Tropical and non-Archimedean limits of degenerating families of volume forms

Sébastien Boucksom and Mattias Jonsson Address: CMLS, École polytechnique
CNRS, Université Paris-Saclay
91128 Palaiseau Cedex
France Email address: sebastien.boucksom@polytechnique.edu Address: Dept of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043
USA Address: Mathematical Sciences
Chalmers University of Technology and University of Gothenburg
SE-412 96 Göteborg
Sweden Email address: mattiasj@umich.edu

Date: August 24, 2026

Abstract.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich–Soibelman and Gross–Wilson, bearing on maximal degenerations of Calabi–Yau manifolds.

2010 Mathematics Subject Classification

Primary: 32Q25, Secondary: 14J32, 14T05, 53C23, 32P05, 14G22

[014P]

Introduction

As is well-known, there is a natural bijection between (smooth, positive) volume forms on a complex manifold and smooth Hermitian metrics on its canonical bundle. Consequently, the data of a smooth family $(\nu_{t})_{t\in{\mathbb{D}}^{*}}$ of volume forms on a holomorphic family $(X_{t})_{t\in{\mathbb{D}}^{*}}$ of compact complex manifolds is equivalent to that of a proper holomorphic submersion $\pi\colon X\to{\mathbb{D}}^{*}$ together with a smooth metric $\psi$ on the relative canonical bundle $K_{X/{\mathbb{D}}^{*}}$ .

We say that the family $(\nu_{t})$ has analytic singularities at $t=0$ if the following conditions hold:

(i)

$\pi\colon X\to{\mathbb{D}}^{*}$ is meromorphic at $0\in{\mathbb{D}}$ in the sense that it extends to a proper, flat map $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ , with ${\mathcal{X}}$ normal;
(ii)

${\mathcal{X}}$ can be chosen so that $K_{X/{\mathbb{D}}^{*}}$ extends to a ${\mathbb{Q}}$ -line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ , and $\psi$ extends continuously to ${\mathcal{L}}$ .

When (i) holds, we call ${\mathcal{X}}$ a model of $X$ . Using resolution of singularities, we can always choose ${\mathcal{X}}$ as an snc model, that is, ${\mathcal{X}}$ is smooth and ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ has simple normal crossing support. To ${\mathcal{X}}$ is then associated a dual complex $\Delta({\mathcal{X}})$ , with one vertex $e_{i}$ for each $E_{i}$ , and a face $\sigma$ for each connected component $Y$ of a non-empty intersection $E_{J}=\bigcap_{i\in J}E_{i}$ with $J\subset I$ .

In the spirit of the Morgan-Shalen topological compactification of affine varieties [MS84], we introduce a natural “hybrid” space

{\mathcal{X}}^{\mathrm{hyb}}:=X\coprod\Delta({\mathcal{X}})

associated to ${\mathcal{X}}$ ; it is equipped with a topology defined in terms of a tropicalization map $X\to\Delta({\mathcal{X}})$ , measuring the logarithmic rate of convergence of local coordinates compatible with ${\mathcal{X}}_{0}$ .

Our first main result says that, after normalizing to unit mass, the volume forms $\nu_{t}$ admit a “tropical” limit inside ${\mathcal{X}}^{\mathrm{hyb}}$ .

[014Q]

Theorem A.

Let $(\nu_{t})_{t\in{\mathbb{D}}^{*}}$ be a family of volume forms on a holomorphic family $X\to{\mathbb{D}}^{*}$ of compact complex manifolds, with analytic singularities at $t=0$ . The asymptotic behavior of the total mass of $\nu_{t}$ is then given by

\nu_{t}(X_{t})\sim c|t|^{2\kappa_{\min}}(\log|t|^{-1})^{d}

with $c\in{\mathbb{R}}_{+}^{*}$ , $\kappa_{\min}\in{\mathbb{Q}}$ and $d\in{\mathbb{N}}^{*}$ , where $d\leq n:=\dim X_{t}$ . Further, given any snc model ${\mathcal{X}}\to{\mathbb{D}}$ of $X\to{\mathbb{D}}^{*}$ such that $K_{X/{\mathbb{D}}^{*}}$ extends to a ${\mathbb{Q}}$ -line bundle on ${\mathcal{L}}$ on ${\mathcal{X}}$ and $\psi$ extends to a continuous metric on ${\mathcal{L}}$ , the rescaled measures

\mu_{t}:=\frac{\nu_{t}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}},

viewed as measures on ${\mathcal{X}}^{\mathrm{hyb}}$ , converge weakly to a Lebesgue type measure $\mu_{0}$ on a $d$ -dimensional subcomplex $\Delta({\mathcal{L}})$ of $\Delta({\mathcal{X}})$ .

The invariant $\kappa_{\min}$ and the subcomplex $\Delta({\mathcal{L}})$ only depend on ${\mathcal{L}}$ (and not on the metric on ${\mathcal{L}}$ ). Consider the logarithmic relative canonical bundle

K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}:=K_{\mathcal{X}}+{\mathcal{X}}_{0,\mathrm{red}}-\pi^{*}(K_{{\mathbb{D}}}+[0])=K_{{\mathcal{X}}/{\mathbb{D}}}+{\mathcal{X}}_{0,\mathrm{red}}-{\mathcal{X}}_{0}

and write $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}={\mathcal{L}}+\sum_{i\in I}a_{i}E_{i}$ with $a_{i}\in{\mathbb{Q}}$ . Setting $\kappa_{i}:=a_{i}/b_{i}$ , we then have $\kappa_{\min}=\min_{i\in I}\kappa_{i}$ , and $\Delta({\mathcal{L}})$ is the subcomplex of $\Delta({\mathcal{X}})$ whose vertices $e_{i}$ correspond to those $i\in I$ achieving the minimum.

On the other hand, the limit measure $\mu_{0}$ does depend on $\psi$ ; it is given by

\mu_{0}=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}(\psi)\right)b_{\sigma}^{-1}\lambda_{\sigma}.

Here, $\sigma$ ranges over the $d$ -dimensional faces of $\Delta({\mathcal{L}})$ , with corresponding strata $Y_{\sigma}\subset{\mathcal{X}}_{0}$ , $\operatorname{Res}_{Y_{\sigma}}(\psi)$ is a naturally defined residual positive measure on $Y_{\sigma}$ , $\lambda_{\sigma}$ is the Lebesgue measure of $\sigma$ normalized by its natural integral affine structure, and $b_{\sigma}\in{\mathbb{Z}}_{>0}$ is an arithmetic coefficient.

The study of the asymptotics of integrals is a very classical subject and has been pursued by many people; see for example the book [AGZV88]. The assertions in Theorem A are closely related to results by Chambert-Loir and Tschinkel (who also worked over general local fields and in an adelic setting). Specifically, the estimate for $\nu_{t}(X_{t})$ , suitably averaged over $t$ , is essentially equivalent to [CLT10, Theorem 1.2]. It also appears in [KS01, §3.1] and is exploited in [BHJ16].

The convergence result for the measures $\mu_{t}$ is also closely related to [CLT10, Corollary 4.8], where, however, the limit measure lives on ${\mathcal{X}}_{0}$ and not on $\Delta({\mathcal{X}})$ .¹¹ 1 A. Chambert-Loir has pointed out that [CLT10, Corollary 4.8] is sufficiently precise, so that when applying it to toric blowups of ${\mathcal{X}}$ one can see the form of the limit measure $\mu_{0}$ in Theorem A. The main new feature of Theorem A is the precise and explicit convergence of the measure $\mu_{t}$ to a “tropical” limit $\mu_{0}$ , living on a simplicial complex.

The following examples illustrate Theorem A. First consider the subvariety

{\mathcal{X}}:=\{(z_{0}^{n+1}+\dots+z_{n}^{n+1})+\varepsilon tz_{0}\cdot\ldots\cdot z_{n}=0\}\subset{\mathbb{C}}\times{\mathbb{P}}^{n},

where $0<\varepsilon\ll 1$ . Write $X:=\mathrm{pr}_{1}^{-1}({\mathbb{C}}^{*})$ . The fiber $X_{t}$ over $t\in{\mathbb{D}}^{*}$ is a Calabi-Yau manifold, and we can choose a nonvanishing holomorphic $n$ -form $\eta_{t}$ on $X_{t}$ to define a smooth metric $\psi$ on $K_{X/{\mathbb{D}}^{*}}$ that extends continuously to ${\mathcal{L}}=K_{{\mathcal{X}}/{\mathbb{D}}}$ . In the terminology of Theorem A we have $\nu_{t}:=2^{-n}i^{n^{2}}\eta_{t}\wedge\overline{\eta}_{t}$ . Here ${\mathcal{X}}_{0}$ is smooth, so $\Delta({\mathcal{X}})$ is a single point. Thus $\nu_{t}(X_{t})\sim c$ for some $c>0$ , and the limit measure $\mu_{0}$ is a point mass.

Now consider instead

{\mathcal{X}}:=\{t\varepsilon(z_{0}^{n+1}+\dots+z_{n}^{n+1})+z_{0}\cdot\ldots\cdot z_{n}=0\}\subset{\mathbb{C}}\times{\mathbb{P}}^{n}.

In this case, $\Delta({\mathcal{L}})=\Delta({\mathcal{X}})$ is a union of $(n+1)$ simplices of dimension $n$ , and topologically a sphere. We have $\nu_{t}(X_{t})\sim c(\log|t|^{-1})^{n}$ and the limit measure is a weighted sum of Lebesgue measures on each simplex. In fact, it is clear by symmetry that the weights are equal; this also follows from Theorem C below.

We also prove a logarithmic version of Theorem A, for a log smooth klt pair $(X,B)$ , and a metric $\psi$ on $K_{(X,B)/{\mathbb{D}}^{*}}$ , see Theorem 8.4.

The space ${\mathcal{X}}^{\mathrm{hyb}}$ and the measure $\mu_{0}$ depend on the choice of snc model ${\mathcal{X}}$ . We obtain a more canonical situation by considering all possible snc models simultaneously. Namely, the set of snc models of $X$ is directed, and in §4 we define a locally compact (Hausdorff) topological space

X^{\mathrm{hyb}}:=\varprojlim_{\mathcal{X}}{\mathcal{X}}^{\mathrm{hyb}},

fibering over ${\mathbb{D}}$ , with central fiber $X^{\mathrm{hyb}}_{0}:=\varprojlim\Delta({\mathcal{X}})$ . For any ${\mathcal{X}}$ , the dual complex $\Delta({\mathcal{X}})$ embeds in the central fiber $X^{\mathrm{hyb}}_{0}$ of $X^{\mathrm{hyb}}$ .

[014R]

Corollary B.

With assumptions and notation as in Theorem A, the measures $\mu_{t}$ , viewed as measures on $X^{\mathrm{hyb}}$ , converge weakly to a measure $\mu_{0}$ . Further, $\mu_{0}$ is a Lebesgue type measure on a $d$ -dimensional complex in $X^{\mathrm{hyb}}_{0}$ .

Now consider the case when $X\to{\mathbb{D}}^{*}$ is projective. As we now explain, the central fiber of $X^{\mathrm{hyb}}$ is then a non-Archimedean space. Namely, $X$ induces a smooth projective variety $X_{K}$ over the non-Archimedean field $K$ of complex formal Laurent series, to which we can associate a Berkovich analytification $X_{K}^{\mathrm{an}}$ . Similarly, any projective snc model ${\mathcal{X}}\to{\mathbb{D}}$ of $X$ induces a projective model ${\mathcal{X}}_{R}$ over the valuation ring $R$ of $K$ . The dual complex $\Delta({\mathcal{X}})$ then has a canonical realization as a compact ${\mathbb{Z}}$ -PA subspace $\operatorname{Sk}({\mathcal{X}})\subset X_{K}^{\mathrm{an}}$ , the skeleton of ${\mathcal{X}}$ . In fact, it is well known (see e.g. [BFJ16]) that there is a homeomorphism $X_{K}^{\mathrm{an}}\overset{\sim}{\to}\varprojlim_{\mathcal{X}}\operatorname{Sk}({\mathcal{X}})$ , so we can identify the central fiber of the space $X^{\mathrm{hyb}}$ with the analytification $X_{K}^{\mathrm{an}}$ . In fact, as shown in Appendix A.6, using ideas from [Berk09], we can view the restriction of $X^{\mathrm{hyb}}\to{\mathbb{D}}$ to a closed subdisc $\overline{{\mathbb{D}}}_{r}$ as the analytification of the base change of $X$ to a suitable Banach ring $A_{r}$ .

Assuming $X\to{\mathbb{D}}^{*}$ is projective, we can describe the limit measure $\mu_{0}$ and its support $\operatorname{Sk}({\mathcal{L}})\simeq\Delta({\mathcal{L}})$ inside $X_{K}^{\mathrm{an}}$ in more detail. The skeleton $\operatorname{Sk}({\mathcal{L}})$ is of purely non-Archimedean nature, and can be seen as a mild generalization of the Kontsevich–Soibelman skeleton introduced in [KS06] and studied in [MN15, NX13, NX16]. The skeletal measure $\mu_{0}$ , on the other hand, depends on both Archimedean and non-Archimedean data. Namely, it is supported on the skeleton $\operatorname{Sk}({\mathcal{L}})$ , but depends on the choice of metric on the restriction of the line bundle ${\mathcal{L}}$ to the central fiber ${\mathcal{X}}_{0}$ (viewed as a complex space) of any snc model ${\mathcal{X}}$ .

We also study both the skeleton and the skeletal measure in the more general case when the model ${\mathcal{X}}$ is allowed to have mild (dlt) singularities.

One major motivation for studying the above general setting comes from degenerations of Calabi–Yau manifolds. Thus suppose $X\to{\mathbb{D}}^{*}$ is a projective holomorphic submersion, meromorphic at $0\in{\mathbb{D}}$ , such that $K_{X/{\mathbb{D}}^{*}}={\mathcal{O}}_{X}$ . Any trivializing section $\eta\in H^{0}(X,K_{X/{\mathbb{D}}^{*}})$ then defines a family $\eta_{t}:=\eta|_{X_{t}}$ of trivializations of $K_{X_{t}}$ , and hence a smooth family of volume forms $\nu_{t}:=|\eta_{t}|^{2}$ with analytic singularities at $t=0$ . Indeed, for any snc model ${\mathcal{X}}\to{\mathbb{D}}$ , $\eta$ extends to a nowhere vanishing section of ${\mathcal{L}}:={\mathcal{O}}_{\mathcal{X}}$ , and $\psi:=\log|\eta|$ defines a smooth metric on ${\mathcal{L}}$ .

The total mass $\nu_{t}(X_{t})=\int_{X_{t}}|\eta_{t}|^{2}$ is then nothing but the $L^{2}$ (or Hodge) metric on the direct image of $K_{X/{\mathbb{D}}^{*}}$ , whose asymptotic behavior at $t=0$ is described in a very precise way by Schmid’s nilpotent orbit theorem [Sch73, Theorem 4.9] (compare for instance [GTZ13b, Proposition 2.1]).

On the other hand, the skeleton $\operatorname{Sk}({\mathcal{L}})$ described above coincides in the current context with the Kontsevich–Soibelman skeleton $\operatorname{Sk}(X)$ [KS06, MN15, NX13]. Its dimension $d$ , which features as the exponent of the log term in the asymptotics of the mass, measures how “bad” the degeneration is. Further, the family $X\to{\mathbb{D}}^{*}$ admits a relative minimal model ${\mathcal{X}}$ , with certain mild (dlt) singularities [KNX15], and the essential skeleton can be identified with the dual complex of ${\mathcal{X}}$ [NX13]. In particular, $d=0$ if and only if $X$ can filled in with a central fiber ${\mathcal{X}}_{0}$ which is a Calabi–Yau variety with canonical singularities.

At the other end of the spectrum, $d=n=\dim X_{t}$ if and only if $X$ is maximally degenerate, i.e. a “large complex structure limit”. In that case, the essential skeleton $\operatorname{Sk}(X)$ is shown to be a pseudomanifold in [NX13]. Building on this, we prove:

[014S]

Theorem C.

Let $X\to{\mathbb{D}}^{*}$ be a smooth projective family of Calabi–Yau varieties, meromorphic at $0\in{\mathbb{D}}$ . Assume that $X$ is maximally degenerate and has semistable reduction. Then the skeletal measure $\mu_{0}$ is a multiple of the integral affine Lesbesgue measure on $\operatorname{Sk}(X)$ .

This theorem also holds in the purely non-Archimedean setting of Calabi–Yau varieties defined over the field of Laurent series. The semistable reduction condition means that $X$ admits an snc model ${\mathcal{X}}$ with ${\mathcal{X}}_{0}$ reduced. This condition is always satisfied after a finite base change.

Theorem C describes measure-theoretic degenerations of Calabi–Yau varieties. Let us briefly discuss the case of metric degenerations. Consider a smooth projective family $X\to{\mathbb{D}}^{*}$ of Calabi–Yau varieties, meromorphic at $0\in{\mathbb{D}}$ , and suppose the family is polarized, that is, we are given a relative ample line bundle $A$ on $X$ . By Yau’s theorem [Yau78], each fiber $X_{t}$ carries a unique Ricci-flat Kähler metric $\omega_{t}$ in the cohomology class of $A_{t}$ .

By [Wan03, Tos15, Taka15], the diameter $D_{t}$ of $(X_{t},\omega_{t})$ remains bounded if and only if $d=0$ , that is, $X$ admits a model ${\mathcal{X}}$ such that ${\mathcal{X}}_{0}$ has klt singularities. In this case, it is shown in [RZ11, RZ13], building in part on [DS14], that $(X_{t},\omega_{t})$ converges in the Gromov-Hausdorff sense to the Calabi–Yau variety ${\mathcal{X}}_{0}$ , endowed with the metric completion of its singular Ricci-flat Kähler metric in the sense of [EGZ09].

The maximally degenerate case $d=n$ is the object of the Kontsevich–Soibelman conjecture [KS06]²² 2 Essentially the same conjecture was stated independently by Gross–Wilson [GW00] and Todorov., which states that $(X_{t},D_{t}^{-2}\omega_{t})$ (which has diameter one) converges in the Gromov-Hausdorff sense to the essential skeleton $\operatorname{Sk}(X)$ endowed with a piecewise smooth metric of Monge-Ampère type, i.e. locally given as the Hessian of a convex function satisfying a real Monge-Ampère equation. This conjecture has been verified for abelian varieties see e.g. [Oda14] but is largely open in general. The “mirror” situation, when one fixes the complex structure and degenerates the cohomology class of the Ricci-flat Kähler metric (along a line segment in the Kähler cone), is better understood [GW00, Tos09, Tos10, GTZ13a, GTZ13b, HT14, TWY14]. By performing a “hyper-Kähler rotation”, this implies a version of the Kontsevich–Soibelman conjecture for special cases of Type III degenerations of K3 surfaces [GW00].

Theorems A and C indicate a possible approach to the Kontsevich–Soibelman conjecture. Indeed, recall that the metric $\omega_{t}$ for $t\in{\mathbb{D}}^{*}$ is constructed as the curvature form of a smooth metric $\phi_{t}$ on $A_{t}$ , where $\phi_{t}$ in turn is obtained as a solution of the complex Monge-Ampère equation $\operatorname{MA}(\phi_{t})=\mu_{t}$ .

On the central fiber $X^{\mathrm{hyb}}_{0}=X_{K}^{\mathrm{an}}$ of $X^{\mathrm{hyb}}$ , it was shown in [BFJ15] that there exists a metric on the line bundle $A_{K}^{\mathrm{an}}$ , unique up to scaling, solving the non-Archimedean Monge-Ampère equation $\operatorname{MA}(\phi_{0})=\mu_{0}$ (at least when $X$ is defined over an algebraic curve). It is now tempting to approach the Kontsevich–Soibelman conjecture by studying the behavior of $\phi_{t}$ as $t\to 0$ . However, this seems to be a delicate issue since there is no a priori reason why the weak continuity at $t=0$ of $t\mapsto\mu_{t}$ would imply continuity of the solutions $t\mapsto\phi_{t}$ .

Instead of Calabi-Yau manifolds, it would be interesting to study degenerating families $X\to{\mathbb{D}}^{*}$ of canonically polarized projective manifolds, where the metric on $K_{X_{t}}$ would be the Kähler-Einstein metric or the Bergman metric, and prove versions of Theorems A and C in this context.

The paper is organized as follows. After recalling various facts in §1 we define in §2 the hybrid space ${\mathcal{X}}^{\mathrm{hyb}}$ associated to an SNC model ${\mathcal{X}}$ . The proof of Theorem A is given in §3. In §4 we define the space $X^{\mathrm{hyb}}$ associated to a degeneration as an inverse limit of the spaces ${\mathcal{X}}^{\mathrm{hyb}}$ , and prove Corollary B. Various notions of skeleta are defined and studied in §5, and in §6 we formalize the notion of a residually metrized model of the canonical bundle, and associate to such an object a positive measure on the relevant Berkovich space. Degenerations of Calabi–Yau varieties are studied in §7 where we prove Theorem C. In §8 we study various extensions, and in Appendix A we recall the Berkovich analytification of a scheme over a Banach ring.

Acknowledgement. We are very grateful to Johannes Nicaise and Chenyang Xu for explaining the behavior of Poincaré residues in the present context. We also thank Vladimir Berkovich, Antoine Chambert-Loir, Antoine Ducros and Charles Favre for useful comments leading up to this work, Bernard Teissier for help with the Hironaka flattening theorem, and Matt Baker and Valentino Tosatti for comments on a preliminary version of this manuscript. Boucksom was supported by the ANR project GRACK. Jonsson was supported by NSF grant DMS-1266207, a grant from the Knut and Alice Wallenberg foundation and a grant from the United States—Israel Binational Science Foundation.

[014T]

1. Preliminaries

The goal of this section is to fix conventions and notation for metrics and measures, and to recall a few basic facts on integral affine structures. We also make a few calculations regarding tropicalizations that will be useful in the proof of Theorem A.

[014U]

1.1. Metrics

We use additive notation for line bundles and metrics over an analytic space $X$ , both in the complex and non-Archimedean setting. This amounts to the following two rules:

(i)

if for $i=1,2$ , $\phi_{i}$ is a metric on a line bundle $L_{i}$ and $a_{i}\in{\mathbb{Z}}$ , then $a_{1}\phi_{1}+a_{2}\phi_{2}$ is a metric on $a_{1}L_{1}+a_{2}L_{2}$ ;
(ii)

a metric on the trivial line bundle ${\mathcal{O}}_{X}$ is of the form $|\cdot|e^{-\phi}$ for a function $\phi$ on $X$ , and we identify the metric with $\phi$ .

If $s$ is a section of a line bundle $L$ on $X$ , then $\log|s|$ stands for the corresponding (possibly singular) metric on $L$ in which $s$ has length 1. For any metric $\phi$ on $L$ , the above rules imply that $\log|s|-\phi$ is a function on $X$ , and

|s|_{\phi}:=|s|e^{-\phi}=\exp(\log|s|-\phi)

is the pointwise length of $s$ in the metric $\phi$ .

A metric on a ${\mathbb{Q}}$ -line bundle $L$ is a collection $(\phi_{m})_{m}$ of metrics on $mL$ , for $m$ sufficiently divisible, such that $\phi_{jm}=j\phi_{m}$ .

The line bundle ${\mathcal{O}}_{X}(D)$ associated to any Cartier divisor $D$ on $X$ comes with a canonical singular metric $\phi_{D}$ , smooth outside $D$ . This fact extends to ${\mathbb{Q}}$ -divisors, by interpreting $\phi_{D}$ as a metric on a ${\mathbb{Q}}$ -line bundle. In the complex case at least, the curvature current of $\phi_{D}$ , correctly normalized, coincides with the integration current on $D$ .

[014V]

1.2. Measures and forms

Any finite-dimensional real vector space $V$ comes equipped with a Lebesgue (or Haar) measure $\lambda$ , uniquely defined up to a multiplicative constant. Any lattice $\Lambda\subset V$ allows us to normalize $\lambda$ by $\lambda(V/\Lambda)=1$ .

To any top-dimensional differential form $\omega$ on a $C^{\infty}$ manifold $X$ is associated a positive measure $|\omega|$ on $X$ . For example, if $\Lambda\subset V$ is a lattice as above, $m_{1},\dots,m_{n}$ is a basis of the dual lattice, then $|dm_{1}\wedge\dots\wedge dm_{n}|$ is Lebesgue measure on $V$ normalized by $\Lambda$ .

If $X$ is a complex manifold of dimension $n$ , and $\Omega$ is a section of $K_{X}$ , that is, a holomorphic $n$ -form, we define $|\Omega|^{2}$ as the positive measure

|\Omega|^{2}:=\frac{i^{n^{2}}}{2^{n}}|\Omega\wedge\bar{\Omega}|.

