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\begin{document}
\frontmatter
\title[Tropical and non-Archimedean limits of volume forms]
{Tropical and non-Archimedean limits of degenerating families of volume forms}
\author[\initial{S.} \lastname{Boucksom}]{\firstname{Sébastien} \lastname{Boucksom}}
\address{CMLS, École polytechnique, CNRS, Université Paris-Saclay\\
91128 Palaiseau Cedex, France}
\email{sebastien.boucksom@polytechnique.edu}
\urladdr{http://sebastien.boucksom.perso.math.cnrs.fr/}
\author[\initial{M.} \lastname{Jonsson}]{\firstname{Mattias} \lastname{Jonsson}}
\address{Department of Mathematics, University of Michigan\\
Ann Arbor, MI 48109-1043, USA\\
and\\
Mathematical Sciences, Chalmers University of Technology
and University of Gothenburg\\
SE-412 96 Göteborg, Sweden}
\email{mattiasj@umich.edu}
\urladdr{http://www.math.lsa.umich.edu/~mattiasj/}
\thanks{S.B.\ was supported by the ANR project GRACK\@. M.J.\ was supported by NSF grants DMS-1266207 and DMS-1600011, a grant from the Knut and Alice Wallenberg foundation and a grant from the United States---Israel Binational Science Foundation.}
\subjclass{32Q25, 14J32, 14T05, 53C23, 32P05, 14G22}
\begin{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.
\end{abstract}
\alttitle{Limites tropicales et non archimédiennes de familles de formes volumes qui dégénèrent}
\begin{altabstract}
Nous étudions le comportement asymptotique de formes volumes dans une famille de variétés complexes compactes qui dégénèrent. Sous des conditions assez générales, nous montrons que les formes volumes convergent en un sens naturel vers une mesure du type de Lebesgue sur un certain complexe simplicial. Ceci fournit en particulier une version en théorie de la mesure d'une conjecture de Kontsevich--Soibelman et Gross--Wilson portant sur les dégénérescences maximales de variétés de Calabi-Yau.
\end{altabstract}
\keywords{Calabi-Yau manifolds, volume forms, degenerations, Berkovich spaces}
\altkeywords{Variétés de Calabi-Yau, formes volumes, dégénérescences, espaces de Berkovich}
\maketitle
\tableofcontents
\mainmatter
\section*{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 $(\nu_t)_{t\in\DD^*}$ of volume forms on a holomorphic family
$(X_t)_{t\in\DD^*}$ of compact complex manifolds is equivalent to
that of a proper holomorphic submersion $\pi\colon X\to\DD^*$
together with a smooth metric $\psi$ on the relative canonical bundle
$K_{X/\DD^*}$.
We say that the family $(\nu_t)$ has
\emph{analytic singularities} at $t=0$ if the following conditions
hold:
\begin{itemize}
\item[(i)]
$\pi\colon X\to\DD^*$ is meromorphic at $0\in\DD$ in the sense
that it extends to a proper, flat map
$\pi\colon\cX\to\DD$, with $\cX$ normal;
\item[(ii)]
$\cX$ can be chosen so that $K_{X/\DD^*}$ extends to a $\QQ$-line
bundle $\cL$ on $\cX$, and $\psi$ extends continuously to $\cL$.
\end{itemize}
When~(i) holds, we call $\cX$ a \emph{model} of $X$.
Using resolution of singularities,
we can always choose $\cX$ as an \emph{snc model}, that is,
$\cX$ is smooth and $\cX_0=\sum_{i\in I} b_i E_i$ has simple normal
crossing support. To $\cX$ is then associated a dual complex $\Delta(\cX)$,
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~\cite{MS}, we introduce a natural ``hybrid'' space
\begin{equation*}
\cX^\hyb:=X\coprod\Delta(\cX)
\end{equation*}
associated to $\cX$; it is equipped with a topology
defined in terms of a tropicalization map $X\to\Delta(\cX)$,
measuring the logarithmic rate of convergence of
local coordinates compatible with $\cX_0$.
Our first main result says that, after normalizing to unit mass,
the volume forms~$\nu_t$ admit a ``tropical'' limit inside $\cX^\hyb$.
\begin{ThmA}\label{th:A}
Let $(\nu_t)_{t\in\DD^*}$ be a family of volume forms on a
holomorphic family $X\to\DD^*$ of compact complex
manifolds, with analytic singularities at $t=0$.
The asymptotic behavior of the total mass of $\nu_t$ is then given by
\begin{equation*}
\nu_t(X_t)\sim c|t|^{2\kappa_{\min}}(\log|t|^{-1})^d
\end{equation*}
with $c\in\RR_+^*$, $\kappa_{\min}\in\QQ$ and $d\in\NN^*$, where $d\le n:=\dim X_t$.
