ and $Y^{+}=(Y,B_{\text{red}})$ . Then $f$ extends to a morphism of log schemes $f^{+}:X^{+}\rightarrow Y^{+}$ .

Lemma 6.1.7. The reflexive sheaves $\unicode[STIX]{x1D714}_{X^{+}/K}^{\log }$ and $\unicode[STIX]{x1D714}_{Y^{+}/K}^{\log }$ are identified via the pullback $(f^{+})^{\ast }$ along the smooth locus of $Y$ .

Proof. From a generalization of [Har77, Proposition IV.2.3] to higher dimension, we have that $\unicode[STIX]{x1D714}_{X/K}=f^{\ast }(\unicode[STIX]{x1D714}_{Y/K})\,\otimes \,{\mathcal{O}}_{X}(R)$ along the smooth locus of $Y$ . It follows that

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{X^{+}/K}^{\log }=\unicode[STIX]{x1D714}_{X/K}\otimes {\mathcal{O}}_{X}(R_{\text{red}})=f^{\ast }(\unicode[STIX]{x1D714}_{Y/K})\otimes {\mathcal{O}}_{X}(R+R_{\text{red}}).\end{eqnarray}$$

In order to study the divisor $R+R_{\text{red}}$ , we consider one irreducible component of $B$ . Let $D_{B}$ be an irreducible component of $B$ , denote by $e$ the ramification index of $f$ at $D_{B}$ , and by $D_{R}$ the support of the preimage of $D_{B}$ . As $\text{mult}_{D_{R}}(R)=e-1$ , $\text{mult}_{D_{R}}(f^{\ast }(D_{B}))=e$ , and $\text{mult}_{D_{R}}(R_{\text{red}})=1$ , we conclude that $R+R_{\text{red}}=f^{\ast }(B)$ , and finally $\unicode[STIX]{x1D714}_{X^{+}/K}^{\log }=(f^{+})^{\ast }(\unicode[STIX]{x1D714}_{Y^{+}/K}^{\log })$ .◻

In particular, Lemma 6.1.7 implies that $G$ -equivariant logarithmic pluricanonical forms on $X$ correspond to logarithmic pluricanonical forms on $Y$ via pullback.

Proposition 6.1.8. Let $\unicode[STIX]{x1D714}$ be a $G$ -invariant $R_{\text{red}}$ -logarithmic $m$ -pluricanonical form on $X^{+}$ and let $\overline{\unicode[STIX]{x1D714}}$ be the corresponding $B_{\text{red}}$ -logarithmic form on $Y^{+}$ via pullback. Let $y$ be a divisorial point of $Y^{\text{an}}$ . Then, for any divisorial point $x\in (f^{\text{an}})^{-1}(y)$ , the weights of $\unicode[STIX]{x1D714}$ at $x$ and of $\overline{\unicode[STIX]{x1D714}}$ at $y$ coincide.

Proof. Let $\mathscr{Y}$ be a model of $Y$ over $R$ such that $y$ has divisorial representation $(\mathscr{Y},E)$ and the model is regular at the generic point of $E$ . Let $\mathscr{X}$ be the normalization of $\mathscr{Y}$ in $K(X)$ : as we observed in paragraph 6.1.5, it is a model of $X$ , regular at generic points of the special fiber $\mathscr{X}_{k}$ . We denote as well by $f:\mathscr{X}\rightarrow \mathscr{Y}$ the extension of $f$ . The preimage of $E$ coincides with the pullback of the Cartier divisor $E$ on $\mathscr{X}$ , hence $f^{-1}(E)$ still defines a codimension- $1$ subset on $\mathscr{X}$ . We denote by $F_{i}$ the irreducible components of $f^{-1}(E)$ and we associate to the $F_{i}$ their corresponding divisorial valuations $x_{i}=(\mathscr{X},F_{i})$ . By Lemma 6.1.3, it is enough to prove the result for one of the $x_{i}$ . We denote it by $x=(\mathscr{X},F)$ and we compare the weights at $y$ and $x$ .