The normalization is chosen so that the measure associated to the form $dz=dx+idy$ on ${\mathbb{C}}$ is Lebesgue measure $|dz|^{2}=|dx\wedge dy|$ on ${\mathbb{C}}\simeq{\mathbb{R}}^{2}$ .

This construction induces a natural bijection between smooth metrics on the canonical bundle $K_{X}$ and (smooth, positive) volume forms on $X$ , which associates to a smooth metric $\psi$ on $K_{X}$ the volume form $e^{2\psi}$ locally defined by

e^{2\psi}:=\frac{i^{n^{2}}|\Omega\wedge\bar{\Omega}|}{2^{n}|\Omega|^{2}_{\psi}}=\frac{|\Omega|^{2}}{|\Omega|^{2}e^{-2\psi}}

for any local section $\Omega$ of $K_{X}$ . If $\psi^{\prime}$ is another metric on $K_{X}$ , then

e^{2\psi^{\prime}}=e^{2(\psi^{\prime}-\psi)}e^{2\psi},

where $e^{2(\psi^{\prime}-\psi)}$ is the usual exponential of the smooth function $2(\psi^{\prime}-\psi)\in C^{\infty}(X)$ . This can be used to make sense of $e^{2\psi}$ as a positive measure for any (possibly singular) metric $\psi$ on $K_{X}$ . Similarly, $e^{2\psi/m}$ is a volume form for every metric $\psi$ on $mK_{X}$ , $m\in{\mathbb{Z}}$ .

Now assume $(X,B)$ is a pair in the sense of the Minimal Model Program, i.e. $X$ is a normal complex space and $B$ is a (not necessarily effective) ${\mathbb{Q}}$ -Weil divisor on $X$ such that

K_{(X,B)}:=K_{X}+B

is a ${\mathbb{Q}}$ -line bundle. Denote by $\phi_{B}$ the canonical singular metric on $B|_{X_{\mathrm{reg}}}$ , viewed as a ${\mathbb{Q}}$ -line bundle. If $\psi$ is smooth metric on the ${\mathbb{Q}}$ -line bundle $K_{(X,B)}$ , then $\psi-\phi_{B}$ is a smooth metric on $K_{X_{\mathrm{reg}}\setminus B}$ , and $e^{2(\psi-\phi_{B})}$ is thus a volume form on $X_{\mathrm{reg}}\setminus B$ .³³ 3 Here and in what follows, we write $X\setminus D$ for the complement of the support of a (not necessarily reduced) divisor $D$ in a complex space $X$ .

A pair $(X,B)$ is subklt if for some (or, equivalently, any) log resolution $\rho\colon X^{\prime}\to X$ of $(X,B)$ , the unique ${\mathbb{Q}}$ -divisor $B^{\prime}$ such that $\rho^{*}K_{(X,B)}=K_{(X^{\prime},B^{\prime})}$ and $\rho_{*}B^{\prime}=B$ has coefficients $<1$ . The pair $(X,B)$ is klt if $B$ is further effective.

[014W]

Lemma 1.1.

For any smooth metric $\psi$ on $K_{(X,B)}$ , $(X,B)$ is subklt if and only if the measure $e^{2(\psi-\phi_{B})}$ has locally finite mass near each point of $X$ .

[014X]

Proof.

With the above notation it is immediate to check that

\rho^{*}e^{2(\psi-\phi_{B})}=e^{2(\rho^{*}\psi-\phi_{B^{\prime}})}.

We are thus reduced to a log smooth pair $(X^{\prime},B^{\prime})$ , i.e. $X^{\prime}$ is smooth and $B^{\prime}$ has snc support, and the proof is then trivial. ∎

When $(X,B)$ is subklt, we may thus view $e^{2(\psi-\phi_{B})}$ as a finite positive (Radon) measure on $X$ , putting no mass on Zariski closed subsets. Such measures are called adapted in [EGZ09, BBEGZ11].

[014Y]

1.3. Integral piecewise affine spaces

The following discussion roughly follows [KKMS, p.59] and [Berk04, §1].

If $P$ is a rational polytope in ${\mathbb{R}}^{n}$ , that is, the convex hull of a finite subset of ${\mathbb{Q}}^{n}$ , denote by $M_{P}\subset C^{0}(P)$ the finitely generated free abelian group obtained by restricting to $P$ affine functions with coefficients in ${\mathbb{Z}}$ (constant term included). Denote by $1_{P}$ the constant function on $P$ with value $1$ , and set

\vec{M}_{P}:=M_{P}/M_{P}\cap{\mathbb{Q}}1_{P}.

Denote also by $b_{P}\in{\mathbb{N}}$ the greatest integer such that $b_{P}^{-1}1_{P}\in M_{P}$ .

The data of $(P,M_{P})$ modulo homeomorphism is called an (abstract) ${\mathbb{Z}}$ -polytope. The functions in $M_{P}$ are called integral affine, or ${\mathbb{Z}}$ -affine.

The evaluation map defines a canonical realization $P\hookrightarrow(M_{P})^{\vee}_{\mathbb{R}}$ as a codimension one rational polytope, with tangent space $T_{P}$ identified with $(\vec{M}_{P})^{\vee}_{\mathbb{R}}$ . Further, the lattice $T_{P,{\mathbb{Z}}}:=\operatorname{Hom}(\vec{M}_{P},{\mathbb{Z}})\subset T_{P}$ yields a normalized Lebesgue measure $\lambda_{P}$ on $P$ .

The main example for us is as follows.

[014Z]

Lemma 1.2.

Given $b_{0},\dots,b_{p}\in{\mathbb{N}}^{*}$ , view

\sigma=\left\{w\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i=0}^{p}b_{i}w_{i}=1\right\}

as a ${\mathbb{Z}}$ -simplex. Then $b_{\sigma}=\gcd(b_{i})$ , and

\operatorname{Vol}(\sigma)=\frac{b_{\sigma}}{p!\prod_{i}b_{i}}.

[0150]

Proof.

Note that $T_{\sigma,{\mathbb{Z}}}=\{w\in{\mathbb{Z}}^{p+1}\mid\sum_{i}b_{i}w_{i}=0\}$ . The linear isomorphism $\phi\colon{\mathbb{R}}^{p+1}\to{\mathbb{R}}^{p+1}$ given by $\phi(w_{j})=(b_{j}w_{j})$ takes $\sigma$ to the standard simplex

\sigma^{\prime}=\{w^{\prime}\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i}w^{\prime}_{j}=1\},

and hence

[T_{\sigma^{\prime},{\mathbb{Z}}}\colon\phi(T_{\sigma,{\mathbb{Z}}})]\operatorname{Vol}(\sigma)=\operatorname{Vol}(\sigma^{\prime})=\frac{1}{p!}.

Write $T_{\sigma^{\prime},{\mathbb{Z}}}$ as the kernel of $\chi\colon{\mathbb{Z}}^{p+1}\to{\mathbb{Z}}$ defined by $\chi(w^{\prime})=\sum_{i}w^{\prime}_{i}$ . Then $\phi(T_{\sigma,{\mathbb{Z}}})=\ker\chi\cap\phi({\mathbb{Z}}^{p+1})$ , $\chi(\phi({\mathbb{Z}}^{p+1}))=\gcd(b_{i}){\mathbb{Z}}$ , and the exact sequence

0\to\frac{\ker\chi}{\ker\chi\cap\phi({\mathbb{Z}}^{p+1})}\to\frac{{\mathbb{Z}}^{p+1}}{\phi({\mathbb{Z}}^{p+1})}\to\frac{{\mathbb{Z}}}{\chi(\phi({\mathbb{Z}}^{p+1}))}\to 0

gives as desired

[T_{\sigma^{\prime},{\mathbb{Z}}}\colon\phi(T_{\sigma,{\mathbb{Z}}})]=\frac{\prod_{i}b_{i}}{\gcd(b_{i})}.

Finally, the first assertion is clear. ∎

[0151]

Remark 1.3.

By setting $w_{0}=b_{0}^{-1}(1-\sum_{i=1}^{p}b_{i}w_{i})$ , we can identify $\sigma$ with the simplex $\sum_{1}^{p}b_{i}w_{i}\leq 1$ in ${\mathbb{R}}_{+}^{p}$ . The normalized Lebesgue measure on $\sigma$ is then given by $\lambda_{\sigma}=b_{\sigma}^{-1}|dw_{1}\wedge\dots\wedge dw_{p}|$ .

A compact rational polyhedron $K$ in ${\mathbb{R}}^{n}$ is a finite union of rational polytopes $P_{i}$ , which may then be arranged so that $P_{i}\cap P_{j}$ is either empty or a common face of $P_{i}$ and $P_{j}$ . We then say that $(P_{i})$ is a subdivision of $K$ , and call the subdivision simplicial if each $P_{i}$ is a simplex. A continuous function on $K$ is integral piecewise affine ( ${\mathbb{Z}}$ -PA for short) if $f|_{P_{i}}\in M_{P_{i}}$ for some subdivision of $K$ . These functions form a subgroup $\operatorname{PA}_{\mathbb{Z}}(K)\subset C^{0}(K)$ , and the data of $(K,\operatorname{PA}_{\mathbb{Z}}(K))$ modulo homeomorphism is called a compact ${\mathbb{Z}}$ -PA space.

The normalized Lebesgue measure of $K$ is defined as

\lambda_{K}=\sum_{\dim P_{i}=\dim K}{\bf 1}_{P_{i}}\lambda_{P_{i}}

for some (and hence any) subdivision into ${\mathbb{Z}}$ -polytopes.

Note that a ${\mathbb{Z}}$ -polytope $P$ can be regarded as a ${\mathbb{Z}}$ -PA space and that $M_{P}\subset\operatorname{PA}_{\mathbb{Z}}(P)$ .

[0152]

1.4. Tropicalizations and polar coordinates

The material in this section is surely well known, but we include the details for lack of a suitable reference. The calculations here are used in the proof of Theorem 3.4 (which implies Theorem A).

Let $N\simeq{\mathbb{Z}}^{p+1}$ be a lattice, $M=\operatorname{Hom}(N,{\mathbb{Z}})$ the dual lattice, ${\mathbb{C}}[M]$ the semigroup ring and $T=\operatorname{Spec}{\mathbb{C}}[M]=N\otimes{\mathbb{C}}^{*}$ the algebraic torus. A basis for $N$ induces a dual basis $(m_{0},\dots,m_{p})$ for $M$ and elements $z_{i}\in{\mathbb{C}}[M]$ , $0\leq i\leq p$ , such that ${\mathbb{C}}[M]={\mathbb{C}}[z_{0}^{\pm 1},\dots,z_{p}^{\pm 1}]$ and $T\simeq({\mathbb{C}}^{*})^{p+1}$ .

Let $\Omega\in H^{0}(T,K_{T})$ be the $T$ -invariant global section given in coordinates by

\Omega=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}.

Note that $\Omega$ is independent of the choice of coordinates, up to a sign. Its associated measure

\rho:=|\Omega|^{2}

is $T$ -invariant, and hence a Haar measure on $T$ .

We can write this measure in (logarithmic) polar coordinates via the canonical tropicalization map $L\colon T\to N_{\mathbb{R}}$ , given in the basis above by

L=(-\log|z_{0}|,\dots,-\log|z_{p}|).

Note that $L$ sits in the exact sequence $1\to K\to T\to N_{\mathbb{R}}$ obtained by tensoring with $N$ the exact sequence $1\to S^{1}\to{\mathbb{C}}^{*}\to{\mathbb{R}}\to 0$ induced by $z\mapsto-\log|z|$ . In particular, $K=N\otimes S^{1}\simeq(S^{1})^{p+1}$ is a compact torus, and $L\colon T\to N_{\mathbb{R}}$ is a principal $K$ -bundle.

On the one hand, let $\omega$ be the translation invariant real $(p+1)$ -form on the tropical torus $N_{\mathbb{R}}\simeq{\mathbb{R}}^{p+1}$ given by

\omega=dm_{0}\wedge\dots\wedge dm_{p}.

This form is again independent of the choice of basis, up to a sign, and its associated measure $\lambda:=|\omega|$ is the Lebesgue (or Haar) measure on $N_{\mathbb{R}}$ normalized by $N$ .

On the other hand, since $L:T\to N_{\mathbb{R}}$ is a principal $K$ -bundle, each fiber $K_{w}=L^{-1}(w)$ has a unique $K$ -invariant probability measure $\rho_{w}$ . Then $\rho$ has a fiber decomposition

\rho=(2\pi)^{p+1}\lambda(dw)\otimes\rho_{w},

i.e.

\int_{T}f\,d\rho=(2\pi)^{p+1}\int_{N_{\mathbb{R}}}\left(\int_{K_{w}}f\,d\rho_{w}\right)\lambda(dw),

(1.1)

for any $f\in C^{0}_{c}(T)$ . Concretely, we can use logarithmic polar coordinates on $T$ :

z_{j}=\exp(-w_{j}+2\pi i\theta_{j})

for $0\leq j\leq p$ ; then $\rho_{w}=|d\theta_{0}\wedge\dots\wedge d\theta_{p}|$ , and

\rho=\left|\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\right|^{2}=(2\pi)^{p+1}|dw_{1}\wedge\dots\wedge dw_{n}|\otimes\rho_{w}.

We will need the same analysis on certain subgroups of $T$ . Fix an element $m\in M$ and let $\chi=\chi^{m}\colon T\to{\mathbb{C}}^{*}$ be the corresponding character. Let $b\in{\mathbb{Z}}_{>0}$ be the largest integer such that $b^{-1}m\in M$ . In the bases above, we can write $m=\sum_{i=0}^{p}b_{i}m_{i}$ and $\chi=\prod_{i}z_{i}^{b_{i}}$ , where $b_{i}\in{\mathbb{Z}}$ ; then $b=\gcd_{i}b_{i}$ . On the other hand, we can pick a basis such that $m=bm_{0}$ and $\chi=z_{0}^{b}$ . This is useful for computations.

For $t\in{\mathbb{C}}^{*}$ , $T_{t}:=\chi^{-1}(t)$ is a complex manifold with $b$ connected components. Note that $T^{\prime}:=X_{1}$ is an algebraic subgroup of $T$ and that $T_{t}$ is a torsor for $T^{\prime}$ for any $t\in{\mathbb{C}}^{*}$ . The $T$ -invariant $(p+1)$ -form $\Omega$ induces in a canonical way a $T^{\prime}$ -invariant $p$ -form $\Omega_{t}$ on $T_{t}$ , obtained as the restriction to $T_{t}$ of any choice of holomorphic $p$ -form $\Omega^{\prime}$ on $T$ such that $\frac{d\chi}{\chi}\wedge\Omega^{\prime}=\Omega$ . In general coordinates as above, we can pick

\Omega^{\prime}=\frac{1}{\#J}\sum_{j\in J}\frac{(-1)^{j}}{b_{j}}\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\widehat{\frac{dz_{j}}{z_{j}}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}},

where $J=\{j\mid b_{j}\neq 0\}$ . In special coordinates, so that $m=bm_{0}$ and $\chi=z_{0}^{b}$ , we then have $\Omega^{\prime}=\frac{1}{b}\frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}$ , and hence

T_{t}=\bigcup_{u^{b}=t}\{z_{0}=u\}{\quad\text{and}\quad}\Omega_{t}=\frac{1}{b}\frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\bigg|_{T_{t}}.

Note that $\rho_{1}:=|\Omega_{1}|^{2}$ is Haar measure on $T^{\prime}$ , whereas $\rho_{t}:=|\Omega_{t}|^{2}$ is a $T^{\prime}$ -invariant measure on $T_{t}$ . In the special case $p=0$ , $T_{t}$ consists of $b$ points, and $\rho_{t}$ gives mass $\frac{1}{b^{2}}$ to each of them.

Next we study the analogous situation in the tropical torus $N_{\mathbb{R}}$ . Viewing $m$ as a linear form on $N_{\mathbb{R}}$ , set $H_{s}:=m^{-1}(s)$ for $s\in{\mathbb{R}}$ . The lattice $N^{\prime}=\operatorname{Ker}m\subset N$ defines an integral affine structure on $H_{s}$ , and hence a normalized Lebesgue measure $\lambda_{s}$ . Note that

|\omega^{\prime}|_{H_{s}}|=\frac{1}{b}\lambda_{s}

for any choice of $p$ -form $\omega^{\prime}$ on $N_{\mathbb{R}}$ such that $dm\wedge\omega^{\prime}=\omega$ . In general coordinates, we pick

\omega^{\prime}=\frac{1}{\#J}\sum_{j\in J}\frac{(-1)^{j}}{b_{j}}dm_{0}\wedge\dots\wedge\widehat{dm_{j}}\wedge\dots\wedge dm_{p},

where $J=\{j\mid b_{j}\neq 0\}$ . In special coordinates, $\omega^{\prime}=\frac{1}{b}dm_{1}\wedge\dots\wedge dm_{p}$ .

Finally we describe $\rho_{t}$ in polar coordinates. The tropicalization map $L\colon T\to N_{\mathbb{R}}$ induces a principal $T^{\prime}\cap K$ -bundle $T_{t}\to H_{s}$ with $s=-\log|t|$ , and hence an invariant probability measure on $\rho_{t,w}$ on each fiber $K_{t,w}:=T_{t}\cap K_{w}$ . We claim that

\rho_{t}=\frac{(2\pi)^{p}}{b}\lambda_{s}(dw)\otimes\rho_{t,w},

i.e.

\int_{T_{t}}f\,d\rho_{t}=\frac{(2\pi)^{p}}{b}\int_{H_{s}}\left(\int_{K_{t,w}}f\,\rho_{t,w}\right)\lambda_{s}(dw),

(1.2)

for any $f\in C^{0}_{c}(T_{t})$ , where $s=\log|t|^{-1}$ .

The proof is essentially the same as that of (1.1). We work in special coordinates, so that $\chi=z_{0}^{b}$ and $m=bm_{0}$ . Then $T_{t}=\{z_{0}^{b}=t\}$ has $b$ connected components $T_{t}^{(l)}$ , $1\leq l\leq b$ , and

\rho_{t}=|\Omega_{t}|^{2}=\frac{1}{b^{2}}\left|\frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\right|^{2}.

The restriction of the tropicalization map to $T_{t}^{(l)}$ amounts to the change of coordinates $z_{j}=u_{j}^{(l)}\exp(-w_{j}+2\pi i\theta_{j})$ for $1\leq j\leq p$ , where the $u_{j}^{(l)}$ are constants with $|u_{j}^{(l)}|=1$ . In these coordinates,

\rho_{t}|_{T_{t}^{(l)}}=\frac{(2\pi)^{p}}{b^{2}}|dm_{1}\wedge\dots\wedge dm_{n}|\otimes|d\theta_{1}\wedge\dots\wedge d\theta_{p}|.

Here $\frac{1}{b}|d\theta_{1}\wedge\dots\wedge d\theta_{p}|$ induces the measure $\rho_{t,w}$ on $K_{t,w}$ , whereas $|dm_{1}\wedge\dots\wedge dm_{s}|$ is Lebesgue measure $\lambda_{s}$ on $H_{s}$ . Hence (1.2) follows.

[0153]

2. The hybrid space associated to an snc model

In this section, we show how to perform a topological surgery in a complex manifold, replacing a simple normal crossing divisor with its dual complex. Our construction is similar to the one used by Morgan-Shalen in [MS84, §I.3], and can even be traced back to the pioneering work of Bergman [Berg71].

[0154]

2.1. The dual complex

Let $D$ be an effective divisor with simple normal crossing (snc) support in a complex manifold ${\mathcal{X}}$ . By definition, $D=\sum_{i\in I}b_{i}E_{i}$ with $b_{i}\in{\mathbb{N}}^{*}$ and $(E_{i})_{i\in I}$ a finite family of smooth irreducible divisors such that

E_{J}:=\bigcap_{i\in J}E_{i}

is either empty or smooth of codimension $|J|$ (with finitely many connected components) for each $\emptyset\neq J\subset I$ . A connected component $Y$ of a non-empty $E_{J}$ is called a stratum. Together with ${\mathcal{X}}\setminus D=E_{\emptyset}$ , the locally closed submanifolds $\mathring{Y}:=Y\setminus\bigcup_{i\in I\setminus J}E_{i}$ define a partition of ${\mathcal{X}}$ .

The dual complex $\Delta(D)$ is the simplicial complex⁴⁴ 4 This is understood in the slightly generalized sense that the intersection of two faces is a union of common faces. defined as follows: to each stratum $Y$ corresponds a simplex

\sigma_{Y}=\left\{w\in{\mathbb{R}}_{+}^{J}\mid\sum_{i\in J}b_{i}w_{i}=1\right\},

and $\sigma_{Y}$ is a face of $\sigma_{Y^{\prime}}$ if and only if $Y^{\prime}\subset Y$ . This description equips $\Delta(D)$ with an integral affine structure, by which we mean a compatible choice of integral affine structures on each simplex $\sigma$ . This further induces a ${\mathbb{Z}}$ -PA structure on $\Delta(D)$ .

We write $Y_{\sigma}$ for the stratum of a face $\sigma$ . Each point $\xi\in D$ belongs to $\mathring{Y_{\xi}}$ for a unique stratum $Y_{\xi}$ , obtained as the connected component of $E_{J_{\xi}}$ containing $\xi$ , with $J_{\xi}=\{i\in I\mid\xi\in E_{i}\}$ . We denote by $\sigma_{\xi}:=\sigma_{Y_{\xi}}$ the corresponding face of $\Delta(D)$ .

[0155]

2.2. The hybrid topology

Next we define a natural topology on the disjoint union

{\mathcal{X}}^{\mathrm{hyb}}:=({\mathcal{X}}\setminus D)\coprod\Delta(D).

Consider a connected open set ${\mathcal{U}}\subset{\mathcal{X}}$ meeting $D$ and local coordinates $z=(z_{0},\dots,z_{n})$ on ${\mathcal{U}}$ . We say that the pair $({\mathcal{U}},z)$ is adapted (to $D$ ) if the following conditions hold:

(i)

if $E_{0},\dots,E_{p}$ are the irreducible components of $D$ intersecting ${\mathcal{U}}$ , then we have ${\mathcal{U}}\cap E_{0}\cap\dots\cap E_{p}={\mathcal{U}}\cap\mathring{Y}$ for a component $Y$ of $E_{0}\cap\dots\cap E_{p}$ ;
(ii)

$z_{i}$ is an equation of $E_{i}\cap{\mathcal{U}}$ with $|z_{i}|<1$ , $0\leq i\leq p$ .

We call $Y=Y_{\mathcal{U}}$ the stratum of ${\mathcal{U}}$ , and denote by

\sigma_{\mathcal{U}}=\left\{w\in{\mathbb{R}}^{p+1}\mid\sum_{i=0}^{p}b_{i}w_{i}=1\right\}

the corresponding face of $\Delta(D)$ . The function $f_{{\mathcal{U}},z}:=\prod_{i=0}^{p}z_{i}^{b_{i}}$ is an equation of $D$ in ${\mathcal{U}}$ , with $|f_{{\mathcal{U}},z}|<1$ , and we get a continuous map $\operatorname{Log}_{{\mathcal{U}}}\colon{\mathcal{U}}\setminus D\to\sigma_{Y}$ by setting

\operatorname{Log}_{{\mathcal{U}}}=\left(\frac{\log|z_{i}|}{\log|f_{\mathcal{U}}|}\right)_{0\leq i\leq p}.

For any two adapted coordinate charts $({\mathcal{U}},z)$ , $({\mathcal{U}}^{\prime},z^{\prime})$ , with the same stratum $Y$ , we have $z^{\prime}_{i}=u_{i}z_{i}$ with $u_{i}$ nonvanishing on ${\mathcal{U}}\cap{\mathcal{U}}^{\prime}$ , for $i=0,\dots,p$ (after a possible reindexing); it follows that

\operatorname{Log}_{{\mathcal{U}}^{\prime}}=\operatorname{Log}_{{\mathcal{U}}}+O\left(\frac{1}{\log|f_{{\mathcal{U}},z}|^{-1}}\right)

(2.1)

locally uniformly on ${\mathcal{U}}\cap{\mathcal{U}}^{\prime}$ . We next show how to globalize this construction.

[0156]

Proposition 2.1.

There exists an open neighborhood ${\mathcal{V}}\subset{\mathcal{X}}$ of $D$ and a continuous map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}\setminus D\to\Delta(D)$ such that for each adapted coordinate chart $({\mathcal{U}},z)$ with ${\mathcal{U}}\subset{\mathcal{V}}$ we have $\operatorname{Log}_{\mathcal{V}}({\mathcal{U}}\setminus D)\subset\sigma_{\mathcal{U}}$ and

\operatorname{Log}_{\mathcal{V}}=\operatorname{Log}_{{\mathcal{U}}}+O\left(\frac{1}{\log|f_{{\mathcal{U}},z}|^{-1}}\right)

(2.2)

uniformly on compact subsets of ${\mathcal{U}}$ .