Further, given any snc model $\cX\to\DD$
of $X\to\DD^*$ such that $K_{X/\DD^*}$ extends to a $\QQ$-line bundle
on $\cL$ on $\cX$ and $\psi$ extends to a continuous metric on $\cL$,
the rescaled measures
\begin{equation*}
\mu_t:=\frac{\nu_t}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^d},
\end{equation*}
viewed as measures on $\cX^\hyb$, converge weakly to a
Lebesgue type measure $\mu_0$ on a $d$-dimensional subcomplex
$\Delta(\cL)$ of $\Delta(\cX)$.
\end{ThmA}
The invariant $\kappa_{\min}$ and the subcomplex $\Delta(\cL)$ only depend on $\cL$ (and not on the metric on $\cL$). Consider the \emph{logarithmic relative canonical bundle}
$$
K^\lo_{\cX/\DD}:=K_\cX+\cX_{0,\red}-\pi^*(K_{\DD}+[0])=K_{\cX/\DD}+\cX_{0,\red}-\cX_0
$$
and write $K^{\lo}_{\cX/\DD}=\cL+\sum_{i\in I} a_i E_i$ with $a_i\in\QQ$. Setting $\kappa_i:=a_i/b_i$, we then have $\kappa_{\min}=\min_{i\in I}\kappa_i$, and $\Delta(\cL)$ is the subcomplex of $\Delta(\cX)$ 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
$$
\mu_0=\sum_{\sigma}\left(\int_{Y_\sigma}\Res_{Y_\sigma}(\psi)\right)b_\sigma^{-1}\lambda_\sigma.
$$
Here, $\sigma$ ranges over the $d$-dimensional faces of $\Delta(\cL)$, with corresponding strata $Y_\sigma\subset\cX_0$, $\Res_{Y_\sigma}(\psi)$ 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 $b_\sigma\in\ZZ_{>0}$ 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~\cite{AGZV}.
The assertions in Theorem~\ref{th: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~\cite[Th.\,1.2]{CLT10}.
It also appears in~\cite[\S3.1]{KS01} and is exploited in~\cite{BHJ2}.
The convergence result for the measures $\mu_t$ is also closely related
to~\cite[Cor.\,4.8]{CLT10}, where, however,
the limit measure lives on $\cX_0$ and not on
$\Delta(\cX)$.\footnote{A.~Chambert-Loir has pointed out
that~\cite[Cor.\,4.8]{CLT10} is sufficiently precise, so that when applying it to
toric blowups of $\cX$ one can see the form of the limit measure $\mu_0$
in Theorem~\ref{th:A}.}
The main new feature of Theorem~\ref{th: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~\ref{th:A}. First consider the subvariety
\begin{equation*}
\cX
:=\{(z_0^{n+1}+\dots+z_n^{n+1})+\epsilon tz_0\cdot\ldots\cdot z_n=0\}\subset\CC\times\PP^n,
\end{equation*}
where $0<\epsilon\ll1$.
Write $X:=\mathrm{pr}_1^{-1}(\CC^*)$.
The fiber $X_t$ over $t\in\DD^*$ 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 $K_{X/\DD^*}$
that extends continuously to $\cL=K_{\cX/\DD}$.
In the terminology of Theorem~\ref{th:A} we have
$\nu_t:=2^{-n}i^{n^2}\eta_t\wedge\overline\eta_t$.
Here $\cX_0$ is smooth, so $\Delta(\cX)$ 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
\begin{equation*}
\cX
:=\{t\epsilon(z_0^{n+1}+\dots+z_n^{n+1})+z_0\cdot\ldots\cdot z_n=0\}\subset\CC\times\PP^n.
\end{equation*}
In this case, $\Delta(\cL)=\Delta(\cX)$ 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~\ref{th:C} below.
We also prove a logarithmic version of Theorem~\ref{th:A}, for a log smooth klt pair $(X,B)$,
and a metric $\psi$ on $K_{(X,B)/\DD^*}$, see Theorem~\ref{T301}.
The space $\cX^\hyb$ and the measure $\mu_0$ depend on the choice of snc model $\cX$.
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~\S\ref{S315} we define a locally compact (Hausdorff) topological space
\begin{equation*}
X^\hyb:=\varprojlim_\cX \cX^\hyb,\vspace*{-3pt}
\end{equation*}
fibering over $\DD$, with central fiber
$X^\hyb_0:=\varprojlim\Delta(\cX)$.
For any $\cX$, the dual complex $\Delta(\cX)$ embeds
in the central fiber $X^\hyb_0$ of $X^\hyb$.
\begin{CorB}\label{cor:B}
With assumptions and notation as in Theorem~\ref{th:A},
the measures~$\mu_t$, viewed as measures on $X^\hyb$,
converge weakly to a measure $\mu_0$.