We recall that for log étale morphisms the sheaves of logarithmic differentials are stable under pullback [Kat89, Proposition 3.12]. Furthermore, it suffices to check that, locally around the generic point of $F,$ the morphism $\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}$ is a log étale morphism of divisorial log structures, to conclude that the weights coincide. For this purpose, we will apply Kato’s criterion for log étaleness [Kat89, Theorem 3.5] to log schemes with respect to the étale topology.

We denote by $\unicode[STIX]{x1D709}_{F}$ the generic point of $F$ and by $\unicode[STIX]{x1D709}_{E}$ the generic point of $E$ . The divisorial log structures on $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ have charts $\mathbb{N}$ at $\unicode[STIX]{x1D709}_{F}$ and $\unicode[STIX]{x1D709}_{E}$ . In the étale topology, the normalization morphism $\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}$ admits a chart induced by $u:\mathbb{N}\rightarrow \mathbb{N}$ , where $1\mapsto m$ for some positive integer $m$ as follows.

Firstly, by the universal property of the fiber product, we have a morphism

$$\begin{eqnarray}\text{Spec}\,{\mathcal{O}}_{\mathscr{X},\unicode[STIX]{x1D709}_{F}}\rightarrow \text{Spec}\,{\mathcal{O}}_{\mathscr{Y},\unicode[STIX]{x1D709}_{E}}\times _{\text{Spec}\,\mathbb{Z}[\mathbb{N}]}\text{Spec}\,\mathbb{Z}[\mathbb{N}]\end{eqnarray}$$

and it corresponds to

$$\begin{eqnarray}{\mathcal{O}}_{\mathscr{Y},\unicode[STIX]{x1D709}_{E}}\otimes _{\mathbb{Z}[\mathbb{N}]}\mathbb{Z}[\mathbb{N}]\rightarrow {\mathcal{O}}_{\mathscr{X},\unicode[STIX]{x1D709}_{F}}.\end{eqnarray}$$

This is a morphism of finite type with finite fibers between regular rings and by [Liu02, Lemma 4.3.20] and [Now97] it is flat and unramified, hence étale. One of the two conditions in Kato’s criterion for log étaleness is then fulfilled. Secondly, the chart $u:\mathbb{N}\mapsto \mathbb{N}$ induces a group homomorphism $u^{\text{gp}}:\mathbb{Z}\mapsto \mathbb{Z}$ ; in particular, it is injective and it has finite cokernel. Then $u$ satisfies the second condition of Kato’s criterion for log étaleness. Therefore we conclude that $\text{wt}_{\overline{\unicode[STIX]{x1D714}}}(y)=\text{wt}_{\unicode[STIX]{x1D714}}(x)$ .◻

Proposition 6.1.9. Let $\unicode[STIX]{x1D714}$ be a $G$ -invariant $R_{\text{red}}$ -logarithmic pluricanonical form on $X$ and let $\overline{\unicode[STIX]{x1D714}}$ be the corresponding $B_{\text{red}}$ -logarithmic form. Then the canonical homeomorphism between $X^{\text{an}}/G$ and $Y^{\text{an}}$ of Proposition 6.1.2 induces the homeomorphism

$$\begin{eqnarray}\text{Sk}(X,\unicode[STIX]{x1D714})/G\simeq \text{Sk}(X/G,\overline{\unicode[STIX]{x1D714}}).\end{eqnarray}$$

Proof. This follows immediately from Corollary 6.1.4 and Proposition 6.1.8.◻

6.1.10. Let $X$ be a smooth $K$ -variety and let $\unicode[STIX]{x1D714}_{X}$ be a pluricanonical form on $X$ . Let $\text{pr}_{j}:X^{n}\rightarrow X$ be the $j$ th canonical projection. Then

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\bigwedge _{1\leqslant j\leqslant n}\text{pr}_{j}^{\ast }\unicode[STIX]{x1D714}_{X}\end{eqnarray}$$

is a pluricanonical form on $X^{n}$ and, moreover, it is invariant under the action of $\mathfrak{S}_{n}$ . We denote by $\overline{\unicode[STIX]{x1D714}}$ the corresponding form on the quotient $X^{n}/\mathfrak{S}_{n}$ as in Lemma 6.1.7.