This will be accomplished by means of a partition of unity, using the following elementary special case of [Cle77, Theorem 5.7].

[0157]

Lemma 2.2.

There exists a family $(({\mathcal{V}}_{\alpha},z_{\alpha}))_{\alpha\in A}$ of adapted coordinate charts, such that $({\mathcal{V}}_{\alpha})_{\alpha}$ forms a locally finite covering of $D$ and such that the strata $Y_{\alpha}$ of the ${\mathcal{V}}_{\alpha}$ satisfy

\bigcap_{\beta\in B}{\mathcal{V}}_{\beta}\neq\emptyset\Longrightarrow\bigcap_{\beta\in B}Y_{\beta}\neq\emptyset

(2.3)

for every finite $B\subset A$ .

[0158]

Proof of Proposition 2.1.

Pick an open cover $({\mathcal{V}}_{\alpha})_{\alpha}$ as in Lemma 2.2, and denote by $\operatorname{Log}_{\alpha}\colon{\mathcal{V}}_{\alpha}\setminus D\to\sigma_{\alpha}$ the corresponding maps. Set ${\mathcal{V}}:=\bigcup_{\alpha}{\mathcal{V}}_{\alpha}$ , and pick a partition of unity $(\chi_{\alpha})$ subordinate to $({\mathcal{V}}_{\alpha})$ . We claim that for each $\xi\in{\mathcal{V}}$ there exists an open neighborhood $W$ of $\xi$ and a face $\sigma_{W}$ of $\Delta(D)$ such that

W\cap\operatorname{supp}\chi_{\alpha}\neq\emptyset\Longrightarrow\sigma_{\alpha}\subset\sigma_{W}

for any $\alpha\in A$ . Indeed, using (2.3) it is easy to see that

W:=\bigcap_{\alpha\mid\xi\in{\mathcal{U}}_{\alpha}}{\mathcal{V}}_{\alpha}\setminus\bigcup_{\alpha\mid\xi\notin\operatorname{supp}\chi_{\beta}}\operatorname{supp}\chi_{\beta}

satisfies this property. By convexity of $\sigma_{W}$ , it follows that $\operatorname{Log}_{\mathcal{V}}:=\sum_{\alpha}\chi_{\alpha}\operatorname{Log}_{{\mathcal{V}}_{\alpha}}$ is well-defined on $W\setminus D$ , and hence yields a continuous map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}\setminus D\to\Delta(D)$ . The last property is a direct consequence of (2.1). ∎

We extend the previous map as

\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}^{\mathrm{hyb}}:=({\mathcal{V}}\setminus D)\cup\Delta(D)\to\Delta(D)

by setting $\operatorname{Log}_{\mathcal{V}}=\operatorname{id}$ on $\Delta(D)$ .

[0159]

Definition 2.3.

The hybrid topology on ${\mathcal{X}}^{\mathrm{hyb}}:=({\mathcal{X}}\setminus D)\cup\Delta(D)$ is defined as the coarsest topology such that:

(i)

${\mathcal{X}}\setminus D\hookrightarrow{\mathcal{X}}^{\mathrm{hyb}}$ is an open embedding;
(ii)

For every open neighborhood ${\mathcal{V}}$ of $D$ in ${\mathcal{X}}$ , the set $({\mathcal{V}}\setminus D)\cup\Delta({\mathcal{X}})$ is open in ${\mathcal{X}}^{\mathrm{hyb}}$ ;
(iii)

$\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}^{\mathrm{hyb}}\to\Delta(D)$ is continuous.

Using (2.2), this definition is easily seen to be independent of the choice of map $\operatorname{Log}_{\mathcal{V}}$ . If $D$ is compact and $K\subset{\mathcal{X}}$ is a compact neighborhood of $D$ , then one easily checks that the corresponding subset $K^{\mathrm{hyb}}=(K\setminus D)\cup\Delta(D)$ is compact (Hausdorff). When $D=b_{0}E_{0}$ has only one irreducible component, $K^{\mathrm{hyb}}$ is simply the Tychonoff one-point compactification of $K\setminus D$ .

[015A]

Example 2.4.

Set ${\mathcal{X}}={\mathbb{D}}^{2}$ and $D=E_{0}+E_{1}$ the union of the coordinate axes, with coordinates $(z_{0},z_{1})$ . Then ${\mathcal{U}}={\mathcal{X}}$ is itself an adapted coordinate chart. In these coordinates, $\operatorname{Log}_{{\mathcal{U}}}\colon{\mathcal{U}}\setminus D\to\sigma_{\mathcal{U}}$ becomes the map $({\mathbb{D}}^{*})^{2}\to[0,1]$ sending $(z_{0},z_{1})$ to $\log|z_{1}|/\log|z_{0}z_{1}|$ . As a consequence, given $t\in{\mathbb{R}}_{+}^{*}$ and $0<\varepsilon\ll 1$ , the closure in ${\mathcal{X}}^{\mathrm{hyb}}$ of the closed subset

F_{\varepsilon}:=\{0<|z_{0}|,|z_{1}|\leq\varepsilon,|z_{0}|^{t+\varepsilon}\leq|z_{1}|\leq|z_{0}|^{t-\varepsilon}\}\subset{\mathbb{D}}^{2}

is given by ${\bar{F}}_{\varepsilon}=F_{\varepsilon}\cup I_{\varepsilon}$ , where $I_{\varepsilon}:=\{t\in[0,1]\mid\frac{t-\varepsilon}{1+t-\varepsilon}\leq t\leq\frac{t+\varepsilon}{1+t+\varepsilon}\}$ . Further, the sets ${\bar{F}}_{\varepsilon}$ , for $0<\varepsilon\ll 1$ form a basis of closed neighborhoods of the point $\frac{t}{1+t}\in[0,1]$ in ${\mathcal{X}}^{\mathrm{hyb}}$ . See Figure 1.

Original source figure — Figure 1. The figure shows the closed subset $F_{\varepsilon}$ in Example 2.4.

[015B]

3. Proof of Theorem A

In this section, we describe in more detail the objects involved in Theorem A, and then provide a proof. We work purely in the complex analytic category here.

[015C]

3.1. Residual measures

Let $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ be an snc degeneration, i.e. a proper, surjective holomorphic map from a connected complex manifold to the unit disc in ${\mathbb{C}}$ , whose restriction to $X:=\pi^{-1}({\mathbb{D}}^{*})$ is a submersion and such that ${\mathcal{X}}_{0}:=\pi^{-1}(0)=\sum_{i\in I}b_{i}E_{i}$ has snc support. Note that $X_{t}:=\pi^{-1}(t)$ is non-singular for $t\in{\mathbb{D}}^{*}$ . The dual complex $\Delta({\mathcal{X}})$ is defined as that of ${\mathcal{X}}_{0}$ ; it is equipped with its natural ${\mathbb{Z}}$ -PA structure. The logarithmic canonical bundle of ${\mathcal{X}}$ is

K^{\mathrm{log}}_{\mathcal{X}}:=K_{\mathcal{X}}+{\mathcal{X}}_{0,\mathrm{red}}.

Setting $K^{\mathrm{log}}_{{\mathbb{D}}}:=K_{\mathbb{D}}+[0]$ , we define the relative logarithmic canonical bundle as

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

Now suppose we are given a ${\mathbb{Q}}$ -line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ . We then have a unique decomposition

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

with $a_{i}\in{\mathbb{Q}}$ . Set $\kappa_{i}:=a_{i}/b_{i}$ and $\kappa_{\min}:=\min_{i}\kappa_{i}$ .

[015D]

Definition 3.1.

We denote by $\Delta({\mathcal{L}})$ the subcomplex of $\Delta({\mathcal{X}})$ such that a face $\sigma$ of $\Delta({\mathcal{X}})$ is in $\Delta({\mathcal{L}})$ if and only if each vertex of $\sigma$ achieves $\min_{i}\kappa_{i}$ .

In general, $\Delta({\mathcal{L}})$ is neither connected nor pure dimensional. We say that a face of $\Delta({\mathcal{L}})$ is maximal if it is not contained in a larger face of $\Delta({\mathcal{L}})$ .

[015E]

Lemma 3.2.

Let $Y\subset{\mathcal{X}}_{0}$ be a stratum corresponding to face $\sigma$ of $\Delta({\mathcal{X}})$ , and denote by $J\subset I$ the set of irreducible components $E_{i}$ cutting out $Y$ . Then

B^{\mathcal{L}}_{Y}:=\sum_{i\notin J}(1-(a_{i}-\kappa_{\min}b_{i}))E_{i}|_{Y}

is a ${\mathbb{Q}}$ -divisor on $Y$ with snc support, and we have a canonical identification

{\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}:=K_{Y}+B^{\mathcal{L}}_{Y}

as ${\mathbb{Q}}$ -line bundles. If we further assume that $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$ , then $B^{\mathcal{L}}_{Y}$ has coefficients $<1$ , so the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt.

[015F]

Proof.

The first point is a simple consequence of the triviality of the normal bundle ${\mathcal{O}}_{{\mathcal{X}}_{0}}({\mathcal{X}}_{0})$ together with the adjunction formula

K_{Y}=(K_{\mathcal{X}}+\sum_{i\in J}E_{i})|_{Y},

canonically realized by Poincaré residues once an order on $J$ has been chosen. When $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$ , each $E_{i}$ meeting $Y$ properly satisfies $\kappa_{i}>\kappa_{\min}$ , which implies that $B^{\mathcal{L}}_{Y}$ has coefficients $<1$ . ∎

If $\psi$ is a continuous metric on ${\mathcal{L}}$ , $\psi|_{Y}$ may thus be viewed as a metric on $K_{(Y,B^{\mathcal{L}}_{Y})}$ . When $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$ , the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt, and Lemma 1.1 applies. This leads to the following notion.

[015G]

Definition 3.3.

Let $Y$ be a stratum corresponding to a maximal face of $\Delta({\mathcal{L}})$ . The residual measure on $Y$ of a continuous metric $\psi$ on ${\mathcal{L}}$ is the (finite) positive measure on $Y$ defined by

\operatorname{Res}_{Y}(\psi):=\exp\left(2(\psi|_{Y}-\phi_{B^{\mathcal{L}}_{Y}})\right).

This measure can be more explicitly described as follows. At each point $\xi\in Y$ , pick local coordinates $(z_{0},\dots,z_{n})$ such that $z_{0},\dots,z_{p}$ are local equations for the components $E_{0},\dots,E_{p}$ of ${\mathcal{X}}_{0}$ that pass through $\xi$ , indexed so that $J=\{0,\dots,d\}$ , where $0\leq d\leq p$ , and such that $t=\prod_{j=0}^{p}z_{j}^{b_{j}}$ The logarithmic form

\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}

is a local trivialization of $K^{\mathrm{log}}_{{\mathcal{X}}}$ , and hence induces a local trivialization $\Omega^{\mathrm{rel}}=\Omega\otimes(dt/t)^{-1}$ of $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}$ . We may then view $\tau:=\prod_{i=0}^{p}z_{i}^{a_{i}}\Omega^{\mathrm{rel}}$ as a local ${\mathbb{Q}}$ -generator of ${\mathcal{L}}$ . Under the identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$ , we have

\tau|_{Y}=\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\operatorname{Res}_{Y}(\Omega)

with

\operatorname{Res}_{Y}(\Omega)=\frac{dz_{d+1}}{z_{d+1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\bigg|_{Y}.

We infer

\operatorname{Res}_{Y}(\psi)=|\tau|^{-2}_{\psi}\prod_{i=d+1}^{p}|z_{i}|^{2(a_{i}-\kappa_{\min}b_{i}-1)}\bigg|\bigwedge_{i=d+1}^{n}dz_{i}\bigg|^{2}.

(3.1)

[015H]

3.2. Statement and first reductions

It will be convenient to introduce the quantity

\lambda(t):=(\log|t|^{-1})^{-1},

for $t\in{\mathbb{D}}^{*}$ . Note that $\lambda(t)\to 0$ as $t\to 0$ .

Let ${\mathcal{X}}^{\mathrm{hyb}}:=X\coprod\Delta({\mathcal{X}})$ be the locally compact hybrid space constructed in §2. It comes with a proper map $\pi\colon{\mathcal{X}}^{\mathrm{hyb}}\to\Delta$ extending $\pi\colon X\to{\mathbb{D}}^{*}$ and such that $\Delta({\mathcal{X}})=\pi^{-1}(0)$ . The next result implies Theorem A in the introduction.

[015I]

Theorem 3.4.

Let $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ be an snc degeneration, ${\mathcal{L}}$ a ${\mathbb{Q}}$ -line bundle on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ , and $\psi$ a continuous metric on ${\mathcal{L}}$ . Define $\kappa_{\min}$ as above, and set $d:=\dim\Delta({\mathcal{L}})$ . Then, viewed as measures on ${\mathcal{X}}^{\mathrm{hyb}}$ ,

\mu_{t}:=\frac{\lambda(t)^{d}}{(2\pi)^{d}|t|^{2\kappa_{\min}}}e^{2\psi_{t}}

converges weakly to

\mu_{0}:=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}(\psi)\right)b_{\sigma}^{-1}\lambda_{\sigma},

where $\sigma$ ranges over the $d$ -dimensional faces of $\Delta({\mathcal{L}})$ . Here $\lambda_{\sigma}$ denotes normalized Lebesgue measure on $\sigma$ and $b_{\sigma}=\gcd_{i\in J}b_{i}$ , where ${\mathcal{X}}_{0}=\sum_{i}b_{i}E_{i}$ and $E_{i}$ , $i\in J$ are the divisors defining $\sigma$ .

We start by making a few reductions. First, we may—and will—assume in what follows that $\kappa_{\min}=0$ . Indeed, $t$ defines a nonvanishing section of ${\mathcal{O}}_{\mathcal{X}}({\mathcal{X}}_{0})$ , and hence a smooth metric $\log|t|$ , so we may replace ${\mathcal{L}}$ and $\psi$ with ${\mathcal{L}}-\kappa_{\min}{\mathcal{X}}_{0}$ and $\psi-\kappa_{\min}\log|t|$ , respectively, and end up with $\kappa_{\min}=0$ .

Since $\min_{i}a_{i}/b_{i}=\kappa_{\min}=0$ , we then have $a_{i}\geq 0$ , with equality if and only if $E_{i}$ corresponds to a vertex of $\Delta({\mathcal{L}})$ .

Next we reduce the assertion of Theorem 3.4 to a local problem. Let $Y\subset{\mathcal{X}}_{0}$ be the stratum of an arbitrary face $\sigma$ of $\Delta({\mathcal{X}})$ , and denote by $E_{0},\dots,E_{p}$ the components of ${\mathcal{X}}_{0}$ cutting out $Y$ , ordered so that

\kappa_{0}=\dots=\kappa_{q}<\kappa_{q+1}\leq\dots\leq\kappa_{p}.

We can then make the identification

\sigma=\left\{w\in{\mathbb{R}}_{+}^{p+1}\mid b\cdot w=1\right\}

with $b=(b_{0},\dots,b_{p})\in{\mathbb{Z}}_{>0}^{p+1}$ . Set $b^{\prime}=(b_{0},\dots,b_{q})\in{\mathbb{Z}}_{>0}^{q+1}$ and

\sigma^{\prime}:=\left\{w^{\prime}\in{\mathbb{R}}_{+}^{q+1}\mid b^{\prime}\cdot w^{\prime}=1\right\}.

Then $\sigma^{\prime}$ is a face of $\sigma$ under the embedding ${\mathbb{R}}_{+}^{q+1}\hookrightarrow{\mathbb{R}}_{+}^{p+1}$ given by $w^{\prime}\to(w^{\prime},0)$ . Let $Y^{\prime}\supset Y$ be the corresponding stratum of ${\mathcal{X}}_{0}$ .

Note that $\sigma$ contains a face of $\Delta({\mathcal{L}})$ if and only if $\kappa_{0}=0$ ; in that case, the face is unique, equal to $\sigma^{\prime}$ (which then implies $q\leq d$ ).

Pick $x\in\mathring{Y}$ , and choose local coordinates $z=(z_{0},\dots,z_{n})$ at $x$ such that $z_{i}$ is a local equation of $E_{i}$ for $0\leq i\leq p$ and

t=\prod_{i=0}^{p}z_{i}^{b_{i}}

We may assume that $z$ is defined on a polydisc ${\mathcal{U}}\simeq{\mathbb{D}}(r)^{p+1}\times{\mathbb{D}}^{n-p}$ with $0<r\ll 1$ . Decompose

z=(z_{0},\dots,z_{n})\in{\mathcal{U}}\simeq{\mathbb{D}}(r)^{p+1}\times{\mathbb{D}}^{n-p}

z=(z^{\prime},z^{\prime\prime},y)\in{\mathbb{D}}(r)^{q+1}\times{\mathbb{D}}(r)^{p-q}\times{\mathbb{D}}^{n-p},

where we view $y$ as a point of ${\mathcal{U}}\cap Y\simeq{\mathbb{D}}^{n-p}$ , and $(z^{\prime\prime},y)$ as a point of ${\mathcal{U}}\cap Y^{\prime}\simeq{\mathbb{D}}(r)^{p-q}\times{\mathbb{D}}^{n-p}$ .

The coordinate chart $({\mathcal{U}},z)$ is adapted to ${\mathcal{X}}_{0}$ in the sense of §2.2, with

\operatorname{Log}_{{\mathcal{U}}}\colon{\mathcal{U}}\setminus{\mathcal{X}}_{0}\to\sigma

given by

\operatorname{Log}_{{\mathcal{U}}}=\left(\frac{\log|z_{i}|}{\log|t|}\right)_{0\leq i\leq p}.

We aim to establish the following result.

[015J]

Lemma 3.5.

Pick $\chi\in C^{0}_{c}({\mathcal{U}})$ . If $\kappa_{0}=0$ and $q=d$ , then

\lim_{t\to 0}(\operatorname{Log}_{{\mathcal{U}}})_{*}(\chi\mu_{t})=\left(\int_{Y^{\prime}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)\right)b_{\sigma^{\prime}}^{-1}\lambda_{\sigma}^{\prime}

in the weak topology of measures on $\sigma$ , with $\sigma^{\prime}$ the unique $d$ -dimensional face of $\Delta({\mathcal{L}})$ contained in $\sigma$ . Otherwise (i.e. if $\kappa_{0}>0$ or $q<d$ ) $(\operatorname{Log}_{{\mathcal{U}}})_{*}(\chi\mu_{t})\to 0$ .

Granted this result, let us show how to prove Theorem 3.4. For $0<r\ll 1$ , ${\mathcal{V}}:=\pi^{-1}(\overline{{\mathbb{D}}}_{r})\subset{\mathcal{X}}$ is an compact neighborhood of ${\mathcal{X}}_{0}$ with a map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}^{\mathrm{hyb}}\to\Delta({\mathcal{X}})$ as in Proposition 2.1. We will use

[015K]

Lemma 3.6.

Let $\mu_{t}$ , $t\in{\mathbb{D}}_{r}$ be a family of probability measures on ${\mathcal{X}}^{\mathrm{hyb}}$ such that $\mu_{t}$ is supported on ${\mathcal{X}}_{t}$ . Then $\lim_{t\to 0}\mu_{t}=\mu_{0}$ if and only if $\lim_{t\to 0}(\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}=\mu_{0}$ . Here the limits are in the sense of weak convergence of measures on ${\mathcal{X}}^{\mathrm{hyb}}$ and $\Delta({\mathcal{X}})$ , respectively.

By Lemma 3.6 we must show that

(\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}\to\mu_{0}=\sum_{\sigma^{\prime}}\left(\int_{Y^{\prime}}\operatorname{Res}_{Y^{\prime}}(\psi)\right)b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}},

where $\sigma^{\prime}$ ranges over $d$ -dimensional simplices in $\Delta({\mathcal{L}})$ . But this is easily seen to follow from Lemma 3.5, using a partition of unity argument as in the proof of Proposition 2.1.

[015L]

Proof of Lemma 3.6.

The direct implication follows from the continuity of $\operatorname{Log}_{\mathcal{V}}$ . For the reverse implication, assume that $\lim_{t\to 0}(\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}=\mu_{0}$ and consider the following three subsets of $C^{0}({\mathcal{V}})$ : $A_{1}$ is the set of functions of the form $\operatorname{Log}_{\mathcal{V}}^{*}\varphi$ , where $\varphi\in C^{0}(\Delta({\mathcal{X}}))$ ; $A_{2}$ is the set of functions of the form $\pi^{*}g$ , where $g\in C^{0}({\mathbb{D}}_{r})$ ; and $A_{3}=C^{0}_{c}({\mathcal{V}}\setminus\Delta({\mathcal{X}}))$ together with the constant function 1. Then the real vector space $A\subset C^{0}({\mathcal{V}})$ spanned by functions of the form $f_{1}f_{2}f_{3}$ , with $f_{i}\in A_{i}$ is easily seen to be an ${\mathbb{R}}$ -algebra that separates points and contains all constant functions. By the Stone-Weierstrass Theorem, $A$ is dense in $C^{0}({\mathcal{V}})$ , so it suffices to prove that $\lim\int f\mu_{t}=\int f\mu_{0}$ for $f\in A$ . By linearity, we may assume $f=f_{1}f_{2}f_{3}$ with $f_{i}\in A_{i}$ . We may further assume $f_{3}=1$ . Write $f_{1}=\operatorname{Log}_{\mathcal{V}}^{*}\varphi$ and $f_{2}=\pi^{*}g$ . Then

\lim_{t\to 0}\int_{X_{t}}f\mu_{t}=\lim_{t\to 0}g(t)\int_{X_{t}}\varphi\circ\operatorname{Log}_{\mathcal{V}}\mu_{t}\\ =\lim_{t\to 0}g(t)\int_{\Delta({\mathcal{X}})}\varphi\ (\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}=g(0)\int_{\Delta({\mathcal{X}})}\varphi\mu_{0}=\int f\mu_{0},

which completes the proof. ∎

[015M]

3.3. Proof of Lemma 3.5

As in §3.1, we introduce the logarithmic form

\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n},

and the corresponding local trivialization $\Omega^{\mathrm{rel}}=\Omega\otimes(dt/t)^{-1}$ of $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}$ . The restriction $\Omega_{t}$ of $\Omega^{\mathrm{rel}}$ to the fiber $U_{t}:={\mathcal{X}}_{t}\cap{\mathcal{U}}$ is a trivializing section of $K_{U_{t}}$ , explicitly given by

\Omega_{t}=\frac{1}{p+1}\sum_{j=0}^{p}\frac{(-1)^{j}}{b_{j}}\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\widehat{\frac{dz_{j}}{z_{j}}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\bigg|_{U_{t}}.

For $t\in{\mathbb{D}}^{*}$ close to 0, consider the map $\operatorname{Log}_{t}\colon U_{t}\to\sigma\times(Y\cap{\mathcal{U}})$ defined by

\operatorname{Log}_{t}=(\operatorname{Log}_{{\mathcal{U}}},y)=\left(\frac{\log|z_{0}|}{\log|t|},\dots,\frac{\log|z_{p}|}{\log|t|},z_{p+1},\dots,z_{n}\right).

Note the similarity to the situation considered in §1.4. More precisely, view $U:={\mathcal{U}}\cap X$ as embedded in $T\times{\mathbb{C}}^{n-p}$ , where $T=({\mathbb{C}}^{*})^{p+1}$ , and consider the character $\chi=\prod_{i=0}^{p}z_{i}^{b_{i}}$ on $T$ . If $L\colon T\to{\mathbb{R}}^{p+1}$ is the tropicalization map, then

\operatorname{Log}_{t}=(\lambda(t)^{-1}L(z^{\prime},z^{\prime\prime}),y).

Each fiber $\operatorname{Log}_{t}^{-1}(w,y)$ is a torsor for the (possibly disconnected) compact Lie group

K=\left\{\theta\in({\mathbb{R}}/{\mathbb{Z}})^{p+1}\mid\sum_{i}b_{i}\theta_{i}=0\right\};

hence carries a unique $K$ -invariant probability measure $\rho_{t,w,y}$ .

The analysis in §1.4 now gives the following expression for the volume form $|\Omega_{t}|^{2}$ on $U_{t}$ in logarithmic polar coordinates:

[015N]

Lemma 3.7.

For $h\in C^{0}_{c}({\mathcal{U}})$ and $t\in{\mathbb{D}}^{*}$ close to 0, we have

\int_{U_{t}}h|\Omega_{t}|^{2}=(2\pi)^{p}\lambda(t)^{-p}\int_{\sigma\times(Y\cap{\mathcal{U}})}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int_{\operatorname{Log}_{t}^{-1}(w,y)}h\,\rho_{t,w,y},

(3.2)

where $dy:=dz_{p+1}\wedge\dots\wedge dz_{n}$ .

As before, view $\tau:=\prod_{i=0}^{p}z_{i}^{a_{i}}\Omega^{\mathrm{rel}}$ as a local ${\mathbb{Q}}$ -generator of ${\mathcal{L}}$ , and set

g:=-\log|\tau|_{\psi}\in C^{0}({\mathcal{U}}).