Further, $\mu_0$ is a Lebesgue type measure on a
$d$-dimensional complex in $X^\hyb_0$.
\end{CorB}
Now consider the case when $X\to\DD^*$ is projective. As we now
explain, the central fiber of $X^\hyb$ is then a \emph{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 $X_K^\an$.
Similarly, any projective snc model $\cX\to\DD$ of $X$ induces a
projective model $\cX_R$ over the valuation ring $R$ of $K$.
The dual complex $\Delta(\cX)$ then has a canonical realization as a
compact $\ZZ$-PA subspace $\Sk(\cX)\subset X_K^\an$, the \emph{skeleton}
of $\cX$. In fact, it is well known (see~\eg\cite{siminag}) that
there is a homeomorphism $X_K^\an\simto\varprojlim_\cX\Sk(\cX)$,
so we can identify the central fiber of the space $X^\hyb$ with
the analytification $X_K^\an$. In fact, as shown in Appendix~\ref{S319}, using
ideas from~\cite{BerkHodge}, we can view the restriction of
$X^\hyb\to\DD$ to a closed subdisc~$\overline{\DD}_r$ as the analytification of the base
change of $X$ to a suitable Banach ring $A_r$.
Assuming $X\to\DD^*$ is projective, we can describe the limit measure $\mu_0$ and its
support $\Sk(\cL)\simeq\Delta(\cL)$ inside $X_K^\an$ in more
detail. The skeleton $\Sk(\cL)$ is of purely non-Archimedean nature,
and can be seen as a mild generalization of the
\emph{Kontsevich--Soibelman skeleton}
introduced in~\cite{KS06} and studied in~\cite{MN,NX13,NX16}.
The \emph{skeletal measure} $\mu_0$, on the other hand, depends on
both Archimedean and non-Archimedean data.
Namely, it is supported on the skeleton $\Sk(\cL)$, but depends on
the choice of metric on the restriction of the line bundle $\cL$ to the central
fiber $\cX_0$ (viewed as a complex space) of any snc model $\cX$.
We also study both the skeleton and the skeletal measure in the more
general case when the model $\cX$ 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 $X\to\DD^*$ is a
projective holomorphic submersion, meromorphic at $0\in\DD$, such that
$K_{X/\DD^*}=\cO_X$.
Any trivializing section $\eta\in H^0(X,K_{X/\DD^*})$ 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 $\cX\to\DD$, $\eta$ extends to a nowhere vanishing section
of $\cL:=\cO_\cX$, and $\psi:=\log|\eta|$ defines a smooth metric on $\cL$.
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
$K_{X/\DD^*}$, whose asymptotic behavior at $t=0$ is described
in a very precise way by Schmid's nilpotent orbit
theorem~\cite[Th.\,4.9]{Sch} (compare for
instance~\cite[Prop.\,2.1]{GTZ13b}).
On the other hand, the skeleton $\Sk(\cL)$ described above coincides
in the current context with the Kontsevich--Soibelman skeleton
$\Sk(X)$~\cite{KS06,MN,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 $X\to\DD^*$
admits a relative minimal model $\cX$, with certain
mild (dlt) singularities~\cite{KNX}, and the essential skeleton can
be identified with the dual complex of $\cX$~\cite{NX13}. In
particular, $d=0$ if and only if $X$ can filled in with a central
fiber $\cX_0$ which is a Calabi--Yau variety with klt singularities.
At the other end of the spectrum, $d=n=\dim X_t$ if and only if $X$
is maximally degenerate, \ie a ``large complex structure limit''.
In that case, the essential skeleton $\Sk(X)$ is shown to be a
pseudomanifold in~\cite{NX13}. Building on this, we prove:
\begin{ThmC}\label{th:C}
Let $X\to\DD^*$ be a smooth projective family
of Calabi--Yau varieties, meromorphic at $0\in\DD$. Assume that $X$
is maximally degenerate and has semistable reduction.
Then the skeletal measure $\mu_0$
is a multiple of the integral affine Lebesgue measure on $\Sk(X)$.
\end{ThmC}
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
$\cX$ with $\cX_0$ reduced. This condition is always
satisfied after a finite base change.
Theorem~\ref{th:C} describes measure-theoretic degenerations of Calabi--Yau
varieties. Let us briefly discuss the case of \emph{metric}
degenerations.
Consider a smooth projective family $X\to\DD^*$ of Calabi--Yau
varieties, meromorphic at $0\in\DD$, and suppose the
family is polarized, that is, we are given a relative ample line
bundle $A$ on $X$. By Yau's theorem~\cite{Yau78}, each fiber $X_t$ carries a
unique Ricci-flat Kähler metric $\omega_t$ in the cohomology
class of $A_t$.