Proposition 6.1.11. Assume that the residue field $k$ is algebraically closed. If $X$ admits a semistable log-regular model or a semistable good minimal dlt model, then the Kontsevich–Soibelman skeleton of the $n$ th symmetric product of $X$ associated to $\overline{\unicode[STIX]{x1D714}}$ is PL homeomorphic to the $n$ th symmetric product of the Kontsevich–Soibelman skeleton of $X$ associated to $\unicode[STIX]{x1D714}_{X}$ .

Proof. Iterating the result of Theorems 5.2.2 and 5.3.5, we have that the projection map defines a PL homeomorphism of Kontsevich–Soibelman skeletons

$$\begin{eqnarray}\text{Sk}(X^{n},\unicode[STIX]{x1D714})\xrightarrow[{}]{{\sim}}\text{Sk}(X,\unicode[STIX]{x1D714}_{X})\times \cdots \times \text{Sk}(X,\unicode[STIX]{x1D714}_{X}).\end{eqnarray}$$

Thus, applying Proposition 6.1.9 with the group $\mathfrak{S}_{n}$ acting on the product $X^{n}$ , we obtain that

$$\begin{eqnarray}\text{Sk}(X^{n}/\mathfrak{S}_{n},\overline{\unicode[STIX]{x1D714}})\simeq \text{Sk}(X^{n},\unicode[STIX]{x1D714})/\mathfrak{S}_{n}\simeq \text{Sk}(X,\unicode[STIX]{x1D714}_{X})^{n}/\mathfrak{S}_{n}.\end{eqnarray}$$

Since the action on the Kontsevich–Soibelman skeleton $\text{Sk}(X^{n},\unicode[STIX]{x1D714})$ is induced from the symmetric action on $X^{n}$ , and the projections $\text{pr}_{j}:X^{n}\rightarrow X$ functorially induce the projections $\overline{\text{pr}_{j}}:\text{Sk}(X,\unicode[STIX]{x1D714}_{X})^{n}\rightarrow \text{Sk}(X,\unicode[STIX]{x1D714})$ , the action of $\mathfrak{S}_{n}$ on $\text{Sk}(X,\unicode[STIX]{x1D714})^{n}$ is exactly by permutations of the components. Thus,

$$\begin{eqnarray}\text{Sk}(X,\unicode[STIX]{x1D714}_{X})^{n}/\mathfrak{S}_{n}\simeq \text{Sym}^{n}(\text{Sk}(X,\unicode[STIX]{x1D714}_{X})).\Box\end{eqnarray}$$

6.2 The essential skeleton of Hilbert schemes of a $\text{K3}$ surface

6.2.1. Let $S$ be an irreducible regular surface. We consider $\text{Hilb}^{n}(S)$ the Hilbert scheme of $n$ points on $S$ : by [Fog68] it is an irreducible regular variety of dimension $2n$ . Moreover, the morphism

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{HC}:\text{Hilb}^{n}(S)\rightarrow S^{n}/\mathfrak{S}_{n}\end{eqnarray}$$

that sends a zero-dimensional scheme $Z\subseteq S$ to its associated zero-cycle $[Z]$ is a birational morphism, called the Hilbert–Chow morphism.

6.2.2. Let $S$ be a $K3$ surface over $K$ , namely $S$ is a complete non-singular variety of dimension  $2$ such that $\unicode[STIX]{x1D6FA}_{S/K}^{2}\simeq {\mathcal{O}}_{S}\text{ and }H^{1}(S,{\mathcal{O}}_{S})=0$ . In particular, $S$ is a variety with trivial canonical line bundle.

Corollary 6.2.3. Assume that the residue field $k$ is algebraically closed. Suppose that $S$ admits a semistable log-regular model or a semistable good minimal dlt model. Then the essential skeleton of the Hilbert scheme of $n$ points on $S$ is PL homeomorphic to the $n$ th symmetric product of the essential skeleton of $S$ ,

$$\begin{eqnarray}\text{Sk}(\text{Hilb}^{n}(S))\xrightarrow[{}]{{\sim}}\text{Sym}^{n}(\text{Sk}(S)).\end{eqnarray}$$

Proof. This follows immediately from Corollary 6.1.11 and the birational invariance of the essential skeleton [MN15, Proposition 4.10.1].◻

Proposition 6.2.4. If the essential skeleton of $S$ is PL homeomorphic to a point, a closed interval or the two-dimensional sphere, then the essential skeleton of $\text{Hilb}^{n}(S)$ is PL homeomorphic to a point, the standard $n$ -simplex or $\mathbb{C}\mathbb{P}^{n}$ , respectively.