By definition, we have $\mu_{t}=(2\pi)^{-d}\lambda(t)^{d}|\Omega_{t}|^{2}/|\Omega_{t}|^{2}_{\psi_{t}}$ , and hence

(2\pi)^{d-p}\lambda(t)^{p-d}|t|^{-2\kappa_{0}}\int\limits_{U_{t}}h\mu_{t}\\ =\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}|t|^{2\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int he^{2g}\,\rho_{t,w,y}\\ =\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}e^{-2\lambda(t)^{-1}\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}b_{\sigma}^{-1}\lambda_{\sigma}\otimes|dy|^{2}\int he^{2g}\,\rho_{t,w,y}

(3.3)

for every $h\in C^{0}_{c}({\mathcal{U}})$ , thanks to Lemma 3.7.

We use the following change of variables. For $t\in{\mathbb{D}}^{*}$ , consider the polytope

\sigma_{t}:=\{(w^{\prime},x^{\prime\prime})\in{\mathbb{R}}_{+}^{q+1}\times{\mathbb{R}}_{+}^{p-q}\mid b^{\prime}\cdot w^{\prime}=1,\ b^{\prime\prime}\cdot x^{\prime\prime}\leq\lambda(t)^{-1}\}\subset\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-q}\subset{\mathbb{R}}_{+}^{p+1},

where $b^{\prime}=(b_{0},\dots,b_{q})$ and $b^{\prime\prime}=(b_{q+1},\dots,b_{p})$ .

[015P]

Lemma 3.8.

The continuous map $Q_{t}\colon\sigma_{t}\to\sigma$ defined by

Q_{t}(w^{\prime},x^{\prime\prime})=\left(\left(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime}\right)w^{\prime},\lambda(t)x^{\prime\prime}\right)

restricts to a homeomorphism between the interior of $\sigma_{t}$ and the interior of $\sigma$ . Further, its inverse maps the Lebesgue measure $b_{\sigma}^{-1}\lambda_{\sigma}$ on $\sigma$ to the measure

(Q_{t}^{-1})_{*}b_{\sigma}^{-1}\lambda_{\sigma}=\left(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime}\right)^{q}\lambda(t)^{p-q}b_{\sigma^{\prime}}^{-1}\lambda^{\prime}_{\sigma^{\prime}}\otimes|dx^{\prime\prime}|,

on $\sigma_{t}$ , where $|dx^{\prime\prime}|$ is Lebesgue measure on ${\mathbb{R}}^{p-q}$ normalized by ${\mathbb{Z}}^{p-q}$ .

[015Q]

Proof.

The first statement is elementary. To prove the second, we must make sure to handle the “multiplicities” $b_{\sigma}$ and $b_{\sigma^{\prime}}$ correctly. Parametrize the interior of $\sigma$ by coordinates $(w_{1},\dots,w_{p})$ using $w_{0}=b_{0}^{-1}(1-\sum_{1}^{p}b_{i}w_{i})$ . By Remark 1.3 we have

b_{\sigma}\lambda_{\sigma}=|dw_{1}\wedge\dots\wedge dw_{p}|

Similarly, we parametrize the interiors of $\sigma^{\prime}$ and $\sigma_{t}$ using coordinates $(w_{1},\dots,w_{q})$ and $(w_{1},\dots,w_{q},x^{\prime\prime}_{q+1},\dots,x^{\prime\prime}_{p})$ , respectively. Then

b_{\sigma^{\prime}}\lambda_{\sigma^{\prime}}=|dw_{1}\wedge\dots\wedge dw_{q}|.

The required formula now follows from an elementary computation. ∎

Using the map $Q_{t}$ and the fact that $\kappa_{i}-\kappa_{0}>0$ for $i>q$ , it is easy to see that

\int_{\sigma}e^{-2\lambda(t)^{-1}\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}\lambda_{\sigma}(dw)=O(\lambda(t)^{p-q}).

By (3.3), it follows that

\mu_{t}(U_{t})=O(\lambda(t)^{d-q}|t|^{2\kappa_{0}}),

(3.4)

and hence $\mu_{t}(U_{t})\to 0$ unless $\kappa_{0}=0$ and $q=d$ , which we henceforth assume. Given $\varphi\in C^{0}(\sigma)$ , our goal is now to show

\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,\mu_{t}\to\left(\int_{\sigma^{\prime}}\varphi b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}\right).\left(\int_{Y^{\prime}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)\right).

(3.5)

Let us first express both sides of (3.5) in logarithmic polar coordinates. We start by the left-hand side. Set $f:=\chi e^{2g}\in C^{0}({\mathcal{U}})$ . By (3.3) and Lemma 3.8 we have

(2\pi)^{d-p}\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,\mu_{t}\\ =\lambda(t)^{d-p}\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}\varphi(w)e^{-2\lambda(t)^{-1}a^{\prime\prime}\cdot w^{\prime\prime}}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int f\,\rho_{t,w,y}\\ =\int\limits_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})}H_{t}(w^{\prime},x^{\prime\prime})b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}(dw^{\prime})\otimes|dx^{\prime\prime}|\otimes|dy|^{2}\int f\,\rho_{t,w^{\prime},x^{\prime\prime},y},

(3.6)

where

H_{t}(w^{\prime},x^{\prime\prime})=\mathbf{1}_{\sigma_{t}}\varphi(Q_{t}(w^{\prime},x^{\prime\prime}))e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime})^{d},

and $\rho_{t,w^{\prime},x^{\prime\prime},y}$ is the same measure as $\rho_{t,w,y}$ via the identification $Q_{t}(w^{\prime},x^{\prime\prime})=w$ .

Note that $\lim_{t\to 0}Q_{t}(w^{\prime},x^{\prime\prime})=(w^{\prime},0)$ , so

\lim_{t\to 0}H_{t}(w^{\prime},x^{\prime\prime})=\mathbf{1}_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}}\varphi(w^{\prime},0)e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}.

Consider the tropicalization map

S\colon Y^{\prime}\cap{\mathcal{U}}\to{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})

given by $S=(-\log|z_{d+1}|,\dots,-\log|z_{p}|,y)$ . Each fiber $S^{-1}(x^{\prime\prime},y)$ is a torsor for the compact torus $({\mathbb{R}}/{\mathbb{Z}})^{p-d}$ and hence carries a unique invariant probability measure $\rho_{x^{\prime\prime},y}$ . As $t\to 0$ , the probability measure $\rho_{t,w^{\prime},x^{\prime\prime},y}$ converges weakly to $\rho_{x^{\prime\prime},y}$ for any $w^{\prime}\in\sigma^{\prime}$ .

By dominated convergence it follows that

\lim_{t\to 0}(2\pi)^{d-p}\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,d\mu_{t}\\ =\int_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})}\varphi(w^{\prime},0)e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}(dw^{\prime})\otimes|dx^{\prime\prime}|\otimes|dy|^{2}\int f\,\rho_{x^{\prime\prime},y}\\ =\left(\int_{\sigma^{\prime}}\varphi b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}\right)\left(\int_{{\mathbb{R}}_{+}^{p-d}}e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}|dx^{\prime\prime}|\int_{Y\cap{\mathcal{U}}}|dy|^{2}\int f\,\rho_{x^{\prime\prime},y}\right).

(3.7)

It only remains to compare the second factor of (3.7) to the second factor in (3.5). To this end, we again use logarithmic polar coordinates. We have

\chi\operatorname{Res}_{Y^{\prime}}(\psi)=f\prod_{i=d+1}^{p}|z_{i}|^{2a_{i}-2}|dz^{\prime\prime}|^{2}\otimes|dy|^{2}.

(3.8)

For $d<j\leq p$ , set $z_{j}=e^{-x_{j}+2\pi i\theta_{j}}$ with $x^{\prime\prime}\in{\mathbb{R}}_{+}^{p-d}$ and $\theta^{\prime\prime}\in({\mathbb{R}}/{\mathbb{Z}})^{p-d}$ . Then

\int_{Y^{\prime}\cap{\mathcal{U}}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)=(2\pi)^{p-d}\int_{{\mathbb{R}}_{+}^{p-d}}e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}|dx^{\prime\prime}|\int_{Y\cap{\mathcal{U}}}|dy|^{2}\int f\,\rho_{x^{\prime\prime},y},

(3.9)

which completes the proof of (3.5), and hence of Theorem 3.4.

[015R]

4. The limit hybrid model

Let $\pi\colon X\to{\mathbb{D}}^{*}$ be a proper submersion, with $X$ a connected complex manifold. Assume that $\pi$ is meromorphic over $0\in{\mathbb{D}}$ in the sense that it admits a model $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ , that is, ${\mathcal{X}}$ is a normal complex space, $\pi$ is a flat proper map, and we are given an isomorphism $X\simeq\pi^{-1}({\mathbb{D}}^{*})$ over ${\mathbb{D}}^{*}$ . We say that ${\mathcal{X}}$ is an snc model (of $X$ ) if ${\mathcal{X}}$ is smooth and the Cartier divisor ${\mathcal{X}}_{0}:=\pi^{-1}(0)$ has simple normal crossing support. Such models always exist by Hironaka’s theorem.

To any snc model ${\mathcal{X}}$ we can associate as in §2 a hybrid space ${\mathcal{X}}^{\mathrm{hyb}}$ , that of course depends on ${\mathcal{X}}$ . In this section we define a canonical hybrid space $X^{\mathrm{hyb}}$ , obtained as the inverse limit of the ${\mathcal{X}}^{\mathrm{hyb}}$ , that does not have this defect. We then prove Theorem B from the introduction.

In the projective case, we show that the both the central fiber $X^{\mathrm{hyb}}_{0}$ and the closed subset $X^{\mathrm{hyb}}_{\overline{{\mathbb{D}}_{r}}}$ can be viewed as analytifications in the sense of Berkovich.

[015S]

4.1. Snc models and simple blowups

Given any two models ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ of $X$ , there is a canonical bimeromorphic map ${\mathcal{X}}^{\prime}\dashrightarrow{\mathcal{X}}$ , and we say that ${\mathcal{X}}^{\prime}$ dominates ${\mathcal{X}}$ if this map is a morphism. Any two models ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ is dominated by a third, for instance the normalization of the graph of ${\mathcal{X}}\dashrightarrow{\mathcal{X}}^{\prime}$ . By Hironaka’s theorem, any model is dominated by an snc model. Thus the set of models forms a directed set, in which snc models are cofinal.

Suppose ${\mathcal{X}}$ is an snc model and that ${\mathcal{X}}^{\prime}$ is another model that dominates ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . As in [KS06, Definition 22] we say that $\rho$ is a simple blowup if it is a blowup along a smooth, connected complex subspace $W$ of ${\mathcal{X}}_{0}$ meeting transversely (or not at all) every irreducible component of ${\mathcal{X}}_{0}$ that does not contain it. In this case, ${\mathcal{X}}^{\prime}$ is also an snc model.

[015T]

Lemma 4.1.

Suppose ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ are snc models and that ${\mathcal{X}}^{\prime}$ dominates ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . Then there exists a third snc model ${\mathcal{X}}^{\prime\prime}$ dominating ${\mathcal{X}}^{\prime}$ , such that the induced map ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is a composition of simple blowups.

We are grateful to Bernard Teissier for help with the following argument.

[015U]

Proof.

By Hironaka’s version of the Chow theorem (in turn a consequence of the flattening theorem), see [Hir75, Corollary 2], there exists a complex manifold ${\mathcal{X}}^{\prime\prime}$ and a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ such that ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$ . Since ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is an isomorphism above $X$ , the construction in [Hir75] further guarantees that ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is an isomorphism above $X$ . Indeed, the proof proceeds by blowing up well-chosen smooth centers contained in the non-flat locus of ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ , see Définition 4.4.3 (2) in loc. cit.

We may therefore assume that ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ itself is projective, and more precisely the blowup of an ideal $I$ cosupported on ${\mathcal{X}}_{0}$ . By the principalization theorem for ideals, there exists a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ that is a composition of simple blowups, such that the pullback of $I$ to ${\mathcal{X}}^{\prime\prime}$ is a principal ideal, see [Kol07, Theorem 3.45] or [Wło09, Theorem 2.0.3]. In particular, ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$ . ∎

[015V]

4.2. Induced maps between dual complexes

Suppose ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}$ are snc models with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . There is then an integral affine map

r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon\Delta({\mathcal{X}}^{\prime})\to\Delta({\mathcal{X}}),

defined as follows. Consider any simplex $\sigma^{\prime}$ of $\Delta({\mathcal{X}}^{\prime})$ and let $Y^{\prime}$ be the corresponding stratum. There exists a unique minimal stratum $Y$ of ${\mathcal{X}}_{0}$ such that $\rho(Y^{\prime})\subset Y$ . Let $\sigma=\sigma_{Y}$ be the corresponding simplex. Let $E_{i}$ , $0\leq i\leq p$ (resp. $E^{\prime}_{j}$ , $0\leq j\leq p^{\prime}$ ) be the irreducible components of ${\mathcal{X}}_{0}$ cutting out $Y$ (resp. $Y^{\prime}$ ). Then

\rho^{*}E_{i}=\sum_{j=0}^{p^{\prime}}a_{ij}E^{\prime}_{j},

for $0\leq i\leq p$ , where $a_{ij}\in{\mathbb{Z}}_{>0}$ .

We can realize the simplex $\sigma$ (resp. $\sigma^{\prime}$ ) as the subset $\{\sum_{i=0}^{p}b_{i}w_{i}=1\}\subset{\mathbb{R}}_{+}^{p+1}$ (resp. $\{\sum_{j=0}^{p^{\prime}}b^{\prime}_{j}w^{\prime}_{j}=1\}\subset{\mathbb{R}}_{+}^{p^{\prime}+1}$ ), where $b_{i}$ (resp. $b^{\prime}_{j}$ ) is the multiplicity of $E_{i}$ in ${\mathcal{X}}_{0}$ (resp. of $E^{\prime}_{j}$ in ${\mathcal{X}}^{\prime}_{0}$ ). The restriction of $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ to $\sigma^{\prime}$ is then given by

w_{i}=\sum_{j=0}^{p^{\prime}}a_{ij}w^{\prime}_{j}.

(4.1)

for $0\leq i\leq p$ . It is clear that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ defines a continuous, integral affine map from $\Delta({\mathcal{X}}^{\prime})$ to $\Delta({\mathcal{X}})$ . Further, if ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}^{\prime\prime}$ are snc models with ${\mathcal{X}}^{\prime\prime}$ dominating ${\mathcal{X}}^{\prime}$ , and ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ , then $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}=r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ .

In general, it may happen that $\rho(Y^{\prime})$ is a strict subvariety of $Y$ , and the linear map defining $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}|_{\sigma^{\prime}}$ could fail to be injective or surjective.

[015W]

Definition 4.2.

With notation as above, we say that $\sigma^{\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ if the restriction $\rho|_{Y^{\prime}}\colon Y^{\prime}\to Y$ is a bimeromorphic morphism and the ${\mathbb{Q}}$ -linear map defining $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}|_{\sigma^{\prime}}$ is an isomorphism. In this case, $\sigma^{\prime}$ and $\sigma$ have the same dimension, and $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ maps $\sigma^{\prime}$ homeomorphically onto a ${\mathbb{Z}}$ -subsimplex of $\sigma$ of the same dimension.

Denote by $A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ the union of all simplices in $\Delta({\mathcal{X}}^{\prime})$ that are active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ . Our goal in this subsection is to prove the following result.

[015X]

Proposition 4.3.

Let ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ be snc models, with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ . Then $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ maps $A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ homeomorphically onto $\Delta({\mathcal{X}})$ .

[015Y]

Corollary 4.4.

The images under $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ of the active simplices in $\Delta({\mathcal{X}}^{\prime})$ form a simplicial ${\mathbb{Z}}$ -subdivision of $\Delta({\mathcal{X}})$ . As a consequence, there exists a unique, ${\mathbb{Z}}$ -PA map $i_{{\mathcal{X}}^{\prime}{\mathcal{X}}}\colon\Delta({\mathcal{X}})\to\Delta({\mathcal{X}}^{\prime})$ such that $i_{{\mathcal{X}}^{\prime}{\mathcal{X}}}(\Delta({\mathcal{X}}))=A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ and $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ i_{{\mathcal{X}}^{\prime}{\mathcal{X}}}=\operatorname{id}$ .

When $\pi$ , $\pi^{\prime}$ and $\rho$ are projective, one can prove Proposition 4.3 using the algebraic tool of valuations. Here we follow an ad hoc approach, based on Lemma 4.1.

[015Z]

Lemma 4.5.

Suppose ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}^{\prime\prime}$ are snc models, with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime\prime}$ dominating ${\mathcal{X}}^{\prime}$ . Let $\sigma^{\prime\prime}$ be a simplex of $\Delta({\mathcal{X}}^{\prime\prime})$ , and let $\sigma^{\prime}$ be the smallest simplex of $\Delta({\mathcal{X}}^{\prime})$ containing $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}(\sigma^{\prime\prime})$ . Then $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ iff $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $\sigma^{\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ . As a consequence, $A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\cap r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}^{-1}(A_{{\mathcal{X}}{\mathcal{X}}^{\prime}})$ .

[0160]

Proof.

To ease notation, set $r^{\prime}:=r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $r:=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ . Let $\sigma$ be the smallest simplex of $\Delta({\mathcal{X}})$ containing $r(\sigma^{\prime})$ . Write $Y$ , $Y^{\prime}$ and $Y^{\prime\prime}$ for the strata of ${\mathcal{X}}_{0}$ , ${\mathcal{X}}^{\prime}_{0}$ and ${\mathcal{X}}^{\prime\prime}_{0}$ corresponding to $\sigma$ , $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ , respectively. The restrictions $r^{\prime}|_{\sigma^{\prime\prime}}\colon\sigma^{\prime\prime}\to\sigma^{\prime}$ and $r_{\sigma^{\prime}}\colon\sigma^{\prime}\to\sigma$ are given by ${\mathbb{Q}}$ -linear maps, and we have induced morphisms $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ .

First suppose that $\sigma^{\prime\prime}$ is active for $r^{\prime}$ and $\sigma^{\prime}$ is active for $r$ . Then $r^{\prime}|_{\sigma^{\prime\prime}}$ and $r|_{\sigma^{\prime}}$ are given by ${\mathbb{Q}}$ -linear isomorphisms; hence so is the composition $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}|_{\sigma^{\prime\prime}}$ . Similarly, the maps $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ are bimeromorphic morphisms; hence so is the composition $Y^{\prime\prime}\to Y$ . It follows that $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ .

Conversely, suppose $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ . Since the map $Y^{\prime\prime}\to Y$ is a bimeromorphic morphism, the map $Y^{\prime\prime}\to Y^{\prime}$ (resp. $Y^{\prime}\to Y$ ) must be injective (resp. surjective). In particular, $\dim Y^{\prime\prime}\leq\dim Y^{\prime}$ and $\dim Y\leq\dim Y^{\prime}$ . Similarly, since the ${\mathbb{Q}}$ -linear map defining $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}|_{\sigma^{\prime\prime}}=r|_{\sigma^{\prime}}\circ r^{\prime}|_{\sigma^{\prime\prime}}$ is an isomorphism, the ${\mathbb{Q}}$ -linear map defining $r^{\prime}|_{\sigma^{\prime\prime}}$ (resp. $r|_{\sigma^{\prime}}$ ) must be injective (resp. surjective). In particular, $\dim\sigma^{\prime\prime}\leq\dim\sigma^{\prime}$ and $\dim\sigma\leq\dim\sigma^{\prime}$ . Now

\dim Y^{\prime\prime}+\dim\sigma^{\prime\prime}=\dim Y^{\prime}+\dim\sigma^{\prime}=\dim Y+\dim\sigma=n-1,

so we infer that $\dim Y^{\prime\prime}=\dim Y^{\prime}=\dim Y$ and $\dim\sigma=\dim\sigma^{\prime}=\dim\sigma^{\prime\prime}$ . This further implies that the maps $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ are bimeromorphic morphisms, and that the ${\mathbb{Q}}$ -linear maps defining $r^{\prime}|_{\sigma^{\prime\prime}}$ and $r|_{\sigma^{\prime}}$ are isomorphisms. Hence $\sigma^{\prime\prime}$ and $\sigma^{\prime}$ are active for $r^{\prime}$ and $r$ , respectively. ∎

[0161]

Lemma 4.6.

(a)

If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is surjective, then so is $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ .
(b)

If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is injective and $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}}^{\prime})$ is surjective, then $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is injective.
(c)

If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ and $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}}^{\prime})$ are both surjective, then so is $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ .
(d)

If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ and $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}}^{\prime})$ are both injective, then so is $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ .

[0162]

Proof.

This is formal consequence of the relations $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\cap r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}^{-1}(A_{{\mathcal{X}}{\mathcal{X}}^{\prime}})$ . For example, let us prove (a). Pick any point $w\in\Delta({\mathcal{X}})$ . The assumption implies that we can find $w^{\prime\prime}\in A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ with $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}(w^{\prime\prime})=w$ . Then $w^{\prime}:=r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}(w^{\prime\prime})\in A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ and $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}(w^{\prime})=w$ . Thus (a) holds. The proofs of (b)–(d) are similar and left to the reader. ∎

[0163]

Lemma 4.7.

The assertions of Proposition 4.3 hold when $\rho$ is a simple blowup.

[0164]

Proof.

This is well known (see e.g. [KS06, p.381]) but we supply a proof for the convenience of the reader. To simplify notation, we set $r:=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ , $A:=A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ , $\Delta:=\Delta({\mathcal{X}})$ and $\Delta^{\prime}:=\Delta({\mathcal{X}}^{\prime})$ .

Let $W$ be the center of the blowup $\rho$ , and $Z$ the smallest stratum of ${\mathcal{X}}_{0}$ containing $W$ . Let $E_{i}$ , $i\in I$ be the irreducible components of ${\mathcal{X}}_{0}$ , $J\subset I$ the subset such that $Z$ is an component of $E_{J}$ , and $\sigma_{Z}$ the simplex defined by $Z$ . Let $E^{\prime}_{i}$ , $i\in I$ be the strict transform of $E_{i}$ to ${\mathcal{X}}^{\prime}$ . Finally, let $E^{\prime}$ be the exceptional divisor of $\rho$ . It corresponds to a vertex $v^{\prime}=v^{\prime}_{E^{\prime}}$ of $\Delta^{\prime}$ .

First assume $W\subsetneq Z$ . In this case, $\Delta^{\prime}$ is obtained from $\Delta$ by “raising a tent over the simplex $\sigma_{Z}$ ”. Let us be more precise. Consider a simplex $\sigma$ of $\Delta$ , corresponding to a stratum $Y$ of ${\mathcal{X}}_{0}$ . By the definition of a simple blowup, $W$ meets every irreducible component of ${\mathcal{X}}_{0}$ transversely (if at all). It follows that $Y$ cannot be contained in $W$ , so $\rho$ is a biholomorphism above a general point of $Y$ . Thus the strict transform $Y^{\prime}$ of $Y$ defines a stratum of ${\mathcal{X}}^{\prime}_{0}$ as well as a simplex $\sigma^{\prime}$ of $\Delta^{\prime}$ , whose vertices correspond to the strict transforms of the vertices of $\sigma$ . In this case, $r$ maps $\sigma^{\prime}$ onto $\sigma$ , and $\rho\colon Y^{\prime}\to Y$ is a bimeromorphic morphism, so $\sigma^{\prime}$ is active for $r$ .

This proves that $r\colon A\to\Delta$ is surjective. To prove injectivity, consider a stratum $Y^{\prime}$ of ${\mathcal{X}}^{\prime}_{0}$ , with corresponding simplex $\sigma^{\prime}$ of $\Delta^{\prime}$ . If $Y^{\prime}$ is not contained in $E^{\prime}$ , then $\rho$ is a biholomorphism at the general point of $Y^{\prime}$ , $Y:=\rho(Y^{\prime})$ is a stratum of ${\mathcal{X}}_{0}$ of the same dimension as $Y^{\prime}$ , and $Y^{\prime}$ is the strict transform of $Y$ . Thus we are in the situation above. On the other hand, if $Y^{\prime}$ is contained in $E^{\prime}$ , then there exist irreducible components $E_{i}$ , $i\in J$ of ${\mathcal{X}}_{0}$ , having strict transforms $E^{\prime}_{i}$ , $i\in J$ , such that $\sigma^{\prime}$ has $v^{\prime}$ and $v^{\prime}_{i}$ , $i\in J$ as vertices. Since $W$ is not a stratum of ${\mathcal{X}}_{0}$ , the smallest stratum $Y$ containing $\rho(Y^{\prime})$ is cut out by $E_{i}$ , $i\in J$ . It follows that $r$ maps the simplex $\sigma^{\prime}$ onto the lower-dimensional simplex $\sigma$ , so $\sigma^{\prime}$ is not active for $r$ . Hence $r\colon A\to\Delta$ is injective.