By~\cite{Wan,Tos15,Taka}, the diameter $D_t$ of $(X_t,\omega_t)$
remains bounded if and only if $d=0$, that is, $X$ admits a
model $\cX$ such that $\cX_0$ has klt singularities.
In this case, it is shown in~\cite{RZ11,RZ13},
building in part on~\cite{DS}, that $(X_t,\omega_t)$
converges in the Gromov-Hausdorff sense to the
Calabi--Yau variety $\cX_0$, endowed with the metric completion
of its singular Ricci-flat Kähler metric in the sense of~\cite{EGZ}.
The maximally degenerate case $d=n$ is the object of the \emph{Kontsevich--Soibelman
conjecture}~\cite{KS06}\footnote{Essentially the same
conjecture was stated independently by Gross--Wilson~\cite{GW00} and
Todorov.}, which states that $(X_t,D_t^{-2}\omega_t)$ (which has
diameter one) \hbox{converges} in the Gromov-Hausdorff sense to the essential
skeleton $\Sk(X)$ endowed with a piecewise smooth metric of
Monge-Ampère type, \ie locally given as the \hbox{Hessian} of a convex
function satisfying a real Monge-Ampère equation.
This conjecture has been verified for abelian varieties see
e.g.~\cite{OdakaCollapse} 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~\cite{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~\cite{GW00}.
Theorems~A and~C indicate a possible approach to the
Kontsevich--Soibelman conjecture. Indeed, recall that the metric
$\omega_t$ for $t\in\DD^*$ 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
$\MA(\phi_t)=\mu_t$.\enlargethispage{.1\baselineskip}%
On the central fiber $X^\hyb_0=X_K^\an$ of $X^\hyb$, it was shown
in~\cite{nama} that there exists a metric on the line bundle
$A_K^\an$, unique up to scaling, solving the
non-Archimedean Monge-Ampère equation
$\MA(\phi_0)=\mu_0$ (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 $t\to0$.
However, this seems to be a delicate issue since there is no a priori
reason why the weak continuity at $t=0$ of $t\mto\mu_t$ would imply continuity
of the solutions $t\mto\phi_t$.
Instead of Calabi-Yau manifolds, it would be interesting to study degenerating
families $X\to\DD^*$ 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~\S\ref{S313} we define
in~\S\ref{sec:hyb} the hybrid space $\cX^\hyb$
associated to an SNC model $\cX$.
The proof of Theorem~\ref{th:A} is given in~\S\ref{S314}.
In~\S\ref{S315} we define the space $X^\hyb$ associated to
a degeneration as an inverse limit of the spaces $\cX^\hyb$,
and prove Corollary~\ref{cor:B}.
Various notions of skeleta are defined and studied in~\S\ref{S316},
and in~\S\ref{sec:skeletal} 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~\S\ref{S317} where
we prove Theorem~\ref{th:C}. In~\S\ref{S312} we study various extensions,
and in the appendix we recall the Berkovich analytification
of a scheme over a Banach ring.
\subsubsection*{Acknowledgements}
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. Finally we thank the referees for useful comments.
\section{Preliminaries}\label{S313}
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~\ref{th:A}.
\subsection{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:
\begin{itemize}
\item[(i)] if for $i=1,2$, $\phi_i$ is a metric on a line bundle $L_i$ and $a_i\in\ZZ$, then $a_1\phi_1+a_2\phi_2$ is a metric on $a_1L_1+a_2L_2$;
\item[(ii)] a metric on the trivial line bundle $\cO_X$ is of the form $|\cdot|e^{-\phi}$ for a function $\phi$ on $X$, and we identify the metric with $\phi$.
\end{itemize}
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
\begin{equation*}
|s|_\phi:=|s|e^{-\phi}=\exp(\log|s|-\phi)
\end{equation*}
is the pointwise length of $s$ in the metric $\phi$.
A metric on a $\QQ$-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 $\cO_X(D)$ associated to any Cartier divisor
$D$ on $X$ comes with a canonical singular metric $\phi_D$,
smooth outside $D$. This fact extends to $\QQ$-divisors,
by interpreting $\phi_D$ as a metric on a $\QQ$-line bundle.
In the complex case at least, the curvature current of $\phi_D$,
correctly normalized, coincides with the integration current on $D$.
\subsection{Measures and forms}\label{S310}
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 a 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
\begin{equation*}
|\Omega|^2:=\frac{i^{n^2}}{2^n}\,|\Omega\wedge\bar\Omega|.
\end{equation*}
The normalization is chosen so that the measure associated to the form $dz=dx+idy$
on $\CC$ is Lebesgue measure $|dz|^2=|dx\wedge dy|$ on $\CC\simeq\RR^2$.