Proof. Applying Corollary 6.2.3, we reduce to the computation of the symmetric product of a point, a closed interval or the sphere $S^{2}$ . Then, the result is trivially true in the first two cases, and follows from [Hat02, §4K] in the third case.◻

6.3 The essential skeleton of generalized Kummer varieties

6.3.1. Let $A$ be an abelian surface over $K$ , namely a complete non-singular, connected group variety of dimension $2$ . Since $A$ is a group variety, the canonical line bundle is trivial and the group structure provides a multiplication morphism $m_{n+1}:A\times A\times \cdots \times A\rightarrow A$ that is invariant under the permutation action of $\mathfrak{S}_{n+1}$ , hence it induces a morphism

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{n+1}:\text{Hilb}^{n+1}(A)\xrightarrow[{}]{\unicode[STIX]{x1D70C}_{HC}}\text{Sym}^{n+1}(A)\rightarrow A\end{eqnarray}$$

by composition with the Hilbert–Chow morphism. Then $K_{n}(A)=\unicode[STIX]{x1D6F4}_{n+1}^{-1}(1)$ is called the $n$ th generalized Kummer variety and is a hyper-Kähler manifold of dimension $2n$ [Bea83].

6.3.2. In [HN17, Proposition 4.3.2], Halle and Nicaise, using Temkin’s generalization of the weight function [Tem16], prove that the essential skeleton of an abelian variety $A$ over $K$ coincides with the construction of a skeleton of $A$ done by Berkovich in [Ber90, Paragraph 6.5]. It follows from this identification and [Ber90, Theorem 6.5.1] that the essential skeleton of $A$ has a group structure, compatible with the group structure on $A^{\text{an}}$ under the retraction $\unicode[STIX]{x1D70C}_{A}$ of $A^{\text{an}}$ onto the essential skeleton, so the diagram

commutes, where $\unicode[STIX]{x1D707}$ denotes the multiplication of $\text{Sk}(A)$ .

Proposition 6.3.3. Assume that the residue field $k$ is algebraically closed. Suppose that $A$ admits a semistable log-regular model or a semistable good minimal dlt model. Then the essential skeleton of the $n$ th generalized Kummer variety is PL homeomorphic to the symmetric quotient of the kernel of the morphism $\unicode[STIX]{x1D707}$ , namely,

$$\begin{eqnarray}\text{Sk}(K_{n}(A))\simeq \text{Sk}(m_{n+1}^{-1}(1)/\mathfrak{S}_{n+1})\simeq \unicode[STIX]{x1D707}_{n+1}^{-1}(1)/\mathfrak{S}_{n+1}.\end{eqnarray}$$

Proof. The first homeomorphism follows from the birational invariance of the essential skeleton [MN15, Proposition 4.10.1]. We write

$$\begin{eqnarray}L=m_{n+1}^{-1}(1)\quad \text{and}\quad \unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D707}_{n+1}^{-1}(1).\end{eqnarray}$$

For any choice of an $\mathfrak{S}_{n+1}$ -invariant generating canonical form on $L$ , it follows from Proposition 6.1.9 that $\text{Sk}(L/\mathfrak{S}_{n+1})\simeq \text{Sk}(L)/\mathfrak{S}_{n+1}$ . We reduce to studying the quotients $\text{Sk}(L)/\mathfrak{S}_{n+1}$ and $\unicode[STIX]{x1D6EC}/\mathfrak{S}_{n+1}$ .