Now assume $W=Z$ is stratum of ${\mathcal{X}}_{0}$ , defining a simplex $\sigma$ with vertices $v_{i}$ , $i\in J$ . In this case, $\Delta^{\prime}$ is obtained from $\Delta$ by a barycentric subdivision of the simplex $\sigma_{Z}$ . Again, let us be more precise. The same argument as above shows that if $Y$ is a stratum of ${\mathcal{X}}_{0}$ that is not contained in $W$ , and $Y^{\prime}$ is the strict transform, then the simplex $\sigma^{\prime}_{Y^{\prime}}$ is active for $r$ and $r(\sigma^{\prime}_{Y^{\prime}})=\sigma_{Y}$ . Further, $\sigma^{\prime}_{Y^{\prime}}$ is the unique simplex in ${\mathcal{X}}^{\prime}_{0}$ that is active for $r$ and whose image under $r$ meets the interior of $\sigma_{Y}$ .

It remains to consider strata of ${\mathcal{X}}_{0}$ contained in $Z$ . This becomes a toroidal calculation. Let $Y$ be such a stratum, cut out by $E_{i}$ , $i\in K$ , where $J\subset K$ . Then $\rho^{-1}(Y)$ consists of $|J|$ strata $Y^{\prime}_{i}$ , $i\in J$ , each cut out by $E^{\prime}$ and $E^{\prime}_{j}$ , $j\in K\setminus\{i\}$ . The restriction $\rho|_{Y^{\prime}_{i}}\colon Y^{\prime}_{i}\to Y$ is a bimeromorphic morphism, and the the corresponding simplex $\sigma^{\prime}_{i}$ is active for $r$ and maps homeomorphically onto a simplex contained in $\sigma_{Y}$ . Further, these simplices $r(\sigma^{\prime}_{i})$ have disjoint interiors and cover $\sigma_{Y}$ . Finally, if $Y^{\prime}$ is a stratum of ${\mathcal{X}}^{\prime}_{0}$ contained in $E=\rho^{-1}(Z)$ , then $Y=\rho(Y^{\prime})$ is a stratum contained in $Z$ , hence $Y^{\prime}=Y^{\prime}_{i}$ is one of the strata above. This completes the proof. ∎

[0165]

Proof of Proposition 4.3.

Since $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is continuous, $A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is compact, and $\Delta({\mathcal{X}})$ is Hausdorff, it suffices to prove that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is bijective.

Using Lemma 4.6 (c)–(d) and Lemma 4.7, one proves by induction on the number of blowups that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is bijective when ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is a composition of simple blowups.

Now consider the general case. Using Lemma 4.1 we find an snc model ${\mathcal{X}}^{\prime\prime}$ dominating both ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ and such that the morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is a composition of simple blowups. Thus $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is bijective. By Lemma 4.6 (a), it follows that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is surjective. Since ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ were arbitrary snc models with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ , it follows that $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is also surjective. It now follows from Lemma 4.6 (b) that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is injective, which completes the proof. ∎

[0166]

4.3. Induced maps between hybrid spaces

To any snc model ${\mathcal{X}}$ of $X$ we associated in §2 a hybrid space ${\mathcal{X}}^{\mathrm{hyb}}=X\coprod\Delta({\mathcal{X}})$ . Let us briefly recall the topology on ${\mathcal{X}}^{\mathrm{hyb}}$ in the present context. Extend $\pi\colon X\to{\mathbb{D}}^{*}$ to a map

\pi\colon{\mathcal{X}}^{\mathrm{hyb}}\to{\mathbb{D}}

by declaring $\pi=0$ on $\Delta({\mathcal{X}})$ . For $0<r\leq 1$ , define ${\mathcal{X}}_{{\mathbb{D}}_{r}}:=\pi^{-1}({\mathbb{D}}_{r})$ . The construction in §2 yields, for $0<r\ll 1$ , a tropicalization map

\log_{\mathcal{X}}\colon{\mathcal{X}}_{{\mathbb{D}}_{r}}\to\Delta({\mathcal{X}})

uniquely defined up to an additive error term of size $O((\log|t|)^{-1})$ . The topology on ${\mathcal{X}}^{\mathrm{hyb}}$ is the coarsest one such that $\log_{\mathcal{X}}$ is continuous, $\pi$ is continuous, and the inclusion $X\subset{\mathcal{X}}^{\mathrm{hyb}}$ is an open embedding.

Now suppose ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}$ are snc models, with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . Define the map $\rho^{\mathrm{hyb}}\colon{\mathcal{X}}^{\prime\mathrm{hyb}}\to{\mathcal{X}}^{\mathrm{hyb}}$ to be the identity on $X\subset{\mathcal{X}}^{\prime}$ and equal to the map $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ on $\Delta({\mathcal{X}}^{\prime})$ defined in §4.2.

[0167]

Proposition 4.8.

The map $\rho^{\mathrm{hyb}}$ is continuous and surjective. Further, we have

\operatorname{Log}_{\mathcal{X}}\circ\rho^{\mathrm{hyb}}=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ\operatorname{Log}_{{\mathcal{X}}^{\prime}}+O((\log|t|)^{-1})

(4.2)

on $X_{{\mathbb{D}}^{*}_{r}}$ for $0<r\ll 1$ .

[0168]

Proof.

Surjectivity follows from Proposition 4.3, and continuity from (4.2) after unwinding the definitions. It remains to establish (4.2). Consider any point $\xi^{\prime}\in{\mathcal{X}}_{0}$ and set $\xi=\pi(\xi^{\prime})$ . We can find adapted coordinate charts $({\mathcal{U}}^{\prime},z^{\prime})$ at $\xi^{\prime}$ on ${\mathcal{X}}^{\prime}$ and $({\mathcal{U}},z)$ at $\xi$ on ${\mathcal{X}}$ such that $\rho({\mathcal{U}}^{\prime})\subset{\mathcal{U}}$ and such that the following holds: $t=\prod_{i=0}^{p}z_{i}^{b_{i}}$ in ${\mathcal{U}}$ , $t=\prod_{j=0}^{p^{\prime}}(z^{\prime}_{j})^{b^{\prime}_{j}}$ in ${\mathcal{U}}^{\prime}$ and $\rho^{*}z_{i}=\prod_{j}(z^{\prime}_{j})^{a_{ij}}$ . Since the map $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is given by (4.1), the result now follows from Proposition 2.1. ∎

[0169]

4.4. The limit hybrid space

Proposition 4.8 allows us to introduce

[016A]

Definition 4.9.

The hybrid space associated to $X$ is the topological space

X^{\mathrm{hyb}}:=\varprojlim_{\mathcal{X}}{\mathcal{X}}^{\mathrm{hyb}},

where ${\mathcal{X}}$ runs over all snc models of $X$ .

Here $X^{\mathrm{hyb}}$ is equipped with the inverse limit topology. The maps $\pi\colon{\mathcal{X}}^{\mathrm{hyb}}\to{\mathbb{D}}$ define a continuous and proper map

\pi\colon X^{\mathrm{hyb}}\to{\mathbb{D}}

We can identify $X$ with the open subset $\pi^{-1}({\mathbb{D}}^{*})$ . Similarly, the compact subset $X^{\mathrm{hyb}}_{0}:=\pi^{-1}(0)$ can be identified with $\varprojlim_{\mathcal{X}}\Delta({\mathcal{X}})$ . For every snc model ${\mathcal{X}}$ we have, by the definition of the inverse limit, a continuous proper map $r_{\mathcal{X}}\colon X^{\mathrm{hyb}}\to{\mathcal{X}}^{\mathrm{hyb}}$ . We also have an embedding $i_{\mathcal{X}}\colon\Delta({\mathcal{X}})\to X^{\mathrm{hyb}}$ of $\Delta({\mathcal{X}})$ onto a closed subset of $X^{\mathrm{hyb}}_{0}$ . It satisfies $r_{\mathcal{X}}\circ i_{\mathcal{X}}=\operatorname{id}$ on $\Delta({\mathcal{X}})$ .

[016B]

Remark 4.10.

It is not clear how to define a map $\operatorname{Log}\colon X^{\mathrm{hyb}}\to\varprojlim_{\mathcal{X}}\Delta({\mathcal{X}})$ , since each tropicalization map $\operatorname{Log}_{\mathcal{X}}$ is only defined on $X_{{\mathbb{D}}^{*}(r)}$ , where $r=r_{\mathcal{X}}$ depends on ${\mathcal{X}}$ . See §4.6 for a substitute in the projective case.

[016C]

4.5. Convergence of measures

For any locally compact Hausdorff space $Z$ , let ${\mathcal{M}}(Z)$ denote the space of signed Radon measures on $Z$ . By definition we have $X^{\mathrm{hyb}}=\varprojlim_{\mathcal{X}}{\mathcal{X}}^{\mathrm{hyb}}$ , and this induces a homeomorphism

{\mathcal{M}}(X^{\mathrm{hyb}})\overset{\sim}{\to}\varprojlim_{\mathcal{X}}{\mathcal{M}}({\mathcal{X}}^{\mathrm{hyb}}).

Theorem 3.4 now implies the following result, which is equivalent to Corollary B in the introduction.

[016D]

Corollary 4.11.

Let $\pi\colon X\to{\mathbb{D}}^{*}$ be a proper submersion that is meromorphic at $0\in{\mathbb{D}}$ , and let $\psi$ be a continuous metric on $K_{X/{\mathbb{D}}^{*}}$ with analytic singularities. Then there exists a positive measure $\mu_{0}$ on $X^{\mathrm{hyb}}_{0}$ such that if $\mu_{t}:=\frac{\lambda(t)^{d}e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi)^{d}}$ , then $\lim_{t\to 0}\mu_{t}=\mu_{0}$ in the sense of weak convergence of measures on $X^{\mathrm{hyb}}$ . Further, there exists a snc model ${\mathcal{X}}\to{\mathbb{D}}$ and a ${\mathbb{Q}}$ -line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ such that $\psi$ extends to a smooth metric on ${\mathcal{L}}$ , and

\mu_{0}:=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}(\psi)\right)b_{\sigma}^{-1}\lambda_{\sigma},

[016E]

4.6. The projective case

Now consider the case when $X\to{\mathbb{D}}^{*}$ is projective.⁵⁵ 5 In the projective case, the existence of the spaces ${\mathcal{X}}^{\mathrm{hyb}}$ and $X^{\mathrm{hyb}}$ was observed by Kontsevich and Soibelman, see [KS06, p.383]. As we now explain, we can then view $X^{\mathrm{hyb}}$ and its central fiber as analytic spaces.

The projectivity assumption means that $X$ can be viewed as a smooth subspace ${\mathbb{P}}^{N}\times{\mathbb{D}}^{*}$ , defined by homogeneous polynomials with coefficients that are holomorphic functions on ${\mathbb{D}}^{*}$ and meromorphic at $0\in{\mathbb{D}}$ .

We can view these coefficients as complex formal Laurent series, that is, elements of the field $K:={\mathbb{C}}(\!({t})\!)$ . Given $r\in(0,1)$ , this field admits a natural non-Archimedean absolute value that is trivial on ${\mathbb{C}}^{*}$ and normalized by $|{t}|=r$ . In other words, we have $|\sum_{j}a_{j}t^{j}|=r^{\min\{j\mid a_{j}\neq 0\}}$ .

Further, the equations defining $X$ now define a smooth projective variety $X_{K}$ over the field $K$ . To this variety we can associate a non-Archimedean space $X_{K}^{\mathrm{an}}$ , namely the Berkovich analytification of $X_{K}$ with respect to non-Archimedean norm on $K$ . This is a connected and locally connected compact (Hausdorff) space.

We claim that $X^{\mathrm{hyb}}_{0}$ is homeomorphic on $X_{K}^{\mathrm{an}}$ . To see this, we note that, for the same reasons as above, every projective snc model ${\mathcal{X}}\to{\mathbb{D}}$ of $X$ defines a projective snc model ${\mathcal{X}}_{R}$ of $X_{K}$ over the valuation ring $R={\mathbb{C}}[\![{t}]\!]$ of $K$ . Further, the dual complex $\Delta({\mathcal{X}})$ of ${\mathcal{X}}$ can be identified with the dual complex $\Delta({\mathcal{X}}_{R})$ of ${\mathcal{X}}_{R}$ . Now, there exists a canonical retraction map $r_{\mathcal{X}}\colon X_{K}\to\Delta({\mathcal{X}}_{K})$ , and we have

X_{K}^{\mathrm{an}}\overset{\sim}{\to}\varprojlim_{{\mathcal{X}}\ \text{projective snc}}\Delta({\mathcal{X}}).

(4.3)

This was announced in [KS06, Theorem 10, p.383]; see e.g. [BFJ16, Corollary 3.2] for details. On the other hand, Lemma 4.1 implies that in $X^{\mathrm{hyb}}_{0}=\varprojlim_{\mathcal{X}}\Delta({\mathcal{X}})$ , we may take the limit over projective snc models. This implies that $X^{\mathrm{hyb}}_{0}\simeq X_{K}^{\mathrm{an}}$ .

Next we analyze the space $X^{\mathrm{hyb}}$ itself, using Appendix A. Fix $0<r<1$ and consider the Banach ring

A_{r}:=\left\{f=\sum_{\alpha\in{\mathbb{Z}}}c_{\alpha}{t}^{\alpha}\in{\mathbb{C}}(\!({t})\!)\ \bigg|\ \|f\|_{\mathrm{hyb}}:=\sum_{\alpha\in{\mathbb{Z}}}\|c_{\alpha}\|_{\mathrm{hyb}}r^{\alpha}<+\infty\right\},

where $\|\cdot\|_{\mathrm{hyb}}$ is the maximum of the usual norm and the trivial norm on ${\mathbb{C}}$ . The Berkovich spectrum ${\mathcal{M}}(A_{r})$ of $A_{r}$ is homeomorphic to $\overline{{\mathbb{D}}}_{r}$ .

Every function that is holomorphic on ${\mathbb{D}}^{*}$ and meromorphic at $0\in{\mathbb{D}}$ defines an element of $A_{r}$ . Hence we can define the base change $X_{A_{r}}\subset{\mathbb{P}}^{N}_{A_{r}}$ using the same homogeneous equations as above. Then $X_{A_{r}}$ is a scheme of finite type over $A_{r}$ , so its analytification $X_{A_{r}}^{\mathrm{An}}$ is a compact Hausdorff space with a continuous map $\pi_{r}$ onto $(\operatorname{Spec}A_{r})^{\mathrm{An}}={\mathcal{M}}(A_{r})\simeq\overline{{\mathbb{D}}}_{r}$ . (In Appendix A.6, this analytification is denoted by $X^{\mathrm{hyb}}$ , but here we use $X_{A_{r}}^{\mathrm{An}}$ for clarity.) We have a homeomorphism

\tau\colon\pi_{r}^{-1}(\overline{{\mathbb{D}}}^{*}_{r})\overset{\sim}{\to}X_{\overline{{\mathbb{D}}}^{*}_{r}}\overset{\sim}{\to}X^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}^{*}_{r}}

(4.4)

and another homeomorphism

\tau_{0}\colon\pi^{-1}(0)\overset{\sim}{\to}X^{\mathrm{hyb}}_{0}\overset{\sim}{\to}X_{K}^{\mathrm{an}}.

(4.5)

[016F]

Proposition 4.12.

The map $\tau\colon X_{A_{r}}^{\mathrm{An}}\to X^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}_{r}}$ is homeomorphism.

[016G]

Proof.

It follows from (4.4) and (4.5) that $\tau$ is a bijection. Since $X_{A_{r}}^{\mathrm{An}}$ is compact and $X_{\overline{{\mathbb{D}}}_{r}}^{\mathrm{hyb}}$ is Hausdorff, it only remains to prove that $\tau$ is continuous. It suffices to show that the corresponding map $\tau_{\mathcal{X}}\colon X_{A_{r}}^{\mathrm{An}}\to{\mathcal{X}}^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}_{r}}$ is continuous for a given snc model ${\mathcal{X}}$ . For this, in turn, it suffices to show that $\operatorname{Log}_{\mathcal{X}}\circ\tau_{\mathcal{X}}$ is continuous near the central fiber.

Consider a coordinate chart $({\mathcal{U}},z)$ adapted to ${\mathcal{X}}_{0}$ in the sense of §2.2. Let $E_{0},\dots,E_{p}$ be the irreducible components of ${\mathcal{X}}_{0}$ intersecting ${\mathcal{U}}$ . Let $\hat{\mathcal{U}}\subset X_{A_{r}}^{\mathrm{An}}$ be the set of seminorms satisfying $|z_{i}|<1$ for $0\leq i\leq p$ . Then we have

	$\displaystyle\operatorname{Log}_{\mathcal{X}}\circ\tau_{\mathcal{X}}$	$\displaystyle=\left(\frac{\log\|z_{i}\|_{\infty}}{\log\|t\|_{\infty}}\right)_{0\leq i\leq p}+O((\log\|t\|_{\infty})^{-1})$
		$\displaystyle=(\log\|z_{i}\|^{-1})_{0\leq i\leq p}+O((\log\|t\|_{\infty})^{-1})$

on $\hat{\mathcal{U}}\setminus\pi^{-1}(0)$ . Now the function $(\log|z_{i}|^{-1})_{i}$ is continuous on $\hat{\mathcal{U}}$ with values in the simplex $\sigma={\mathbb{R}}_{+}^{p+1}\cap\{\sum_{0}^{p}b_{i}w_{i}=1\}\subset\Delta({\mathcal{X}})$ . This completes the proof, since we can cover a neighborhood of the central fiber in $X_{A_{r}}^{\mathrm{An}}$ with sets of the type $\hat{{\mathcal{U}}}$ . ∎

[016H]

5. Berkovich spaces and skeleta

Our goal in this section and the next is to study the limit measure $\mu_{0}$ appearing in Corollary B in more detail. This measure lives on a Berkovich space and its support has an integral piecewise affine structure.

In this section we undertake a fairly general study of metrics on the canonical bundle of a projective variety defined over a discretely valued field of residue characteristic zero. To such a metric is associated a skeleton, a subset of the underlying Berkovich space. In the setting of Corollary B, the skeleton will be the support of the measure $\mu_{0}$ .

The material here has overlap with [MN15, NX13] and also draws on [Tem14], but we present some details for the convenience of the reader.

Until further notice, $X$ denotes a smooth proper variety over the field $K:=k(\!({t})\!)$ of formal Laurent series with coefficients in an algebraically closed field $k$ of characteristic $0$ . We set $n:=\dim X$ and denote by $X^{\mathrm{an}}$ the Berkovich analytification of $X$ with respect the non-Archimedean absolute value $|\cdot|=r^{\operatorname{ord}_{0}}$ on $K$ , for some fixed $r\in(0,1)$ .

While $X^{\mathrm{an}}$ comes equipped with a structure sheaf, we shall merely consider it as a topological space. Since $X$ is proper, $X^{\mathrm{an}}$ is compact. There is a natural continuous surjective map $X^{\mathrm{an}}\to X$ such that the preimage of a (scheme) point $\xi\in X$ is identified with the set of real-valued valuations⁶⁶ 6 Here we use additive terminology; the multiplicative norm associated to $v$ is $r^{v}$ . $v$ on the residue field of $\xi$ satisfying $v|_{k^{*}}\equiv 0$ and $v({t})=1$ . In particular, the preimage $X^{\operatorname{val}}$ of the generic point of $X$ consists of real-valued valuations of the function field $F(X)$ .

[016I]

5.1. Models

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

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

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

v_{E}:=b_{E}^{-1}\operatorname{ord}_{E}

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

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

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

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

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

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

[016J]

5.2. Model metrics

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

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

\inf_{X^{\mathrm{an}}}\phi_{D}=\min_{E}\phi_{D}(v_{E})

(5.1)

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

[016K]

5.3. Log canonical divisors

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

(i)

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

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

(iii)

the relative log canonical divisor

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

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

[016L]

Example 5.1.

Assume that ${\mathcal{X}}$ is snc, and write as above ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ . Pick a closed point $\xi\in{\mathcal{X}}_{0}$ , and denote by $J=\{0,\dots,p\}\subset I$ the set of components of ${\mathcal{X}}_{0}$ passing through $\xi$ . We may choose a regular system of parameters $z_{0},\dots,z_{n}\in{\mathcal{O}}_{{\mathcal{X}},\xi}$ such that $z_{i}$ is a local equation of $E_{i}$ for $0\leq i\leq p$ , i.e. ${t}=uz_{0}^{b_{0}}\dots z_{p}^{b_{p}}$ for some unit $u\in{\mathcal{O}}^{*}_{{\mathcal{X}},\xi}$ . The logarithmic form

\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}

is then a local generator of $K^{\mathrm{log}}_{{\mathcal{X}}}$ , and induces a local generator

\Omega^{\mathrm{rel}}:=\Omega\otimes(\frac{d{t}}{{t}})^{-1}

of $K^{\mathrm{log}}_{{\mathcal{X}}/S}$ .

[016M]

Remark 5.2.

When ${\mathcal{X}}$ is snc, ${\mathcal{O}}_{\mathcal{X}}(K^{\mathrm{log}}_{{\mathcal{X}}/S})$ coincides with the relative logarithmic dualizing sheaf $\omega_{{\mathcal{X}}^{+}/S^{+}}$ of [NX13, (3.2.2)]. When ${\mathcal{X}}$ is regular, ${\mathcal{O}}_{\mathcal{X}}(K_{\mathcal{X}})$ is described in [dFEM11, Appendix A] as the determinant of the locally free sheaf $\Omega^{\prime}_{{\mathcal{X}}/k}\subset\Omega^{1}_{{\mathcal{X}}/k}$ of special differentials, corresponding to derivations $D$ of ${\mathcal{O}}_{\mathcal{X}}$ such that $D(f)=f^{\prime}({t})d{t}$ for $f\in k[\![{t}]\!]$ .

[016N]

5.4. Log discrepancies

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

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

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

K^{\mathrm{log}}_{{\mathcal{X}}^{\prime}}=\rho^{*}K^{\mathrm{log}}_{\mathcal{X}}+\sum_{E}b_{E}A_{\mathcal{X}}(v_{E})E,

(5.2)

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

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

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

[016P]

Example 5.3.

Assume that $\dim X=1$ , and let ${\mathcal{X}}$ be a dlt model. Each irreducible component $E_{i}$ is then a smooth curve. At a point $\xi\in E_{i}\cap E_{j}$ , $i\neq j$ , ${\mathcal{X}}$ is snc. At a closed point $\xi\in\mathring{E}_{i}$ , ${\mathcal{X}}$ is either regular, or has a cyclic quotient singularity.

[016Q]

Example 5.4.

If ${\mathcal{X}}$ is toroidal, then ${\mathcal{X}}$ is lc, and ${\mathcal{X}}$ is dlt if and only if it is snc. Following [dFKX12, KNX15], we could say that an lc model ${\mathcal{X}}$ is qdlt (for quotient of dlt) if its lc centers are contained in a toroidal open subset ${\mathcal{U}}\subset{\mathcal{X}}$ .

[016R]

Example 5.5.

If ${\mathcal{X}}$ is any model such that ${\mathcal{X}}_{0}$ has klt singularities (and hence is reduced), then ${\mathcal{X}}$ is dlt, by inversion of adjunction.

[016S]

5.5. The skeleton of a dlt model

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

\sigma=\left\{w\in{\mathbb{R}}_{+}^{J}\mid\sum_{i\in J}b_{i}w_{i}=1\right\},

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

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

\operatorname{emb}_{\mathcal{X}}\colon\Delta({\mathcal{X}})\to{\mathcal{X}}^{\mathrm{an}}

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

v(F(X)^{*})=M_{\sigma}(w):=\{f(w)\mid f\in M_{\sigma}\}.

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

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

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

X^{\mathrm{an}}\overset{\sim}{\to}\varprojlim_{\mathcal{X}}\operatorname{Sk}({\mathcal{X}}),

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

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

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

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

\bigcup_{{\mathcal{X}}\ \text{snc}}\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}

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

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

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

[016T]

5.6. From log discrepancies to Temkin’s metric

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

(i)

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

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

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

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

(5.3)

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

[016U]

Proposition 5.6.

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

[016V]

Lemma 5.7.

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

[016W]

Proof.

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

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

[016X]

Proof of Proposition 5.6.

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

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

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

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

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

(5.4)

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

[016Y]

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

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

[016Z]

Definition 5.8.

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

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

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

[0170]

Definition 5.9.

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

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

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

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

[0171]

Proposition 5.10.

Assume that $\psi$ is a model metric on $K_{X}^{\mathrm{an}}$ , determined by a model ${\mathcal{L}}$ of $K_{X}$ on a proper dlt model ${\mathcal{X}}$ of $X$ . Then $\operatorname{Sk}(\psi)\subset\operatorname{Sk}({\mathcal{X}})$ , and $\kappa=A_{X}-\psi$ is affine on each face of $\Delta({\mathcal{X}})$ . In particular,

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

(5.5)

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

[0172]

Proof.