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
\begin{equation*}
e^{2\psi}
:=\frac{i^{n^2}|\Omega\wedge\bar\Omega|}{2^n|\Omega|^2_\psi}
=\frac{|\Omega|^2}{|\Omega|^2 e^{-2\psi}}
\end{equation*}
for any local section $\Omega$ of $K_X$. If $\psi'$ is another metric on $K_X$, then
$$
e^{2\psi'}=e^{2(\psi'-\psi)} e^{2\psi},
$$
where $e^{2(\psi'-\psi)}$ is the usual exponential of the smooth
function $2(\psi'-\psi)\in C^\infty(X)$. 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
$mK_X$, $m\in\ZZ$.
Now assume $(X,B)$ is a \emph{pair} in the sense of the Minimal
Model Program, \ie $X$ is a normal complex space and $B$ is a (not
necessarily effective) $\QQ$-Weil divisor on $X$ such that
\begin{equation*}
K_{(X,B)}:=K_X+B
\end{equation*}
is a $\QQ$-line bundle. Denote by $\phi_B$ the canonical singular metric on
$B|_{X_\reg}$, viewed as a $\QQ$-line bundle. If $\psi$ is smooth
metric on the $\QQ$-line bundle $K_{(X,B)}$, then $\psi-\phi_B$ is a smooth
metric on $K_{X_\reg\setminus B}$, and $e^{2(\psi-\phi_B)}$ is thus a
volume form on $X_\reg\setminus B$.\footnote{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 \emph{subklt} if for some
(or, equivalently, any) log resolution $\rho\colon X'\to X$ of $(X,B)$, the unique
$\QQ$-divisor $B'$ such that $\rho^*K_{(X,B)}=K_{(X',B')}$ and $\rho_*B'=B$ has
coefficients $<1$. The pair $(X,B)$ is \emph{klt} if $B$ is further effective.
\begin{lem}\label{lem:subklt}
For any continuous 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 $X$.
\end{lem}
\begin{proof}
With the above notation it is immediate to check that
$$
\rho^*e^{2(\psi-\phi_B)}=e^{2(\rho^*\psi-\phi_{B'})}.
$$
We are thus reduced to a log smooth pair $(X',B')$, \ie $X'$ is smooth
and $B'$ has snc support, and the proof is then trivial.
\end{proof}
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 \emph{adapted} in~\cite{EGZ,BBEGZ}.
\subsection{Integral piecewise affine spaces}\label{S301}
The following discussion roughly follows~\cite[p.\,59]{KKMS} and~\cite[\S1]{BerkContr}.
If $P$ is a rational polytope in $\RR^n$, that is,
the convex hull of a finite subset of~$\QQ^n$,
denote by $M_P\subset C^0(P)$ the finitely generated free abelian group obtained by restricting to $P$ affine functions with
coefficients in $\ZZ$ (constant term included). Denote by~$1_P$ the constant function on $P$ with value $1$, and set
$$
\vec{M}_P:=M_P/M_P\cap\QQ 1_P.
$$
Denote also by $b_P\in\NN$ the greatest integer such that
$b_P^{-1}1_P\in M_P$.
The data of $(P,M_P)$ modulo homeomorphism is called an
\emph{(abstract) $\ZZ$-polytope}. The functions in $M_P$ are called
\emph{integral affine}, or $\ZZ$-affine.
The evaluation map defines a canonical realization
$P\hto (M_P)^{^\vee}_\RR$ as a codimension one rational polytope,
with tangent space $T_P$ identified with $(\vec{M}_P)^{^\vee}_\RR$.
Further, the lattice $T_{P,\ZZ}:=\Hom(\vec{M}_P,\ZZ)\subset T_P$
yields a normalized Lebesgue measure $\lambda_P$ on $P$.
The main example for us is as follows.
\begin{lem}\label{lem:integral} Given $b_0,\dots,b_p\in\NN^*$, view
$$
\sigma=\bigl\{w\in\RR_+^{p+1}\bigm|\ts\sum_{i=0}^p b_i w_i=1\bigr\}
$$
as a $\ZZ$-simplex. Then $b_\sigma=\gcd(b_i)$, and
$$
\vol(\sigma)=\frac{b_\sigma}{p!\prod_i b_i}.
$$
\end{lem}
\begin{proof}
Note that $T_{\sigma,\ZZ}=\{w\in\ZZ^{p+1}\mid\sum_i b_i w_i=0\}$.