Let $\mathfrak{S}_{n}^{\prime }$ and $\mathfrak{S}_{n}^{\prime \prime }$ be the subgroups of $\mathfrak{S}_{n+1}$ of the permutations that fix $n$ and $n+1$ , respectively. Then $\mathfrak{S}_{n+1}$ is generated by the two subgroups, so its action on $\text{Sk}(L)$ and $\unicode[STIX]{x1D6EC}$ is completely determined by the actions of these subgroups. We consider the isomorphisms

$$\begin{eqnarray}\displaystyle & f_{n}:L\xrightarrow[{}]{{\sim}}A^{n}\quad (z_{1},\ldots ,z_{n+1})\mapsto (z_{1},\ldots ,z_{n-1},z_{n+1}), & \displaystyle \nonumber\\ \displaystyle & f_{n+1}:L\xrightarrow[{}]{{\sim}}A^{n}\quad (z_{1},\ldots ,z_{n+1})\mapsto (z_{1},\ldots ,z_{n-1},z_{n}). & \displaystyle \nonumber\end{eqnarray}$$

Then $f_{n}$ is $\mathfrak{S}_{n}^{\prime }$ -equivariant, $f_{n+1}$ is $\mathfrak{S}_{n}^{\prime \prime }$ -equivariant and the morphism $\unicode[STIX]{x1D713}$

is equivariant with respect to the action of $\mathfrak{S}_{n}^{\prime \prime }$ on the source and of $\mathfrak{S}_{n}^{\prime }$ on the target. Hence, we obtain a commutative diagram of equivariant isomorphisms. We denote by $\overline{f}_{n}$ , $\overline{f}_{n+1}$ and $\overline{\unicode[STIX]{x1D713}}$ the isomorphisms induced on the essential skeletons. By Theorems 5.2.2 and 5.3.5 we can identify $\text{Sk}(A^{n})$ with $\text{Sk}(A)^{n}$ . Thus, we have the following commutative diagram.

Then the action of $\mathfrak{S}_{n+1}$ on $\text{Sk}(L)$ is induced by the isomorphisms $\overline{f}_{n}$ and $\overline{f}_{n+1}$ from the actions of $\mathfrak{S}_{n}^{\prime \prime }$ and $\mathfrak{S}_{n}^{\prime }$ on $\text{Sk}(A)^{n}$ and these actions are compatible as $\overline{\unicode[STIX]{x1D713}}$ is equivariant.

In a similar way, $\unicode[STIX]{x1D6EC}$ is isomorphic to $n$ copies of $\text{Sk}(A)$ and comes equipped with an action of $\mathfrak{S}_{n+1}$ . So we have equivariant projections $g_{n}$ and $g_{n+1}$ with respect to $\mathfrak{S}_{n}^{\prime }$ and $\mathfrak{S}_{n}^{\prime \prime }$ . The equivariant morphism that completes and makes the diagram commutative is $\overline{\unicode[STIX]{x1D713}}$ . Finally, we have the equivariant commutative diagram

and we conclude that the quotients $\text{Sk}(L)/\mathfrak{S}_{n+1}$ and $\unicode[STIX]{x1D6EC}/\mathfrak{S}_{n+1}$ are homeomorphic.◻

Proposition 6.3.4. If the essential skeleton of $A$ is PL homeomorphic to a point, the circle $S^{1}$ or the torus $S^{1}\times S^{1}$ , then the essential skeleton of $K_{n}(A)$ is PL homeomorphic to a point, the standard $n$ -simplex or $\mathbb{C}\mathbb{P}^{n}$ , respectively.

Proof. The case of the point is trivial. For the circle $S^{1}$ , it follows directly from [Mor67, Theorem]. To prove the result for the torus $S^{1}\times S^{1}$ , we apply [Loo76/77, Theorem 3.4]: the action of the symmetric group corresponds to the root system of $A_{n}$ , the highest root is the sum of the simple roots, each with coefficient $1$ , and so the quotient is the complex projective space of dimension  $n$ .◻

6.4 Remarks on hyper-Kähler varieties

6.4.1. The cases we consider in Propositions 6.2.4 and 6.3.4 are motivated by the work of Kulikov, Persson, and Pinkham. In [Kul77] and [PP81], they consider degenerations over the unit complex disk, of surfaces such that some power of the canonical bundle is trivial. They prove that, after base change and birational transformations, any such degeneration can be arranged to be semistable with trivial canonical bundle, namely a Kulikov degeneration. Then they classify the possible special fibers of Kulikov degenerations according to the type of degeneration.