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

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

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

[0173]

5.8. Residual boundaries

The following construction plays a crucial role for the understanding of the limit measure appearing in Corollary B.

Consider a model metric ${\mathcal{L}}$ of $K_{X}$ defined on a proper dlt model ${\mathcal{X}}$ . Following §3.1 we explain how to associate a subklt pair $(Y,B^{\mathcal{L}}_{Y})$ to each stratum $Y$ of ${\mathcal{X}}_{0}$ corresponding to a maximal simplex in $\Delta({\mathcal{L}})$ .

Let us first recall a few facts about adjunction. When ${\mathcal{X}}$ is an snc model, each stratum $Y$ comes with a boundary $B_{Y}:=\sum_{i\notin J_{Y}}E_{i}\cap Y$ . Here $(Y,B_{Y})$ is log smooth, and

K^{\mathrm{log}}_{{\mathcal{X}}/S}\big|_{Y}=K_{(Y,B_{Y})}:=K_{Y}+B_{Y},

(5.6)

the identification being provided by Poincaré residues. When ${\mathcal{X}}$ is merely dlt, each stratum $Y$ is normal, and comes with a canonically defined effective ${\mathbb{Q}}$ -divisor $B_{Y}$ such that $(Y,B_{Y})$ is dlt and still satisfies (5.6) (cf. [Kol13, 4.19]). We have

B_{Y}=\sum_{i\notin J_{Y}}E_{i}\cap Y+B^{\prime}_{Y}

where $B^{\prime}_{Y}$ is an effective ${\mathbb{Q}}$ -divisor supported in the complement of ${\mathcal{X}}_{\mathrm{snc}}$ .

[0174]

Example 5.11.

For each $i$ , $E_{i}\cap({\mathcal{X}}\setminus{\mathcal{X}}_{\mathrm{snc}})$ contains finitely many prime divisors $F_{ik}$ of $E_{i}$ . At the generic point of $F_{ik}$ , ${\mathcal{X}}$ has cyclic quotient singularities, and

B_{E_{i}}=\sum_{j\neq i}E_{j}\cap E_{i}+\sum_{k}\left(1-\frac{1}{m_{ik}}\right)F_{ik}

with $m_{ik}$ the order of the corresponding cyclic groups, cf. [Kol13, 3.36.3].

Now let $\psi$ be a model metric on $K_{X}^{\mathrm{an}}$ , determined by a model ${\mathcal{L}}$ of $K_{X}$ on a proper dlt model ${\mathcal{X}}$ of $X$ . Introduce as before the function $\kappa:=A_{X}-\psi$ on $X^{\mathrm{an}}$ , and note that the ${\mathbb{Q}}$ -Cartier divisor

D:=K^{\mathrm{log}}_{{\mathcal{X}}/S}-{\mathcal{L}}-\kappa_{\min}{\mathcal{X}}_{0}=\sum_{i}(\kappa(v_{E_{i}})-\kappa_{\min})b_{i}E_{i}

is effective.

[0175]

Lemma 5.12.

If $Y$ is a stratum of ${\mathcal{X}}_{0}$ corresponding to a face $\sigma$ of $\Delta({\mathcal{L}})$ , then $Y\not\subset\operatorname{supp}D$ . It follows that the ${\mathbb{Q}}$ -Cartier divisor

B^{\mathcal{L}}_{Y}:=B_{Y}-D|_{Y}

is well-defined, and we have a canonical identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$ as ${\mathbb{Q}}$ -line bundles. Further, if $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$ , then the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt.

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

[0176]

Proof.

The first two points are clear. When $\sigma$ is a maximal face, each $E_{i}$ meeting $Y$ satisfies $\kappa(v_{E_{i}})>\kappa_{\min}$ . As a result, $D|_{Y}$ contains each lc center $E_{i}\cap Y$ of $(Y,B_{Y})$ , which yields the last assertion. ∎

[0177]

5.9. Skeleta and base change

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

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

X^{\prime\mathrm{an}}/G\overset{\sim}{\to}X^{\mathrm{an}}.

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

\phi_{\rho^{*}D}=mp^{*}\phi_{D}.

(5.7)

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

[0178]

Lemma 5.13.

We have $p^{-1}(\operatorname{Sk}({\mathcal{X}}))=\operatorname{Sk}({\mathcal{X}}^{\prime})$ . Further, for each face $\sigma$ of $\Delta({\mathcal{X}})$ , there exist positive integers $e_{\sigma}$ , $f_{\sigma}$ and $g_{\sigma}$ satisfying

e_{\sigma}=\frac{m}{\gcd(m,b_{\sigma})}{\quad\text{and}\quad}f_{\sigma}g_{\sigma}=\gcd(m,b_{\sigma})

and such that the following properties hold: $p^{-1}(\sigma)$ is a union of $g_{\sigma}$ faces $\sigma^{\prime}_{\alpha}$ of $\Delta({\mathcal{X}}^{\prime})$ , and these are permuted by $G$ . For each $\alpha$ :

(a)

$p$ induces a ${\mathbb{Q}}$ -affine isomorphism $\sigma^{\prime}_{\alpha}\overset{\sim}{\to}\sigma$ ;
(b)

$p$ induces a generically finite map $Y_{\sigma^{\prime}_{\alpha}}\to Y_{\sigma}$ , of degree $f_{\sigma}$ ;
(c)

$mp^{*}M_{\sigma}\subset M_{\sigma^{\prime}_{\alpha}}$ , and $[M_{\sigma^{\prime}_{\alpha}}:mp^{*}M_{\sigma}]=e_{\sigma}$ .

Furthermore, we have:

(i)

$M_{\sigma^{\prime}_{\alpha}}=p^{*}\left(mM_{\sigma}+{\mathbb{Z}}1_{\sigma}\right)$ ;
(ii)

$\operatorname{Vol}(\sigma^{\prime}_{\alpha})=m^{\dim\sigma}\operatorname{Vol}(\sigma)$ ;
(iii)

$b_{\sigma^{\prime}_{\alpha}}=b_{\sigma}/\gcd(m,b_{\sigma})$ .

[0179]

Proof.

The proof uses the toroidal theory of [KKMS] together with elementary ramification theory of valuations [ZS75].

Let $\sigma$ be the face of $\Delta({\mathcal{X}})$ corresponding to an irreducible component $Y$ of $E_{0}\cap\dots\cap E_{p}$ . Set $b_{i}=\operatorname{ord}_{E_{i}}({t})$ . With the identification

\sigma=\{w\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i}b_{i}w_{i}=1\},

the integral affine structure $M_{\sigma}$ is given by the lattice ${\mathbb{Z}}^{p+1}$ . Note that $b_{\sigma}=\gcd_{i}b_{i}$ .

Given a closed point $\xi\in\mathring{Y}$ , we can find local coordinates $z_{0},\dots,z_{n}$ in the formal completion $\widehat{\mathcal{O}}_{{\mathcal{X}},\xi}\simeq k[\![z_{0},\dots,z_{n}]\!]$ such that ${t}=\prod_{i=0}^{p}z_{i}^{b_{i}}$ . A toric computation (cf. [KKMS, pp.98–102]) shows that $\xi$ has $\gcd(m,b_{\sigma})$ preimages $\xi^{\prime}_{\alpha}$ in ${\mathcal{X}}^{\prime}_{0}$ , with ${\mathcal{X}}^{\prime}$ formally isomorphic, at each $\xi^{\prime}_{\alpha}$ , to the product of ${\mathbb{A}}_{k}^{n-p}$ with the affine toric $k$ -variety corresponding to the cone ${\mathbb{R}}_{+}^{p+1}\subset{\mathbb{R}}^{p+1}$ with lattice

M^{\prime}:={\mathbb{Z}}^{p+1}+{\mathbb{Z}}\left(\frac{b_{0}}{m},\dots,\frac{b_{p}}{m}\right).

It follows that $p^{-1}(\sigma)$ is the union of the corresponding faces $\sigma^{\prime}_{\alpha}$ of $\Delta({\mathcal{X}}^{\prime})$ , each isomorphic to

\sigma^{\prime}=\left\{w^{\prime}\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i}b_{i}w^{\prime}_{i}=m\right\},

with integral affine structure induced by $M^{\prime}$ . Now $p$ restricts to a homeomorphism $\sigma^{\prime}_{\alpha}\overset{\sim}{\to}\sigma$ given by $w=w^{\prime}/m$ . Thus $M_{\sigma^{\prime}_{\alpha}}=mp^{*}M_{\sigma}+{\mathbb{Z}}1_{\sigma^{\prime}_{\alpha}}$ . This implies (i), and (ii)–(iii) easily follow.

Now note that

[M_{\sigma^{\prime}_{\alpha}}^{\prime}:mp^{*}M_{\sigma}]=[mp^{*}M_{\sigma}+{\mathbb{Z}}1_{\sigma^{\prime}_{\alpha}}:mp^{*}M_{\sigma}]\\ =[{\mathbb{Z}}^{p+1}+{\mathbb{Z}}(\frac{b_{0}}{m},\dots,\frac{b_{p}}{m}):{\mathbb{Z}}^{p+1}]=\frac{m}{\gcd(m,b_{\sigma})}=:e_{\sigma}.

It remains to analyze the degree $f_{\sigma}$ of the restriction $Y_{\sigma^{\prime}_{\alpha}}\to Y_{\sigma}$ . For this we use ramification theory.

The function field $F(X^{\prime})=F(X)({t}^{1/m})$ is a Galois extension of $F(X)$ of degree $m$ , with Galois group $G$ . For any valuation $v^{\prime}\in X^{\prime\operatorname{val}}$ , we have $v^{\prime}|_{F(X)}=mp(v^{\prime})$ .

Let $v\in X^{\mathrm{an}}$ be a valuation corresponding to a point $w\in\sigma$ . Assume $w$ is “general” in the sense that $\dim_{\mathbb{Q}}\sum_{i=0}^{p}{\mathbb{Q}}w_{i}=p$ . The point $w$ has $g_{\sigma}$ preimages $w^{\prime}_{\alpha}$ under $p$ , one in each $\sigma^{\prime}_{\alpha}$ , and the valuations $v^{\prime}_{\alpha}:=m^{-1}w^{\prime}_{\alpha}$ are all the extensions of $v$ to $F(X^{\prime})$ . Let us compute the residue degree and ramification index of these extensions.

The residue fields of $v$ and $v^{\prime}_{\alpha}$ are exactly the function fields of $Y$ and $Y^{\prime}_{\alpha}$ , respectively, so the residue degree of the extension $v^{\prime}_{\alpha}$ of $v$ is equal to $f_{\sigma}$ .

The value group $\Gamma_{v}=v(F(X))$ of $v$ is given by $\Gamma_{v}=\sum_{i=0}^{p}{\mathbb{Z}}w_{i}$ . Similarly, the value group of $v^{\prime}_{\alpha}$ is given by $\Gamma_{v^{\prime}_{\alpha}}=\frac{1}{m}{\mathbb{Z}}+\frac{1}{m}\sum_{i=0}^{p}{\mathbb{Z}}w^{\prime}_{i}=\frac{1}{m}{\mathbb{Z}}+\sum_{i=0}^{p}{\mathbb{Z}}w_{i}$ . It follows that the ramification index of the extension $v^{\prime}_{\alpha}$ of $v$ is given by

[\Gamma_{v^{\prime}_{\alpha}}:\Gamma_{v}]=[\frac{1}{m}{\mathbb{Z}}+\sum_{i=0}^{p}{\mathbb{Z}}w_{i}:\sum_{i=0}^{p}{\mathbb{Z}}w_{i}]=\gcd({\mathbb{Z}}\cap m\sum_{i=0}^{p}{\mathbb{Z}}w_{i})=\frac{m}{\gcd(m,b_{\sigma})}=e_{\sigma}.

By [ZS75, p.77] we now have $e_{\sigma}f_{\sigma}g_{\sigma}=m$ , which completes the proof. ∎

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

[017A]

Lemma 5.14.

Let $\psi$ be a continuous metric on $K_{X}^{\mathrm{an}}$ , $\psi^{\prime}$ the metric on $K_{X^{\prime}}^{\mathrm{an}}\simeq p^{*}K_{X}^{\mathrm{an}}$ corresponding to $p^{*}\psi$ , and set $\kappa^{\prime}:=A_{X^{\prime}}-\psi^{\prime}$ . Then $\kappa^{\prime}=mp^{*}\kappa$ . As a consequence, $\operatorname{Sk}(\psi^{\prime})=p^{-1}\operatorname{Sk}(\psi)$ and $\kappa^{\prime}_{\min}=m\kappa_{\min}$ .

[017B]

Proof.

By [Gub98, Theorem 7.12] (see also [BFJ16, Corollary 2.3]), model metrics are dense in the set of continuous metrics on $K_{X}^{\mathrm{an}}$ . Hence we may assume $\psi$ is a model metric. Using (5.3), it is enough to show that $\kappa^{\prime}(v^{\prime})=m\kappa(p(v^{\prime}))$ for a divisorial valuation $v^{\prime}\in X^{\prime\mathrm{div}}$ . Let ${\mathcal{X}}$ be an snc model with $p(v^{\prime})\in\operatorname{Sk}({\mathcal{X}})$ , and such that $\psi=\phi_{\mathcal{L}}$ for a model ${\mathcal{L}}$ of $K_{X}$ on ${\mathcal{X}}$ . Since the normalized base change ${\mathcal{X}}^{\prime}$ of ${\mathcal{X}}$ is toroidal, we can choose a toroidal modification ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}^{\prime}$ with ${\mathcal{X}}^{\prime\prime}$ snc. The induced morphism $\rho\colon{\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is toroidal; hence it satisfies the log ramification formula

mK^{\mathrm{log}}_{{\mathcal{X}}^{\prime\prime}/S^{\prime}}=\rho^{*}K^{\mathrm{log}}_{{\mathcal{X}}/S}.

By (5.7), we infer $\phi_{K^{\mathrm{log}}_{{\mathcal{X}}^{\prime\prime}/S^{\prime}}}-\psi^{\prime}=p^{*}(\phi_{K^{\mathrm{log}}_{{\mathcal{X}}/S}}-\psi)$ , which gives the desired result since $v^{\prime}\in\operatorname{Sk}({\mathcal{X}}^{\prime\prime})$ , $p(v^{\prime})\in\operatorname{Sk}({\mathcal{X}})$ imply $A_{{\mathcal{X}}^{\prime\prime}}(v^{\prime})=A_{{\mathcal{X}}}(p(v^{\prime}))=0$ . ∎

[017C]

6. Skeletal measures

From now on, we assume that $k={\mathbb{C}}$ , and that $X$ is a smooth projective variety over the non-Archimedean field $K={\mathbb{C}}(\!({t})\!)$ . Our goal is to construct measures of the types appearing in Theorem A and Corollary B.

[017D]

6.1. Residually metrized models

As explained above, to any model ${\mathcal{L}}$ of a line bundle $L$ on $X$ , defined on a proper dlt model ${\mathcal{X}}$ of $X$ , we can associate a skeleton $\operatorname{Sk}({\mathcal{L}})\subset\operatorname{Sk}({\mathcal{X}})\subset X^{\mathrm{an}}$ . To produce a measure on $\operatorname{Sk}({\mathcal{L}})$ we need additional data.

[017E]

Definition 6.1.

Let $L$ be a line bundle on $X$ . A residually metrized model of $L$ is a pair ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ where ${\mathcal{L}}$ is a model of $L$ , determined on a proper dlt model ${\mathcal{X}}$ of $X$ , and $\psi_{0}$ is a continuous Hermitian metric on ${\mathcal{L}}_{0}:={\mathcal{L}}|_{{\mathcal{X}}_{0}}$ , viewed as a holomorphic line bundle over the complex space ${\mathcal{X}}_{0}$ . A residually metrized model metric $\psi^{\#}$ on $L$ is an equivalence class of such pairs, modulo pull-back to a higher model.

[017F]

Example 6.2.

If $L$ is trivial, then any choice of trivialization $s\in H^{0}(X,L)$ defines a residually metrized model metric $\psi^{\#}$ on $L$ , determined on any model ${\mathcal{X}}$ by ${\mathcal{L}}={\mathcal{O}}_{{\mathcal{X}}}$ and $\psi_{0}$ the trivial metric on ${\mathcal{O}}_{{\mathcal{X}}_{0}}$ .

[017G]

6.2. Residual measures

Let ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ be a residually metrized model of $K_{X}$ , determined on a proper dlt model ${\mathcal{X}}$ . If $Y$ is a stratum corresponding to a top-dimensional face of $\Delta({\mathcal{L}})$ , Lemma 5.12 shows that the restriction of $\psi_{0}$ to ${\mathcal{L}}|_{Y}$ induces a Hermitian metric $\psi_{Y}$ on $K_{(Y,B^{\mathcal{L}}_{Y})}:=K_{Y}+B^{\mathcal{L}}_{Y}$ , with $(Y,B^{\mathcal{L}}_{Y})$ subklt. By Lemma 1.1, we may thus introduce:

[017H]

Definition 6.3.

Let $Y$ be a stratum corresponding to a top-dimensional face of $\Delta({\mathcal{L}})$ . The residual measure of ${\mathcal{L}}^{\#}$ on $Y$ is the (finite) positive measure

\operatorname{Res}_{Y}({\mathcal{L}}^{\#}):=\exp\left(2(\psi_{Y}-\phi_{B^{\mathcal{L}}_{Y}})\right).

This definition is of course compatible with one in §3.1, and can be more explicitly described as follows. Let $\xi$ be a (closed) point of $Y\cap{\mathcal{X}}_{\mathrm{snc}}$ , index the irreducible components $E_{0},\dots,E_{p}$ passing through $\xi$ so that $Y$ is a component of $\bigcap_{0\leq i\leq d}E_{i}$ with $d=\dim\Delta({\mathcal{L}})\leq p$ . In the notation of Example 5.1, the Poincaré residue

\operatorname{Res}_{Y}(\Omega)=\left(\frac{dz_{d+1}}{z_{d+1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\right)\bigg|_{Y}

is a generator of $K_{(Y,B_{Y})}=K^{\mathrm{log}}_{{\mathcal{X}}/S}\big|_{Y}$ . Setting $a_{i}:=\kappa(v_{E_{i}})b_{i}\in{\mathbb{Q}}$ , we have

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

and we may thus view

\tau:={t}^{\kappa_{\min}}\prod_{i=0}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}={t}^{\kappa_{\min}}\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}

as a local ${\mathbb{Q}}$ -generator of ${\mathcal{L}}$ . Further, $B^{\mathcal{L}}_{Y}=\sum_{i=d+1}^{p}(1-(a_{i}-\kappa_{\min}b_{i}))E_{i}|_{Y}$ , and $\tau|_{Y}$ corresponds to

\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\operatorname{Res}_{Y}(\Omega)

under the identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$ . We arrive at

	$\displaystyle\operatorname{Res}_{Y}({\mathcal{L}}^{\#})$	$\displaystyle=\frac{\prod_{i=d+1}^{p}\|z_{i}\|^{2(a_{i}-\kappa_{\min}b_{i})}}{\left\|{t}^{\kappa_{\min}}\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}\right\|^{2}_{\psi_{0}}}\left\|\operatorname{Res}_{Y}(\Omega)\right\|^{2}$
		$\displaystyle=\frac{\prod_{i=d+1}^{p}\|z_{i}\|^{2(a_{i}-\kappa_{\min}b_{i}-1)}}{\left\|{t}^{\kappa_{\min}}\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\Omega^{\mathrm{rel}}\right\|^{2}_{\psi_{0}}}\|dz_{d+1}\wedge\dots\wedge dz_{n}\|^{2}.$		(6.1)

[017I]

6.3. Measures on dual complexes

We now define measures associated to residually metrized model metrics.

[017J]

Definition 6.4.

Let ${\mathcal{L}}^{\#}$ be a residually metrized model of $K_{X}$ , determined on a proper dlt model ${\mathcal{X}}$ of $X$ . To ${\mathcal{L}}^{\#}$ we associate a positive measure $\mu_{{\mathcal{L}}^{\#}}$ on $\Delta({\mathcal{L}})\subset\Delta({\mathcal{X}})$ defined by

\mu_{{\mathcal{L}}^{\#}}=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\lambda_{\sigma},

where $\sigma$ runs over the top-dimensional faces of $\Delta({\mathcal{L}})$ .

By Lemma 1.2, we have

\mu_{{\mathcal{L}}^{\#}}(\sigma)=\frac{\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})}{d!\prod_{i\in J}b_{i}}

for each face $\sigma$ corresponding to a component of some $E_{J}$ .

[017K]

6.4. Skeletal mesures on Berkovich spaces

Now consider a residually metrized model metric $\psi^{\#}$ on $K_{X}$ . Pick any representative ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ for $\psi^{\#}$ , where ${\mathcal{L}}$ is a model of $K_{X}$ determined on a proper dlt model ${\mathcal{X}}$ of $X$ , and where $\psi_{0}$ is a continuous metric on ${\mathcal{L}}_{0}:={\mathcal{L}}|_{{\mathcal{X}}_{0}}$ .

[017L]

Definition 6.5.

The skeletal measure $\mu_{\psi^{\#}}$ is the image of the measure $\mu_{{\mathcal{L}}^{\#}}$ under the embedding $\Delta({\mathcal{L}})\hookrightarrow X^{\mathrm{an}}$ . We view it as a positive measure on $X^{\mathrm{an}}$ , supported on the skeleton $\operatorname{Sk}(\psi^{\#}):=\operatorname{Sk}(\phi_{\mathcal{L}})$ .

This definition makes sense, in view of the following result.

[017M]

Lemma 6.6.

The skeletal measure $\mu_{{\mathcal{L}}^{\#}}$ is independent of the choice of representative ${\mathcal{L}}^{\#}$ for $\psi^{\#}$ .

[017N]

Proof.

Let ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ be proper dlt models of $X$ , with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ via a proper birational morphism $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . Let ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ be a residually metrized model of $K_{X}$ consisting of a model ${\mathcal{L}}$ of $K_{X}$ determined on ${\mathcal{X}}$ and a continuous metric $\psi_{0}$ on ${\mathcal{L}}_{0}$ . Set ${\mathcal{L}}^{\prime}=\rho^{*}{\mathcal{L}}$ , $\psi^{\prime}_{0}=\rho^{*}\psi_{0}$ and ${\mathcal{L}}^{\prime\#}=({\mathcal{L}}^{\prime},\psi^{\prime}_{0})$ . We must prove that $\mu_{{\mathcal{L}}^{\prime\#}}=\mu_{{\mathcal{L}}^{\#}}$ .

Let $\sigma^{\prime}$ be a top-dimensional face of $\Delta({\mathcal{L}}^{\prime})$ , $Y^{\prime}$ the associated stratum of ${\mathcal{X}}^{\prime}_{0}$ , $Y$ the minimal stratum of ${\mathcal{X}}_{0}$ containing $\rho(Y^{\prime})$ and $\sigma=\sigma_{Y}$ the associated simplex of $\Delta({\mathcal{X}})$ . Then $\sigma$ and $\sigma^{\prime}$ have the same dimension, and if we (somewhat abusively) identify $\sigma$ and $\sigma^{\prime}$ with their images in $\operatorname{Sk}(\phi_{\mathcal{L}})\subset X^{\mathrm{an}}$ , then $\sigma^{\prime}$ is a rational subsimplex of $\sigma$ . It suffices to prove that $\mu_{{\mathcal{L}}^{\prime\#}}(\sigma^{\prime})=\mu_{{\mathcal{L}}^{\#}}(\sigma^{\prime})$ .

Now $\rho$ restricts to a birational morphism of $Y^{\prime}\to Y$ , so since $\lambda_{\sigma}|_{\sigma^{\prime}}=\lambda_{\sigma^{\prime}}$ and $b_{\sigma}=b_{\sigma^{\prime}}$ , it suffices to prove that $\operatorname{Res}_{Y^{\prime}}({\mathcal{L}}^{\prime\#})=\rho^{*}\operatorname{Res}_{Y}({\mathcal{L}}^{\#})$ . But this is formal. Indeed, we have $(\rho|_{Y})_{*}(B_{Y^{\prime}}^{{\mathcal{L}}^{\prime}})=B_{Y}^{\mathcal{L}}$ and we can identify $(\rho|_{Y})^{*}K_{(Y,B_{Y}^{\mathcal{L}})}$ with $K_{(Y^{\prime},B_{Y^{\prime}}^{{\mathcal{L}}^{\prime}})}$ in such a way that the restriction of $\psi^{\prime}_{0}$ to ${\mathcal{L}}^{\prime}|_{Y^{\prime}}=K_{(Y^{\prime},B_{Y^{\prime}}^{{\mathcal{L}}^{\prime}})}$ coincides with the pullback under $\rho|_{Y^{\prime}}$ of the restriction of $\psi_{0}$ to ${\mathcal{L}}|_{Y}=K_{(Y,B_{Y}^{\mathcal{L}})}$ . ∎

[017P]

6.5. Behavior under base change

Fix $m\in{\mathbb{Z}}_{>0}$ . As before, denote by $X^{\prime}$ the base change of $X$ to $K^{\prime}={\mathbb{C}}(\!({t}^{1/m})\!)$ , with induced map $p\colon X^{\prime\mathrm{an}}\to X^{\mathrm{an}}$ .