The linear isomorphism $\phi\colon\RR^{p+1}\to\RR^{p+1}$ given by
$\phi(w_j)=(b_j w_j)$ takes $\sigma$ to the standard simplex
$$
\sigma'=\bigl\{w'\in\RR_+^{p+1}\bigm|\ts\sum_i w'_j=1\bigr\},
$$
and hence
$$
[T_{\sigma',\ZZ}\colon\phi(T_{\sigma,\ZZ})]\vol(\sigma)=\vol(\sigma')=\frac{1}{p!}.
$$
Write $T_{\sigma',\ZZ}$ as the kernel of $\chi\colon\ZZ^{p+1}\to\ZZ$ defined
by $\chi(w')=\sum_i w'_i$. Then
\[
\phi(T_{\sigma,\ZZ})=\ker\chi\cap\phi(\ZZ^{p+1}),\quad\chi(\phi(\ZZ^{p+1}))=\gcd(b_i)\ZZ,
\]
and the exact sequence
$$
0\to\frac{\ker\chi}{\ker\chi\cap\phi(\ZZ^{p+1})}\to
\frac{\ZZ^{p+1}}{\phi(\ZZ^{p+1})}\to\frac{\ZZ}{\chi(\phi(\ZZ^{p+1}))}\to 0
$$
gives as desired
$$
[T_{\sigma',\ZZ}\colon\phi(T_{\sigma,\ZZ})]=\frac{\prod_i b_i}{\gcd(b_i)}.
$$
Finally, the first assertion is clear.
\end{proof}
\begin{rmk}\label{R301}
By setting $w_0=b_0^{-1}(1-\sum_{i=1}^pb_iw_i)$, we can identify $\sigma$
with the simplex $\sum_1^p b_iw_i\le 1$ in $\RR_+^p$. The normalized
Lebesgue measure on $\sigma$ is then given by
$\lambda_\sigma=b_\sigma^{-1}|dw_1\wedge\dots\wedge dw_p|$.
\end{rmk}
A \emph{compact rational polyhedron} $K$ in $\RR^n$ 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 \emph{subdivision} of
$K$, and call the subdivision \emph{simplicial} if each $P_i$ is a
simplex. A continuous function on $K$ is \emph{integral piecewise affine}
($\ZZ$-PA for short) if $f|_{P_i}\in M_{P_i}$ for some subdivision of
$K$. These functions form a subgroup $\PA_\ZZ(K)\subset C^0(K)$,
and the data of $(K,\PA_\ZZ(K))$ modulo homeomorphism is called a
\emph{compact $\ZZ$-PA space}.
The \emph{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 $\ZZ$-polytopes.
Note that a $\ZZ$-polytope $P$ can be regarded as a $\ZZ$-PA space
and that $M_P\subset\PA_\ZZ(P)$.
\subsection{Tropicalizations and polar coordinates}\label{S302}
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~\ref{thm:genconv} (which implies Theorem~\ref{th:A}).
Let $N\simeq\ZZ^{p+1}$ be a lattice, $M=\Hom(N,\ZZ)$ the dual lattice,
$\CC[M]$ the semigroup ring and $T=\Spec\CC[M]=N\otimes\CC^*$ the algebraic torus.
A basis for $N$ induces a dual basis
$(m_0,\dots,m_p)$ for $M$ and elements $z_i\in\CC[M]$, $0\le i\le p$,
such that $\CC[M]=\CC[z_0^{\pm1},\dots,z_p^{\pm1}]$ and $T\simeq(\CC^*)^{p+1}$.
Let $\Omega\in H^0(T,K_T)$ be the $T$-invariant global section
given in coordinates by
\begin{equation*}
\Omega=\frac{dz_0}{z_0}\wedge\dots\wedge\frac{dz_p}{z_p}.
\end{equation*}
Note that $\Omega$ is independent of the choice of coordinates, up
to a sign. Its associated measure
\begin{equation*}
\rho:=|\Omega|^2
\end{equation*}
is $T$-invariant, and hence a Haar measure on $T$.
We can write this measure in (logarithmic) polar coordinates via the canonical
\emph{tropicalization map} $L\colon T\to N_\RR$, given in the basis above by
\begin{equation*}
L=(-\log|z_0|,\dots,-\log|z_p|).
\end{equation*}
Note that $L$ sits in the exact sequence $1\to K\to T\to N_\RR$ obtained by tensoring with~$N$ the exact sequence $1\to S^1\to\CC^*\to\RR\to 0$ induced by $z\mto-\log|z|$. In~particular, $K=N\otimes S^1\simeq(S^1)^{p+1}$ is a compact torus, and $L\colon T\to N_\RR$ is a principal $K$-bundle.
On the one hand, let $\omega$ be the translation invariant real $(p+1)$-form on the
tropical torus $N_\RR\simeq\RR^{p+1}$ given by
\begin{equation*}
\omega=dm_0\wedge\dots\wedge dm_p.
\end{equation*}
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 $N_\RR$ normalized by $N$.