We recall that the monodromy operator $T$ on $\text{H}^{2}(X_{t},\mathbb{Q})$ of the fibers $X_{t}$ of a Kulikov degeneration is unipotent, so we denote by $\unicode[STIX]{x1D708}$ the nilpotency index of $\log (T)$ , namely the positive integer such that $\log (T)^{\unicode[STIX]{x1D708}}=0$ and $\log (T)^{(\unicode[STIX]{x1D708}-1)}\neq 0$ . The type of the Kulikov degeneration is defined as the nilpotency index $\unicode[STIX]{x1D708}$ and called type I, II or III accordingly.

It follows from [Kul77, Theorem II], that the dual complex of the special fiber of a Kulikov degeneration of a $K3$ surface is a point, a closed interval or the sphere $S^{2}$ according to the respective type. For a degeneration of abelian surfaces, the dual complex of the special fiber is homeomorphic to a point, the circle $S^{1}$ or the torus $S^{1}\times S^{1}$ according to the three types (see an overview of these results in [FM83]). In all cases, the dimension of the dual complex is equal $\unicode[STIX]{x1D708}-1$ , hence determined by the type.

6.4.2. Hilbert schemes of $K3$ surfaces and generalized Kummer varieties represent two families of examples of hyper-Kähler varieties. For a semistable degeneration of hyper-Kähler manifolds over the unit disk, it is possible to define the type as the nilpotency index of the monodromy operator on the second cohomology group. It naturally extends the definition for Kulikov degenerations.

In [KLSV17], Kollár, Laza, Saccà, and Voisin study the essential skeleton of a degeneration of hyper-Kähler manifolds in terms of the type. More precisely, in their Theorem 0.10, given a minimal dlt degeneration of $2n$ -dimensional hyper-Kähler manifolds, firstly they prove that the dual complex of the special fiber has dimension $(\unicode[STIX]{x1D708}-1)n$ , where $\unicode[STIX]{x1D708}$ denotes the type of the degeneration. Secondly, they prove that, in the type III case, the dual complex is a simply connected closed pseudo-manifold with the rational homology of $\mathbb{C}\mathbb{P}^{n}$ .

From this perspective, Propositions 6.2.4 and 6.3.4 confirm and strengthen their result for the specific cases of Hilbert schemes and generalized Kummer varieties. In particular, we turn the rational cohomological description of the essential skeleton (Theorem 0.10(ii)) into a topological characterization.

For Hilbert schemes associated to some type II degenerations of $K3$ surfaces, a complementary proof of our result is due to Gulbrandsen, Halle, Hulek, and Zhang; see [GHH16, GHHZ18]. Their approach is based on the method of expanded degenerations, which first appeared in [Li01], and on the construction of suitable GIT quotients, in order to obtain an explicit minimal dlt degeneration for the associated family of Hilbert schemes.

6.4.3. The structure of the essential skeleton of a degeneration of hyper-Kähler manifolds is relevant in the context of mirror symmetry and in view of the work of Kontsevich and Soibelman [KS01, KS06]. The Strominger–Yau–Zaslow (SYZ) fibration [SYZ96] is a conjectural geometric explanation for the phenomenon of mirror symmetry and, roughly speaking, asserts the existence of a special Lagrangian fibration, such that mirror pairs of manifolds with trivial canonical bundle should admit fiberwise dual special Lagrangian fibrations. Moreover, the expectation is that, for type III degenerations of $2n$ -dimensional hyper-Kähler manifolds, the base of the SYZ fibration is $\mathbb{C}\mathbb{P}^{n}$ (see, for instance, [Hwa08]).

The most relevant fact from our perspective is that Kontsevich and Soibelman predict that the base of the Lagrangian fibration of a type III degeneration is homeomorphic to the essential skeleton. So, the outcomes on the topology of the essential skeleton we obtain in Propositions 6.2.4 and 6.3.4 match the predictions of mirror symmetry about the occurrence of $\mathbb{C}\mathbb{P}^{n}$ in the type III case.