[017Q]

Theorem 6.7.

Let $\psi^{\#}$ be a residually metrized model metric on $K_{X}$ , and let $\psi^{\prime\#}$ be its pull-back to $X^{\prime}$ . Then

p_{*}\mu_{\psi^{\prime\#}}=m^{d}\mu_{\psi^{\#}}

with $d=\dim\operatorname{Sk}(\psi^{\#})$ .

[017R]

Proof.

Pick a representative ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ of $\psi^{\#}$ such that ${\mathcal{L}}$ is defined on a proper snc model ${\mathcal{X}}$ . Let ${\mathcal{X}}^{\prime}$ be the normalized base change by ${t}={t}^{\prime m}$ .

Let $\sigma$ be a $d$ -dimensional face of $\Delta({\mathcal{L}})$ . By Lemma 5.13, $p^{-1}(\sigma)$ is the union of $g_{\sigma}$ distinct isomorphic faces $\sigma^{\prime}_{\alpha}$ of $\Delta({\mathcal{X}})$ such that

b_{\sigma^{\prime}_{\alpha}}=b_{\sigma}/\gcd(m,b_{\sigma})

(6.2)

\operatorname{Vol}(\sigma^{\prime}_{\alpha})=m^{d}\operatorname{Vol}(\sigma).

(6.3)

Further, the induced map $Y^{\prime}_{\sigma^{\prime}_{\alpha}}\to Y$ is generically finite, of degree $f_{\sigma}$ independent of $\alpha$ , and we have $f_{\sigma}g_{\sigma}=\gcd(m,b_{\sigma})$ . Pick a toroidal modification ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}^{\prime}$ with ${\mathcal{X}}^{\prime\prime}$ snc, denote by $\rho\colon{\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ the composition, and set ${\mathcal{L}}^{\prime\prime}:=\rho^{*}{\mathcal{L}}$ .

Each face $\sigma^{\prime}_{\alpha}$ above is subdivided into simplices $\sigma^{\prime\prime}_{\alpha\beta}$ of $\Delta({\mathcal{L}}^{\prime\prime})$ of dimension $d$ , each corresponding to a stratum $Y^{\prime\prime}_{\alpha\beta}$ of ${\mathcal{X}}^{\prime\prime}_{0}$ , and $\rho|_{Y^{\prime\prime}_{\alpha\beta}}\colon Y^{\prime\prime}_{\alpha\beta}\to Y$ is generically finite, of degree $f_{\sigma}$ . Further, (6.2) implies that

b_{\sigma^{\prime\prime}_{\alpha\beta}}=b_{\sigma^{\prime}_{\alpha}}=b_{\sigma}/\gcd(m,b_{\sigma})\quad\text{for all $\alpha,\beta$}.

(6.4)

We shall need the following result:

[017S]

Lemma 6.8.

With notation as above, we have, for all $\alpha$ , $\beta$ :

\operatorname{Res}_{Y^{\prime\prime}_{\alpha\beta}}({\mathcal{L}}^{\prime\prime\#})=\gcd(m,b_{\sigma})^{-2}\rho^{*}\operatorname{Res}_{Y}({\mathcal{L}}^{\#}).

Grant this result for the moment. Lemma 6.8 implies

\int_{Y^{\prime\prime}_{\alpha\beta}}\operatorname{Res}_{Y^{\prime\prime}_{\alpha\beta}}({\mathcal{L}}^{\prime\prime\#})=f_{\sigma}\gcd(m,b_{\sigma})^{-2}\int_{Y}\operatorname{Res}_{Y}({\mathcal{L}}^{\#}),

and hence

(p_{*}\mu^{\prime})(\sigma)=\sum_{\alpha,\beta}\mu^{\prime}(\sigma^{\prime\prime}_{\alpha\beta})=\sum_{\alpha,\beta}\left(\int_{Y^{\prime\prime}_{\alpha\beta}}\operatorname{Res}_{Y^{\prime\prime}_{\alpha\beta}}({\mathcal{L}}^{\prime\prime\#})\right)b_{\sigma^{\prime\prime}_{\alpha\beta}}^{-1}\operatorname{Vol}(\sigma^{\prime\prime}_{\alpha\beta})\\ =f_{\sigma}\gcd(m,b_{\sigma})^{-2}\left(\int_{Y}\operatorname{Res}_{Y}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\gcd(m,b_{\sigma})\sum_{\alpha}\operatorname{Vol}(\sigma^{\prime}_{\alpha})\\ =m^{d}\left(\int_{Y}\operatorname{Res}_{Y}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\operatorname{Vol}(\sigma)=m^{d}\mu(\sigma),

thanks to (6.3) and (6.4). ∎

[017T]

Proof of Lemma 6.8.

Pick a closed point $\xi^{\prime\prime}\in\mathring{Y^{\prime\prime}}$ and set $\xi=\rho(\xi^{\prime\prime})\in\mathring{Y}$ . We use the notation at the end of §6.2 with $p=d$ . Namely, pick local coordinates $(z_{i})_{0\leq i\leq n}$ at $\xi$ and $(z^{\prime\prime}_{j})_{0\leq j\leq n}$ at $\xi^{\prime\prime}$ such that $E_{i}=\{z_{i}=0\}$ for $0\leq i\leq d$ and $E^{\prime\prime}_{j}=\{z_{j}^{\prime\prime}=0\}$ for $0\leq j\leq d$ . We have $\rho^{*}z_{i}=u_{i}\prod_{j=0}^{d}(z^{\prime\prime}_{j})^{c_{ij}}$ for $0\leq i\leq d$ , where $c_{ij}\in{\mathbb{Z}}_{\geq 0}$ and $u_{i}\in{\mathcal{O}}_{{\mathcal{X}}^{\prime\prime},\xi^{\prime\prime}}$ is a unit. Further, by Lemma 5.13, the matrix $(c_{ij})$ has determinant $\pm e_{\sigma}$ , where $e_{\sigma}=m/\gcd(m,b_{\sigma})$ .

Set

\Omega_{1}:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{d}}{z_{d}}{\quad\text{and}\quad}\Omega_{2}:=dz_{d+1}\wedge\dots\wedge dz_{n},

and define $\Omega_{1}^{\prime\prime}$ , $\Omega^{\prime\prime}_{2}$ similarly. Then $\Omega:=\Omega_{1}\wedge\Omega_{2}$ and $\Omega^{\prime\prime}:=\Omega^{\prime\prime}_{1}\wedge\Omega_{2}^{\prime\prime}$ are local ${\mathbb{Q}}$ -generators of $K^{\mathrm{log}}_{\mathcal{X}}$ and $K^{\mathrm{log}}_{{\mathcal{X}}^{\prime\prime}}$ at $\xi$ and $\xi^{\prime\prime}$ , respectively. Further,

\operatorname{Res}_{Y}(\Omega)=\Omega_{2}|_{Y}{\quad\text{and}\quad}\operatorname{Res}_{Y^{\prime\prime}}(\Omega^{\prime\prime})=\Omega^{\prime\prime}_{2}|_{Y^{\prime\prime}}.

Now

\rho^{*}\Omega_{1}=\pm e_{\sigma}\Omega^{\prime\prime}_{1}+\frac{1}{z_{0}^{\prime\prime}\dots z^{\prime\prime}_{d}}\tilde{\Omega}^{\prime\prime}_{1},

where $\tilde{\Omega}^{\prime\prime}_{1}$ is a regular $(d+1)$ -form vanishing at $\xi^{\prime\prime}$ , and

\rho^{*}\Omega_{2}=q\Omega^{\prime\prime}_{2}+\tilde{\Omega}^{\prime\prime}_{2},

where $q\in{\mathcal{O}}_{{\mathcal{X}},\xi^{\prime\prime}}$ and $\tilde{\Omega}^{\prime\prime}_{2}$ is a regular $(n-d)$ -form at $\xi^{\prime\prime}$ satisfying $\Omega^{\prime\prime}_{1}\wedge\tilde{\Omega}^{\prime\prime}_{2}=0$ . On the one hand, this leads to

(\rho|_{Y^{\prime\prime}})^{*}\operatorname{Res}_{Y}(\Omega)=(\rho|_{Y^{\prime\prime}})^{*}(\Omega_{2}|_{Y})=q\Omega^{\prime\prime}_{2}|_{Y^{\prime\prime}}=q\operatorname{Res}_{Y^{\prime\prime}}(\Omega^{\prime\prime}).

On the other hand, we also get

\rho^{*}\Omega=\pm qe_{\sigma}(1+h)\Omega^{\prime\prime},

with $q$ as above and $h\in{\mathcal{O}}_{{\mathcal{X}}^{\prime\prime},\xi^{\prime\prime}}$ vanishing along $Y^{\prime\prime}$ .

Define $\Omega^{\mathrm{rel}}$ and $\Omega^{{}^{\prime\prime}\mathrm{rel}}$ by $\frac{d{t}}{{t}}\otimes\Omega^{\mathrm{rel}}=\Omega$ and $\frac{d{t}^{\prime}}{{t}^{\prime}}\otimes\Omega^{{}^{\prime\prime}\mathrm{rel}}=\Omega^{\prime\prime}$ , respectively. Then

m\frac{d{t}^{\prime}}{{t}^{\prime}}\otimes\rho^{*}\Omega^{\mathrm{rel}}=\rho^{*}(\frac{d{t}}{{t}})\otimes\rho^{*}\Omega^{\mathrm{rel}}=\rho^{*}\Omega=\pm qe_{\sigma}(1+h)\Omega^{\prime\prime}=\pm qe_{\sigma}(1+h)\frac{d{t}^{\prime}}{{t}^{\prime}}\otimes\Omega^{{}^{\prime\prime}\mathrm{rel}},

so that

\rho^{*}\Omega^{\mathrm{rel}}=\pm\frac{e_{\sigma}}{m}q(1+h)\Omega^{{}^{\prime\prime}\mathrm{rel}}.

As a consequence,

\rho^{*}|{t}^{\kappa_{\min}}\Omega^{\mathrm{rel}}|_{\psi_{0}}=\frac{e_{\sigma}}{m}|q||(1+h)||({t}^{\prime})^{\kappa^{\prime}_{\min}}\Omega^{{}^{\prime\prime}\mathrm{rel}}|_{\psi^{\prime}_{0}}.

Since $h$ vanishes along $Y^{\prime\prime}$ , this finally leads to

(\rho|_{Y^{\prime\prime}})^{*}\operatorname{Res}_{Y}({\mathcal{L}}^{\#})=\frac{(\rho|_{Y^{\prime\prime}})^{*}|\operatorname{Res}_{Y}(\Omega)|^{2}}{(\rho|_{Y^{\prime\prime}})^{*}|{t}^{\kappa_{\min}}\Omega^{\mathrm{rel}}|^{2}_{\psi_{0}}}=\frac{|(\rho|_{Y^{\prime\prime}})^{*}(\Omega_{2}|_{Y})|^{2}}{\frac{e^{2}_{\sigma}}{m^{2}}|q|^{2}|({t}^{\prime})^{\kappa^{\prime}_{\min}}\Omega^{{}^{\prime\prime}\mathrm{rel}}|_{\psi^{\prime}_{0}}^{2}}\\ =\left(\frac{m}{e_{\sigma}}\right)^{2}\frac{|\operatorname{Res}_{Y^{\prime\prime}}(\Omega^{\prime\prime})|^{2}}{|({t}^{\prime})^{\kappa^{\prime}_{\min}}\Omega^{{}^{\prime\prime}\mathrm{rel}}|_{\psi^{\prime}_{0}}^{2}}=\left(\frac{m}{e_{\sigma}}\right)^{2}\operatorname{Res}_{Y^{\prime\prime}}({\mathcal{L}}^{{}^{\prime\prime}\#}),

which completes the proof since $e_{\sigma}=\frac{m}{\gcd(b_{\sigma},m)}$ . ∎

[017U]

7. The Calabi–Yau case

As in §6, we assume that $X$ is a smooth projective variety over ${\mathbb{C}}(\!({t})\!)$ . Now we further assume that $K_{X}$ is trivial. Pick a trivializing section $\eta\in H^{0}(X,K_{X})$ , and denote by $\log|\eta|$ the associated model metric on $K_{X}^{\mathrm{an}}$ , determined on any model ${\mathcal{X}}$ by ${\mathcal{L}}={\mathcal{O}}_{\mathcal{X}}$ , with $\eta$ providing the identification ${\mathcal{L}}|_{X}\simeq K_{X}$ . Denote also by $\log|\eta|^{\#}$ the residually metrized model metric induced by the trivial Hermitian metric $\psi_{0}=0$ on ${\mathcal{O}}_{{\mathcal{X}}_{0}}$ .

The function $\kappa:=A_{X}-\log|\eta|=-\log|\eta|_{A_{X}}$ coincides with the weight function of [MN15, NX13]. By definition, the Kontsevich–Soibelman skeleton of $X$ is $\operatorname{Sk}(X):=\operatorname{Sk}(\log|\eta|^{\#})$ . It is indeed independent of the choice of $\eta$ , since any other trivializing section of $K_{X}$ is of the form $\eta^{\prime}=f\eta$ with $f\in{\mathbb{C}}(\!({t})\!)^{*}$ , and hence $\kappa^{\prime}=\kappa+\operatorname{ord}_{0}(f)$ .

[017V]

7.1. Topology of the skeleton

By [NX13, Theorem 4.2.4], the ${\mathbb{Z}}$ -PA-space $\operatorname{Sk}(X)$ is connected, of pure dimension $d$ , and is a deformation retract of $X^{\mathrm{an}}$ . Further, $\operatorname{Sk}(X)$ is a pseudomanifold with boundary, i.e. for some (or, equivalently, any) triangulation $\Delta$ of $\operatorname{Sk}(X)$ , we have:

(a)

Non-branching property: every $(d-1)$ -simplex of $\Delta$ is contained in at most two $d$ -simplices
(b)

Strong connectedness: every pair of $n$ -simplices $\sigma$ , $\sigma^{\prime}$ is joined by a chain of $n$ -simplices $\sigma=\sigma_{1},\dots,\sigma_{N}=\sigma^{\prime}$ with $\sigma_{i}$ and $\sigma_{i+1}$ sharing a common $(n-1)$ -face.

In the maximally degenerate case $d=n$ , $\operatorname{Sk}(X)$ is even a pseudomanifold, i.e. (a) is replaced by

(a’)

every $(n-1)$ -simplex of $\Delta$ is contained in exactly two $n$ -simplices.

See also [KX15] for even more precise results on the structure of $\operatorname{Sk}(X)$ . For example, $\operatorname{Sk}(X)$ is homeomorphic to a sphere if $n\leq 3$ .

[017W]

7.2. The skeletal measure

Consider the skeletal measure $\mu_{\log|\eta|^{\#}}$ on $\operatorname{Sk}(X)$ . Choose an snc model ${\mathcal{X}}$ , and write as usual ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ . The form $\eta$ defines an identification $K^{\mathrm{log}}_{{\mathcal{X}}/S}=\sum_{i\in I}a_{i}E_{i}$ , and Proposition 5.10 yields

\kappa_{\min}=\min_{i}\frac{a_{i}}{b_{i}}.

(7.1)

If $\kappa_{\min}\in{\mathbb{Z}}$ , then

\omega:=\frac{d{t}}{{t}^{\kappa_{\min}+1}}\wedge\eta

is a logarithmic form on ${\mathcal{X}}$ . For each face $\sigma$ of $\Delta({\mathcal{L}})$ , ordering the set $J\subset I$ of components cutting out the stratum $Y=Y_{\sigma}$ yields a well-defined Poincaré residue $\operatorname{Res}_{Y}(\omega)$ . By Lemma 5.12, $\operatorname{Res}_{Y}(\omega)$ is a rational section of $K_{Y}$ , with divisor

-B^{\mathcal{L}}_{Y}=\sum_{i\notin J}(a_{i}-\kappa_{\min}b_{i}-1)E_{i}|_{Y}

When $\sigma$ is a maximal face, $\operatorname{Res}_{Y}(\omega)$ is thus a holomorphic form on $Y$ ; using the formulas in §6.2, it is easy to see that the residual measure on $Y$ is given by

\operatorname{Res}_{Y}(\log|\eta|^{\#})=|\operatorname{Res}_{Y}(\omega)|^{2}.

The following result corresponds to Theorem C in the introduction.

[017X]

Theorem 7.1.

Assume that $X$ is maximally degenerate, i.e. $\dim\operatorname{Sk}(X)=n$ . If $X$ has semistable reduction, then the skeletal measure $\mu_{\log|\eta|^{\#}}$ is a multiple of the integral Lebesgue measure of $\operatorname{Sk}(X)$ .

[017Y]

Proof.

Let ${\mathcal{X}}$ be a semistable model, i.e. ${\mathcal{X}}$ is snc with ${\mathcal{X}}_{0}$ reduced. By (7.1), we have $\kappa_{\min}\in{\mathbb{Z}}$ . Since some non-empty $E_{J}$ might have several components, the dual complex $\Delta({\mathcal{X}})$ is possibly not a triangulation of $\operatorname{Sk}(X)$ . However, the barycentric subdivision $\Delta^{\prime}$ of $\Delta({\mathcal{X}})$ is a triangulation; the corresponding toroidal modification ${\mathcal{X}}^{\prime}$ is snc, with ${\mathcal{X}}^{\prime}_{0}$ is possibly non-reduced, but $b_{\sigma}=1$ for each $n$ -simplex $\sigma$ of $\Delta^{\prime}$ . Applying the above discussion to ${\mathcal{X}}^{\prime}$ , we infer

\mu_{\log|\eta|^{\#}}=\sum_{\sigma}|\operatorname{Res}_{y_{\sigma}}(\omega)|^{2}\lambda_{\sigma},

with $\sigma$ ranging over the $n$ -dimensional faces of $\Delta^{\prime}$ , with corresponding strata $y_{\sigma}\in{\mathcal{X}}^{\prime}_{0}$ reduced to single points. It will thus be enough to show that $|\operatorname{Res}_{y_{\sigma}}(\omega)|$ is independent of $\sigma$ .

By the strong connectedness property, any two $n$ -simplices $\sigma$ , $\sigma^{\prime}$ of $\Delta^{\prime}$ can be joined by a chain of $n$ -simplices $\sigma=\sigma_{1},\dots,\sigma_{N}=\sigma^{\prime}$ with $\sigma_{i}$ and $\sigma_{i+1}$ sharing a common $(n-1)$ -face $\tau_{i}$ . Denoting by $y_{i}=y_{\sigma_{i}}$ and $Y_{i}=Y_{\tau_{i}}$ the corresponding strata in ${\mathcal{X}}^{\prime}$ , we thus have $y_{i},y_{i+1}\in Y_{i}$ . Further, the Poincaré residue $\operatorname{Res}_{Y_{i}}(\omega)$ has poles precisely at $y_{i},y_{i+1}$ , since any other pole would correspond to an $n$ -simplex of $\Delta^{\prime}$ containing $\tau_{i}$ , contradicting the non-branching property. Since $\operatorname{Res}_{y_{i}}\operatorname{Res}_{Y}(\omega)=\operatorname{Res}_{y_{i}}(\omega)$ , the residue theorem applied to the Riemann surface $Y_{i}$ yields $\operatorname{Res}_{y_{i}}(\omega)+\operatorname{Res}_{y_{i+1}}(\omega)=0$ , and hence $|\operatorname{Res}_{y_{1}}(\omega)|=\dots=|\operatorname{Res}_{y_{N}}(\omega)|$ . ∎

[017Z]

Remark 7.2.

Theorem 7.1 fails in general when $X$ does not have semistable reduction. Indeed, the semistable reduction theorem [KKMS] shows that the base change $p\colon X^{\prime}\to X$ to ${\mathbb{C}}(\!({t}^{1/m})\!)$ has semistable reduction for some $m$ divisible enough. By Lemma 5.14, $\dim\operatorname{Sk}(X^{\prime})=n$ , and $\mu_{\log|\eta^{\prime}|^{\#}}$ is thus a multiple of the integral Lebesgue measure $\lambda^{\prime}$ of $\operatorname{Sk}(X^{\prime})$ , by Theorem 7.1. By Theorem 6.7, $\mu_{\log|\eta|^{\#}}=m^{-n}p_{*}\lambda^{\prime}$ . However, $p_{*}\lambda^{\prime}$ is not proportional to the integral Lebesgue measure $\lambda$ of $\operatorname{Sk}(X)$ in general. Indeed, for each $n$ -simplex $\sigma$ of $\Delta({\mathcal{L}})$ , Lemma 5.13 shows that $(p_{*}\lambda^{\prime})_{\sigma}=m^{n}b_{\sigma}\lambda_{\sigma}$ , and $b_{\sigma}$ is in general not independent of $\sigma$ .

[0180]

8. Extensions

In this section we extend the main results in various directions.

[0181]

8.1. A singular version of Theorem A

Let $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ be a projective, flat holomorphic map of a normal complex space onto the disc, with $X:=\pi^{-1}({\mathbb{D}}^{*})$ smooth over ${\mathbb{D}}^{*}$ . Since $\pi$ is projective, it defines a smooth projective variety $X_{{\mathbb{C}}(\!({t})\!)}$ over ${\mathbb{C}}(\!({t})\!)$ , as well as a model ${\mathcal{X}}_{{\mathbb{C}}[\![{t}]\!]}$ .

Let ${\mathcal{L}}$ be a ${\mathbb{Q}}$ -line bundle on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$ , and $\psi$ a continuous Hermitian metric on ${\mathcal{L}}$ . This data induces a continuous Hermitian metric $\psi_{t}$ on $K_{X_{t}}$ for $t\in{\mathbb{D}}^{*}$ , as well as a residually metrized model ${\mathcal{L}}^{\#}$ of $K_{X_{{\mathbb{C}}(\!({t})\!)}}$ , the model given by ${\mathcal{L}}_{{\mathbb{C}}[\![{t}]\!]}$ and the metric by the restriction of $\psi$ to ${\mathcal{L}}_{0}={\mathcal{L}}|_{{\mathcal{X}}_{0}}$ . Thus we obtain a skeletal measure $\mu_{{\mathcal{L}}^{\#}}$ on $X^{\mathrm{an}}_{{\mathbb{C}}(\!({t})\!)}$ .

Denote by ${\mathcal{L}}^{\prime}$ (resp. $\psi^{\prime}$ ) the pull-back of ${\mathcal{L}}$ (resp. $\psi$ ) to a log resolution ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . By invariance of skeletal measures under pull-back, we have $\mu_{{\mathcal{L}}^{\prime\#}}=\mu_{{\mathcal{L}}^{\#}}$ , and Theorem 3.4 therefore implies:

[0182]

Theorem 8.1.

The rescaled measures

\mu_{t}:=\frac{e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}},

viewed as measures on ${\mathcal{X}}^{\prime\mathrm{hyb}}$ , converge weakly to $\mu_{{\mathcal{L}}^{\#}}$ .

[0183]

Corollary 8.2.

If ${\mathcal{X}}$ (i.e. the pair $({\mathcal{X}},{\mathcal{X}}_{0,\mathrm{red}})$ ) is dlt, then

\lim_{t\to 0}\frac{\int_{{\mathcal{X}}_{t}}e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}}=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\operatorname{Vol}(\sigma),

where $\sigma$ runs over the $d$ -dimensional faces of $\Delta({\mathcal{L}})$ .

When $d=0$ , this implies the following slight generalization of [Li13, Lemma 1].

[0184]

Corollary 8.3.

Assume that ${\mathcal{X}}_{0}$ has klt singularities (and hence ${\mathcal{X}}$ is dlt by inversion of adjunction). Let $\psi$ be a continuous metric on $K_{{\mathcal{X}}/{\mathbb{D}}}$ . Then $t\mapsto\int_{{\mathcal{X}}_{t}}e^{2\psi_{t}}$ is continuous at $t=0$ .

[0185]

8.2. Corollary B for pairs

Suppose $(X,B)$ is a projective subklt pair over ${\mathbb{D}}^{*}$ that is meromorphic at $0\in{\mathbb{D}}$ .