On the other hand, since $L:T\to N_\RR$ 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
$$
\rho=(2\pi)^{p+1}\lambda(dw)\otimes\rho_w,
$$
\ie
\begin{equation}\label{e302}
\int_Tf\,d\rho
=(2\pi)^{p+1}\int_{N_\RR}\left(\int_{K_w}f\,d\rho_w\right)\lambda(dw),
\end{equation}
for any $f\in C^0_c(T)$. Concretely, we can use logarithmic polar coordinates on $T$:
\begin{equation*}
z_j=\exp(-w_j+2\pi i\theta_j)
\end{equation*}
for $0\le j\le p$; then $\rho_w=|d\theta_0\wedge\dots\wedge
d\theta_p|$,
and
\begin{equation*}
\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.
\end{equation*}
We will need the same analysis on certain subgroups of $T$.
Fix an element $m\in M$ and let $\chi=\chi^m\colon T\to\CC^*$
be the corresponding character. Let $b\in\ZZ_{>0}$ be the largest
integer such that $b^{-1}m\in M$.
In the bases above, we can write
$m=\sum_{i=0}^pb_im_i$ and $\chi=\prod_iz_i^{b_i}$,
where $b_i\in\ZZ$; then $b=\gcd_ib_i$.
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 $t\in\CC^*$, $T_t:=\chi^{-1}(t)$ is a complex manifold
with $b$ connected components. Note that $T':=X_1$ is an algebraic subgroup of $T$
and that $T_t$ is a torsor for $T'$ for any $t\in\nobreak\CC^*$. The $T$-invariant $(p+1)$-form $\Omega$ induces in a canonical way a $T'$\nobreakdash-invariant $p$\nobreakdash-form~$\Omega_t$ on $T_t$, obtained as the restriction to $T_t$ of any choice of holomorphic $p$\nobreakdash-form~$\Omega'$ on $T$ such that
$\frac{d\chi}{\chi}\wedge\Omega'=\Omega$.
In general coordinates as above, we can pick
\begin{equation*}
\Omega'
=\frac1{\#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},
\end{equation*}
where $J=\{j\mid b_j\ne 0\}$.
In special coordinates, so that $m=bm_0$ and $\chi=z_0^b$,
we then have $\Omega'=\frac1b\frac{dz_1}{z_1}\wedge\dots\wedge\frac{dz_p}{z_p}$, and hence
\begin{equation*}
T_t=\bigcup_{u^b=t}\{z_0=u\}
\qand
\Omega_t
=\frac1b\,\frac{dz_1}{z_1}\wedge\dots\wedge\frac{dz_p}{z_p}
\bigg|_{T_t}.
\end{equation*}
Note that $\rho_1:=|\Omega_1|^2$ is Haar measure on $T'$, whereas
$\rho_t:=|\Omega_t|^2$ is a $T'$-invariant measure on $T_t$.
In the special case $p=0$, $T_t$ consists of $b$ points, and $\rho_t$
gives mass~$\sfrac1{b^2}$ to each of them.
Next we study the analogous situation in the tropical torus $N_\RR$.
Viewing $m$ as a linear form on $N_\RR$, set $H_s:=m^{-1}(s)$ for $s\in\RR$.
The lattice $N'=\Ker m\subset N$ defines an integral affine
structure on $H_s$, and hence a normalized Lebesgue measure $\lambda_s$. Note that
\begin{equation*}
|\omega'|_{H_s}|=\frac1b\,\lambda_s
\end{equation*}
for any choice of $p$-form $\omega'$ on $N_\RR$ such that
$dm\wedge\omega'=\omega$. In general coordinates, we pick
\begin{equation*}
\omega'
=\frac1{\#J}\sum_{j\in J}\frac{(-1)^j}{b_j}\,
dm_0\wedge\dots\wedge\widehat{dm_j}
\wedge\dots\wedge dm_p,
\end{equation*}
where $J=\{j\mid b_j\ne 0\}$.
In special coordinates, $\omega'=(\sfrac1b) dm_1\wedge\dots\wedge dm_p$.
Finally we describe $\rho_t$ in polar coordinates.
The tropicalization map $L\colon T\to N_\RR$ induces a principal $T'\cap K$-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
$$
\rho_t=\frac{(2\pi)^p}{b}\,\lambda_s(dw)\otimes\rho_{t,w},
$$
\ie
\begin{equation}\label{e301}
\int_{T_t}f\,d\rho_t=\frac{(2\pi)^p}{b}\int_{H_s}\biggl(\int_{K_{t,w}}f\,\rho_{t,w}\biggr)\lambda_s(dw),
\end{equation}
for any $f\in C^0_c(T_t)$, where $s=\log|t|^{-1}$.
The proof is essentially the same as that of~\eqref{e302}.
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 $T_t^{(\ell)}$, $1\le
\ell\le b$, and
\begin{equation*}
\rho_t
=|\Omega_t|^2
=\frac1{b^2}\left|\frac{dz_1}{z_1}\wedge\dots\wedge\frac{dz_p}{z_p}\right|^2.