By Bertini’s theorem (see [Kol97, 4.8] and also below), the pair $(X_{t},B_{t})$ is subklt for all $t\in{\mathbb{D}}^{*}$ outside a discrete subset $Z$ . Let $\psi$ be a continuous metric on $K_{{(X,B)}/{\mathbb{D}}^{*}}$ . As explained in §1.2, $\psi$ induces a finite positive measure $e^{2(\psi_{t}-\phi_{B_{t}})}$ on $X_{t}$ for $t\in{\mathbb{D}}^{*}\setminus Z$ .

Assume that $\psi$ has analytic singularities in the sense that there exists a flat projective map ${\mathcal{X}}\to{\mathbb{D}}$ extending $X\to{\mathbb{D}}^{*}$ , with ${\mathcal{X}}$ normal, and a ${\mathbb{Q}}$ -line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{(X,B)/{\mathbb{D}}^{*}}$ such that $\psi$ extends continuously to ${\mathcal{L}}$ .

Our assumptions imply that $X$ is defined over the Banach ring $A_{r}$ described in Appendix A for $0<r\ll 1$ . Let $X^{\mathrm{hyb}}$ be the analytification of the base change $X_{A_{r}}$ . Recall that $X^{\mathrm{hyb}}$ naturally fibers over $\overline{{\mathbb{D}}}_{r}$ , with $X^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}^{*}_{r}}\simeq X_{\overline{{\mathbb{D}}}^{*}_{r}}$ and $X^{\mathrm{hyb}}_{0}\simeq X_{{\mathbb{C}}(\!({t})\!)}^{\mathrm{an}}$ .

[0186]

Theorem 8.4.

The pair $(X_{t},B_{t})$ is klt for $0<|t|\ll 1$ . Further, there exist $\kappa_{\min}\in{\mathbb{Q}}$ and $d\in{\mathbb{N}}^{*}$ such that the rescaled measures

\mu_{t}:=\frac{e^{2\psi_{t}}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}},

viewed as measures on $X^{\mathrm{hyb}}$ , converge weakly, as $t\to 0$ , to a finite positive measure $\mu_{0}$ on $X^{\mathrm{hyb}}_{0}=X_{{\mathbb{C}}(\!({t})\!)}^{\mathrm{an}}$ .

A special case of Theorem 8.4 is the log Calabi–Yau setting, when the ${\mathbb{Q}}$ -line bundle $K_{(X,B)/{\mathbb{D}}^{*}}$ is trivial. In general, we are not able to give a very precise description of the limit measure $\mu_{0}$ , but the proof will show that $\mu_{0}$ is a skeletal measure when the pair $(X,B)$ is log smooth.

[0187]

Proof of Theorem 8.4.

Let us first treat the case when $(X,B)$ is log smooth. In this case we need not assume that $X\to{\mathbb{D}}^{*}$ is projective. It follows from the normal crossings condition that $(X_{t},B_{t})$ is subklt for $0<|t|\ll 1$ . After reparametrizing we may assume this is true for all $t\in{\mathbb{D}}^{*}$ , that is, $Z=\emptyset$ . Set

\nu_{t}=e^{2(\psi_{t}-\phi_{B_{t}})}.

This is a positive measure on $X_{t}$ , smooth outside the support of $B_{t}$ . Pick an snc model $({\mathcal{X}},{\mathcal{B}})$ of $(X,B)$ , where ${\mathcal{B}}$ is the closure of $B$ in ${\mathcal{X}}$ , such that $\psi$ extends to a continuous metric on a ${\mathbb{Q}}$ -line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{(X,B)/{\mathbb{D}}^{*}}$ .

We can then prove a version of Theorem A inside the hybrid space ${\mathcal{X}}^{\mathrm{hyb}}$ . By letting ${\mathcal{X}}$ vary, we obtain Theorem 8.4 as a consequence, just as Corollary B follows from Theorem A.

The proof is very similar to the proof of Theorem A, so we will only indicate the modifications needed. Let us write

K^{\mathrm{log}}_{({\mathcal{X}},{\mathcal{B}})/{\mathbb{D}}}={\mathcal{L}}+\sum_{i\in I}a_{i}E_{i}

with $a_{i}\in{\mathbb{Q}}$ . Set $\kappa_{i}:=a_{i}/b_{i}$ and $\kappa_{\min}:=\min_{i}\kappa_{i}$ . Here ${\mathcal{X}}_{0}=\sum_{i}b_{i}E_{i}$ as before.

Define $\Delta({\mathcal{L}})$ as the subcomplex of $\Delta({\mathcal{X}})$ spanned by the vertices such that $\kappa_{i}=\kappa_{\min}$ . This will be the support of the measure $\mu_{0}$ . For every stratum $Y$ corresponding to a maximal face of $\Delta({\mathcal{L}})$ , define a subklt pair $(Y,B_{Y}^{\mathcal{L}})$ using

B^{\mathcal{L}}_{Y}:={\mathcal{B}}|_{Y}+\sum_{i\notin J}(1-(a_{i}-\kappa_{\min}b_{i}))E_{i}|_{Y}

The residual measure $\operatorname{Res}_{Y}(\psi)$ is given by

\operatorname{Res}_{Y}(\psi):=\exp\left(2(\psi|_{Y}-\phi_{B^{\mathcal{L}}_{Y}})\right).

Finally set

\mu_{0}:=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}(\psi)\right)b_{\sigma}^{-1}\lambda_{\sigma},

where $\sigma$ ranges over the $d$ -dimensional faces of $\Delta({\mathcal{L}})$ , with $d=\dim\Delta({\mathcal{L}})$ .

We then prove a version of Theorem 3.4. Namely, if

\mu_{t}:=\frac{\lambda(t)^{d}}{(2\pi)^{d}|t|^{2\kappa_{\min}}}e^{2(\psi_{t}-\phi_{B_{t}})}.

then we show that $\mu_{t}$ converges to $\mu_{0}$ in ${\mathcal{X}}^{\mathrm{hyb}}$ as $t\to 0$ . This is done via a local convergence result as in Lemma 3.5. Namely, given a point $\xi\in{\mathcal{X}}_{0}$ , we choose local coordinates $(z_{0},\dots,z_{n})$ at $\xi$ as in §3.2, but further require that these coordinates also cut out the irreducible components of ${\mathcal{B}}$ containing $\xi$ . More precisely, there exist $m$ with $p\leq m\leq n$ such that these irreducible components are given by $B_{i}=\{z_{i}=0\}$ for $p<i\leq m$ . Also set $c_{i}:=\operatorname{ord}_{B_{i}}({\mathcal{B}})<1$ .

A local ${\mathbb{Q}}$ -generator for ${\mathcal{L}}$ at $\xi$ is then given by

\tau=\prod_{i=0}^{p}z_{i}^{a_{i}}\prod_{i=p+1}^{m}z_{i}^{-c_{i}}\ \Omega^{\mathrm{rel}},

with $\Omega^{\mathrm{rel}}$ as before. For a stratum $Y$ corresponding to a $d$ -dimensional simplex in $\Delta({\mathcal{L}})$ , the residual measure is given by

\operatorname{Res}_{Y}(\psi)=|\tau|^{-2}_{\psi}\prod_{i=d+1}^{p}|z_{i}|^{2(a_{i}-\kappa_{\min}b_{i}-1)}\prod_{i=p+1}^{m}|z_{i}|^{-2c_{i}}\bigg|\bigwedge_{i=d+1}^{n}dz_{i}\bigg|^{2}.

(8.1)

The measure $\mu_{t}$ can be written near $\xi$ as

\mu_{t}=\frac{\lambda(t)^{d}}{(2\pi)^{d}}\frac{\prod_{i=p+1}^{m}|z_{i}|^{-2c_{i}}|\Omega_{t}|^{2}}{\big|\prod_{i=p+1}^{m}z_{i}^{-c_{i}}\Omega_{t}\big|^{2}_{\psi_{t}}}.

The proof now proceeds exactly as in §3.3 except that we need to insert a factor $\prod_{i=p+1}^{m}|z_{i}|^{-2c_{i}}$ in the last two lines of (3.3) and (3.6), the second line and the second factor of the last line of (3.7), and the right-hand sides of (3.8) and (3.9). This completes the proof in the log smooth case.

Now we consider the general case, assuming $X\to{\mathbb{D}}^{*}$ is projective. Pick a log resolution $q\colon(X^{\prime},B^{\prime})\to(X,B)$ . Since $(X^{\prime}_{t},B^{\prime}_{t})$ is subklt for $0<|t|\ll 1$ , the same is true for $(X_{t},B_{t})$ . We have an induced continuous map $q^{\mathrm{hyb}}\colon(X^{\prime})^{\mathrm{hyb}}\to X^{\mathrm{hyb}}$ . By what precedes, there exist $\kappa\in{\mathbb{Q}}$ and $d\in{\mathbb{N}}$ such that the measure $\mu^{\prime}_{t}:=\frac{e^{2(\psi^{\prime}_{t}-\phi_{B^{\prime}_{t}})}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}}$ on $X^{\prime}_{t}=X^{\mathrm{hyb}}$ converges to a nonzero positive measure $\mu^{\prime}_{0}$ on $(X^{\prime})^{\mathrm{hyb}}$ . By continuity, it follows that $\mu_{t}=q^{\mathrm{hyb}}_{*}\mu^{\prime}_{t}$ converges to the nonzero positive measure $\mu_{0}=q^{\mathrm{hyb}}_{*}\mu^{\prime}_{0}$ on $X^{\mathrm{hyb}}$ . This completes the proof. ∎

[0188]

8.3. Degenerations of Ricci-flat Kähler manifolds

Let $M$ be a Ricci-flat Kähler manifold, i.e. a compact Kähler manifold with trivial first Chern class $c_{1}(M)\in H^{2}(M,{\mathbb{C}})$ . Then $M$ carries a canonical probability measure $\mu$ , given by $\mu=e^{2\psi}/\int_{M}e^{2\psi}$ where $\psi$ is a Hermitian metric on $K_{M}$ with curvature $0$ (and hence unique up to a constant).

By the Calabi-Yau theorem, each Kähler $(1,1)$ -class on $M$ further contains a unique Ricci-flat Kähler metric $\omega$ , characterized by

\frac{\omega^{n}}{\int_{M}\omega^{n}}=\mu.

Recall also that $K_{M}$ is torsion, i.e. $rK_{M}\simeq{\mathcal{O}}_{M}$ for some positive integer $r$ . Indeed, this is a consequence of the Beauville-Bogomolov theorem [Beau83, Bog74], which implies that $M$ admits a finite étale cover $p:M^{\prime}\to M$ with $K_{M^{\prime}}=p^{*}K_{M}$ trivial. A trivializing section $\eta$ of $rK_{M}$ defines a metric $\psi=\tfrac{1}{r}\log|\eta|$ on $K_{M}$ as above, and hence $\mu=|\eta|^{2/r}/\int|\eta|^{2/r}$ .

As a consequence of Theorem A, we shall prove:

[0189]

Theorem 8.5.

Let $\pi\colon X\to{\mathbb{D}}^{*}$ be a holomorphic family of Calabi-Yau Kähler manifolds $X_{t}$ , meromorphic at $t=0$ , and let $\mu_{t}$ be the corresponding family of canonical probability measures. For any snc model ${\mathcal{X}}$ , $\mu_{t}$ converges in ${\mathcal{X}}^{\mathrm{hyb}}$ to a skeletal measure $\mu_{0}$ supported in $\Delta({\mathcal{X}})$ .

[018A]

Proof.

As recalled above, $K_{X_{t}}$ is torsion for each fixed $t$ . Equivalently, $h^{0}(X_{t},rK_{X_{t}})=1$ for some positive integer $r$ . Since $t\mapsto h^{0}(X_{t},rK_{X_{t}})$ is upper semicontinuous in the Zariski topology, it follows that $rK_{X_{t}}$ is trivial for a fixed $r$ independent of $t$ . Given any snc model $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ , $\pi_{*}{\mathcal{O}}(rK_{{\mathcal{X}}/{\mathbb{D}}})$ is torsion free of rank one, and hence a line bundle. The choice of a trivializing section yields a holomorphic section $\eta$ of $K_{{\mathcal{X}}/{\mathbb{D}}}$ , inducing a holomorphic family $\eta_{t}$ of trivializing sections of $rK_{X_{t}}$ for $t\neq 0$ . As a consequence, the family of volume forms $\nu_{t}:=|\eta_{t}|^{2/r}$ has analytic singularities at $t=0$ , and the result is thus a consequence of Theorem A, since $\mu_{t}=\nu_{t}/\nu_{t}(X_{t})$ . ∎

[018B]

Appendix A Berkovich spaces over Banach rings

In this appendix we review the construction of the analytification of a scheme of finite type defined over a Banach ring. The main reference for this is [Berk09]; see also [Poi10, Poi13a, Jon16]. For suitable choices of Banach rings, this leads to spaces that contain both Archimedean and non-Archimedean data.

[018C]

A.1. Berkovich spectra

Let $A$ be a Banach ring, that is, a commutative ring that is complete with respect to a submultiplicative norm $\|\cdot\|$ . The Berkovich spectrum ${\mathcal{M}}(A)$ is the set of all bounded multiplicative seminorms on $A$ . In other words, a point $x\in{\mathcal{M}}(A)$ corresponds to a function $|\cdot|_{x}\colon A\to{\mathbb{R}}_{\geq 0}$ such that $|\cdot|_{x}\leq\|\cdot\|$ , $|1|_{x}=1$ , $|f+g|_{x}\leq|f|_{x}+|g|_{x}$ and $|fg|_{x}=|f|_{x}|g|_{x}$ for $f,g\in A$ . The spectrum is a nonempty, compact Hausdorff space with respect to the topology of pointwise convergence.

For $x\in{\mathcal{M}}(A)$ , denote by ${\mathfrak{p}}_{x}$ the kernel of $|\cdot|_{x}$ . This is a prime ideal of $A$ , and $|\cdot|_{x}$ defines a multiplicative norm on $A/{\mathfrak{p}}_{x}$ . The completion of the fraction field of $A/{\mathfrak{p}}_{x}$ with respect to this norm is a valued field ${\mathcal{H}}(x)$ . We write $f(x)$ for the image of $f\in A$ in ${\mathcal{H}}(x)$ ; then $|f(x)|=|f|_{x}$ . The assignment $x\mapsto{\mathfrak{p}}_{x}$ yields a map ${\mathcal{M}}(A)\to\operatorname{Spec}(A)$ that is continuous for the Zariski topology $\operatorname{Spec}(A)$ .

[018D]

Example A.1.

If $k$ is a valued field (i.e. a field with a multiplicative norm), then ${\mathcal{M}}(k)$ is a singleton.

[018E]

Example A.2.

When $A$ is a complex Banach algebra, the Gelfand-Mazur Theorem implies that the Berkovich spectrum agrees with the maximal ideal spectrum.

[018F]

A.2. Analytification of a scheme

To any scheme $X$ of finite type over a Banach ring $A$ , Berkovich associates an analytification⁹⁹ 9 We use the term analytification even though we shall only consider $X^{\mathrm{An}}$ as a topological space. In particular, $X^{\mathrm{An}}$ only depends on the reduced scheme structure of $X$ . $X^{\mathrm{An}}$ , a locally compact topological space with a continuous morphism $X^{\mathrm{An}}\to{\mathcal{M}}(A)$ , defined as follows.

When $X=\operatorname{Spec}B$ is affine, with $B$ a finitely generated $A$ -algebra, $X^{\mathrm{An}}$ is defined as the set of multiplicative seminorms $|\cdot|_{x}$ on $B$ whose restriction to $A$ is bounded by the given norm on $A$ , i.e. belongs to ${\mathcal{M}}(A)$ . The topology on $X^{\mathrm{An}}$ is the weakest one for which $x\mapsto|f|_{x}=|f(x)|$ is continuous for every $f\in B$ .

In the general case, the analytification $X^{\mathrm{An}}$ is defined by gluing together the analytifications of an affine open cover, and yields a covariant functor $X\mapsto X^{\mathrm{An}}$ . If $X\hookrightarrow Y$ is an open (resp. closed) embedding, then so is $X^{\mathrm{An}}\hookrightarrow Y^{\mathrm{An}}$ . If $X\to Y$ is surjective, then so is $X^{\mathrm{An}}\to Y^{\mathrm{An}}$ .

The topological space $X^{\mathrm{An}}$ is Hausdorff (resp. compact) if $X$ is separated (resp. projective). The assignment $x\mapsto{\mathfrak{p}}_{x}$ above globalizes to a continuous map

\pi\colon X^{\mathrm{An}}\to X,

where $X$ is equipped with the Zariski topology.

When $A$ is a valued field, it is more common to write $X^{\mathrm{an}}$ instead of $X^{\mathrm{An}}$ [Berk90].

[018G]

Example A.3.

For $A={\mathbb{C}}$ , the Gelfand-Mazur theorem shows that $X^{\mathrm{An}}$ coincides with the usual analytification of $X$ , i.e. the set $X({\mathbb{C}})$ of complex points of $X$ endowed with the euclidean topology.

[018H]

A.3. The hybrid norm on ${\mathbb{C}}$

Denote by ${\mathbb{C}}_{\mathrm{hyb}}$ the Banach field $({\mathbb{C}},\|\cdot\|_{\mathrm{hyb}})$ , where the hybrid norm is defined as

\|\cdot\|_{\mathrm{hyb}}:=\max\{|\cdot|_{0},|\cdot|_{\infty}\},

with $|\cdot|_{0}$ the trivial absolute value and $|\cdot|_{\infty}$ the usual absolute value.

The elements of the Berkovich spectrum ${\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})$ are of the form $|\cdot|_{\infty}^{\rho}$ for $\rho\in[0,1]$ , interpreted as the trivial absolute value $|\cdot|_{0}$ for $\rho=0$ . This yields a homeomorphism ${\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})\simeq[0,1]$ .

[018I]

A.4. Hybrid geometry over ${\mathbb{C}}$

If $X$ is a scheme of finite type over ${\mathbb{C}}$ , we denote by $X^{\operatorname{hol}}=X({\mathbb{C}})$ its analytification with respect to the usual absolute value $|\cdot|_{\infty}$ , by $X^{\mathrm{an}}_{0}$ its analytification with respect to the trivial absolute value, and by $X^{\mathrm{hyb}}$ its analytification with respect to the hybrid norm $\|\cdot\|_{\mathrm{hyb}}$ .

From the structure morphism $X\to\operatorname{Spec}{\mathbb{C}}$ we obtain a continuous map $\lambda\colon X^{\mathrm{hyb}}\to{\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})\simeq[0,1]$ . The fiber $\lambda^{-1}(\rho)$ is equal to the analytification of $X$ with respect to the multiplicative norm $|\cdot|_{\infty}^{\rho}$ on ${\mathbb{C}}$ . In particular, we have canonical identifications $\lambda^{-1}(1)\simeq X^{\operatorname{hol}}$ and $\lambda^{-1}(0)\simeq X^{\mathrm{An}}_{0}$ . For $0<\rho\leq 1$ , the fiber $\lambda^{-1}(\rho)$ is also homeomorphic to $X^{\operatorname{hol}}$ . In fact, we have a a homeomorphism

\lambda^{-1}\left((0,1]\right)\simeq(0,1]\times X^{\operatorname{hol}},

see [Berk09, Lemma 2.1].

[018J]

A.5. The hybrid circle

Now consider the hybrid circle of radius $r\in(0,1)$ , that is, $C_{\mathrm{hyb}}(r):=\{|{t}|=r\}\subset{\mathbb{A}}^{1,\mathrm{hyb}}=(\operatorname{Spec}{\mathbb{C}}[{t}])^{\mathrm{hyb}}$ . By [Poi10, Prop 2.1.1], this is compact and realized as the Berkovich spectrum of the Banach ring

A_{r}:=\left\{f=\sum_{\alpha\in{\mathbb{Z}}}c_{\alpha}{t}^{\alpha}\in{\mathbb{C}}(\!({t})\!)\ \bigg|\ \|f\|_{\mathrm{hyb}}:=\sum_{\alpha\in{\mathbb{Z}}}\|c_{\alpha}\|_{\mathrm{hyb}}r^{\alpha}<+\infty\right\}.

Since $\|c_{\alpha}\|_{\mathrm{hyb}}\geq|c_{\alpha}|_{\infty}$ , every $f\in A_{r}$ defines a continuous function $f^{\operatorname{hol}}$ on the punctured closed disc $\overline{{\mathbb{D}}}^{*}_{r}$ that is holomorphic on ${\mathbb{D}}^{*}_{r}$ and meromorphic at 0.

[018K]

Proposition A.4.

There is a homeomorphism $\overline{{\mathbb{D}}}_{r}\overset{\sim}{\to}{\mathcal{M}}(A_{r})\simeq C_{\mathrm{hyb}}(r)$ , that maps $z\in\overline{{\mathbb{D}}}_{r}\subset{\mathbb{C}}$ to the seminorm on $A_{r}$ defined by

|f|=\begin{cases}r^{\operatorname{ord}_{0}(f)}&\ \text{if $z=0$}\\ r^{\frac{\log|f^{\operatorname{hol}}(z)|_{\infty}}{\log|z|_{\infty}}}\ &\text{otherwise},\end{cases}

(A.1)

and via which the map $\lambda\colon C_{\mathrm{hyb}}(r)\to[0,1]$ is given by $\lambda(z)=\frac{\log r}{\log|z|_{\infty}}$ .

[018L]

Proof.

The map $\tau\colon\overline{{\mathbb{D}}}_{r}\to{\mathcal{M}}(A_{r})$ given by (A.1) is clearly well defined. It is also continuous on $\overline{{\mathbb{D}}}^{*}_{r}$ . To prove continuity at $0$ , we note that for each $f\in A_{r}$ , we can write $f^{\operatorname{hol}}=z^{\operatorname{ord}_{0}(f)}u$ , where $u$ is a continuous function on $\overline{{\mathbb{D}}}_{r}$ that is holomorphic on ${\mathbb{D}}_{r}$ with $u(0)\neq 0$ . As a consequence, we get $\lim_{z\to 0}\frac{\log|f^{\operatorname{hol}}(z)|_{\infty}}{\log|z|_{\infty}}=\operatorname{ord}_{0}(f)$ .

Now, for each $\rho\in(0,1]$ , $\lambda^{-1}(\rho)\subset C_{\mathrm{hyb}}(r)$ can be identified with the circle of radius $r$ with respect to the absolute value $|\cdot|_{\infty}^{\rho}$ , while $\lambda^{-1}(0)$ is the non-Archimedean absolute value $r^{-\operatorname{ord}_{0}}$ on ${\mathbb{C}}(\!({t})\!)$ . This proves that the map $\tau$ above is bijective, and hence a homeomorphism by compactness. ∎

[018M]

Remark A.5.

When $r<s$ , the identity gives a bounded map from $A_{s}$ to $A_{r}$ , and $\varinjlim_{r\to 0}A_{r}$ is the fraction field of ${\mathcal{O}}_{{\mathbb{C}},0}$ , i.e. the ring of meromorphic germs at the origin of ${\mathbb{C}}$ .

[018N]

A.6. Geometry over the hybrid circle

Let now $X$ be a scheme of finite type over $A_{r}$ . We will associate to $X$ three kinds of analytic spaces.

First, since $X$ is obtained by gluing together finitely many affine schemes cut out by polynomials with coefficients holomorphic on ${\mathbb{D}}^{*}_{r}\subset{\mathbb{C}}$ and meromorphic at $0$ , we can associate to $X$ in a functorial way a complex analytic space $X^{\operatorname{hol}}$ over ${\mathbb{D}}^{*}_{r}$ , which we call its holomorphic analytification.

Second, since $A_{r}$ is contained in ${\mathbb{C}}(\!({t})\!)$ , we may also consider the base change $X_{{\mathbb{C}}(\!({t})\!)}$ and its non-Archimedean analytification $X^{\mathrm{an}}_{{\mathbb{C}}(\!({t})\!)}$ with respect to the non-Archimedean absolute value $r^{\operatorname{ord}_{0}}$ on ${\mathbb{C}}(\!({t})\!)$ .

Third, we denote by $X^{\mathrm{hyb}}$ the analytification of $X$ as a scheme of finite type over the Banach ring $A_{r}$ , and call it the hybrid analytification of $X$ . In view of Proposition A.4, it comes with a continuous structure map

\pi\colon X^{\mathrm{hyb}}\to\overline{{\mathbb{D}}}_{r}\simeq{\mathcal{M}}(A_{r}),

Recall further that $X^{\mathrm{hyb}}$ is locally compact, Hausdorff if $X$ is separated, and compact if $X$ is proper over $A_{r}$ . The discussion above implies:

[018P]

Lemma A.6.

We have canonical homeomorphisms

\pi^{-1}(0)\simeq X_{{\mathbb{C}}(\!({t})\!)}^{\mathrm{an}}{\quad\text{and}\quad}\pi^{-1}({\mathbb{D}}^{*}_{r})\simeq X^{\operatorname{hol}}

(A.2)

compatible with the projection to ${\mathbb{D}}_{r}$ .

In §4 we give a topological description of $X^{\mathrm{hyb}}$ .

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