\end{equation*}
The restriction of the tropicalization map to $T_t^{(\ell)}$ amounts to the change of coordinates
$z_j=u_j^{(\ell)}\exp(-w_j+2\pi i\theta_j)$ for $1\le j\le p$, where the
$u_j^{(\ell)}$ are constants with $|u_j^{(\ell)}|=1$. In these coordinates,
\begin{equation*}
\rho_t|_{T_t^{(\ell)}}
=\frac{(2\pi)^p}{b^2}\,|dm_1\wedge\dots\wedge dm_n|\otimes
|d\theta_1\wedge\dots\wedge d\theta_p|.
\end{equation*}
Here $(\sfrac1b)|d\theta_1\wedge\dots\wedge d\theta_p|$ 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~\eqref{e301} follows.
\section{The hybrid space associated to an snc model}\label{sec:hyb}
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~\cite[\S I.3]{MS}, and can even be traced back to the pioneering work of Bergman~\cite{Berg}.
\subsection{The dual complex}\label{sec:dual}
Let $D$ be an effective divisor with simple normal crossing (snc) support in a
complex manifold $\cX$. By definition, $D=\sum_{i\in I} b_i E_i$ with $b_i\in\NN^*$
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 $\emptyset\ne J\subset I$.
A connected component $Y$ of a non-empty $E_J$ is called a
\emph{stratum}. Together with $\cX\setminus D=E_\emptyset$, the locally closed
submanifolds $\mathring{Y}:=Y\setminus\bigcup_{i\in I\setminus J} E_i$
define a partition of $\cX$.
The \emph{dual complex} $\Delta(D)$ is the simplicial complex\footnote{This is understood in the slightly generalized sense that the intersection of two faces is a \emph{union} of common faces.} defined as follows: to each stratum $Y$ corresponds a simplex
$$
\sigma_Y=\left\{w\in\RR_+^J\mid\ts\sum_{i\in J} b_i w_i=1\right\},
$$
and $\sigma_{Y}$ is a face of $\sigma_{Y'}$ if and only if $Y'\subset Y$.
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 $\ZZ$-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 $\Delta(D)$.
\subsection{The hybrid topology}\label{sec:hybtop}
Next we define a natural topology on the disjoint union
$$
\cX^\hyb:=(\cX\setminus D)\coprod\Delta(D).
$$
Consider a connected open set $\cU\subset \cX$ meeting $D$ and
local coordinates $z=(z_0,\dots,z_n)$ on $\cU$.
We say that the pair $(\cU,z)$
is \emph{adapted} (to $D$) if the following conditions hold:
\begin{itemize}
\item[(i)]
if $E_0,\dots,E_p$ are the irreducible components of $D$
intersecting $\cU$, then
\hbox{$\cU\cap E_0\cap\dots\cap E_p$}, if nonempty, equals
$\cU\cap\mathring{Y}$
for a component $Y$ of $E_0\cap\dots\cap E_p$;
\item[(ii)]
$z_i$ is an equation of $E_i\cap \cU$ with $|z_i|<1$, $0\le i\le p$.
\end{itemize}
We call $Y=Y_\cU$ the stratum of $\cU$, and denote by
$$
\sigma_\cU=\left\{w\in\RR^{p+1}\mid\ts\sum_{i=0}^p b_i w_i=1\right\}
$$
the corresponding face of $\Delta(D)$.
The function $f_{\cU,z}:=\prod_{i=0}^p z_i^{b_i}$ is an equation of $D$
in $\cU$, with $|f_{\cU,z}|<1$,
and we get a continuous map $\Log_{\cU}\colon \cU\setminus D\to\sigma_Y$ by setting
$$
\Log_{\cU}=\left(\frac{\log|z_i|}{\log|f_\cU|}\right)_{0\le i\le p}.
$$
For any two adapted coordinate charts $(\cU,z)$, $(\cU',z')$, with the same stratum $Y$,
we have $z'_i=u_i z_i$ with $u_i$ nonvanishing on $\cU\cap \cU'$, for
$i=0,\dots,p$ (after a possible reindexing); it follows that
\begin{equation}\label{equ:Log'}
\Log_{\cU'}=\Log_{\cU}+O\left(\frac{1}{\log|f_{\cU,z}|^{-1}}\right)
\end{equation}
locally uniformly on $\cU\cap \cU'$. We next show how to globalize this construction.
\begin{prop}\label{prop:globlog}
There exists an open neighborhood
$\cV\subset \cX$ of $D$ and a continuous map
$\Log_\cV\colon \cV\setminus D\to\Delta(D)$ such that for each adapted
coordinate chart $(\cU,z)$ with $\cU\subset \cV$ we have
$\Log_\cV(\cU\setminus D)\subset\sigma_\cU$ and
\begin{equation}\label{equ:globlog}
\Log_\cV=\Log_{\cU}+O\left(\frac{1}{\log|f_{\cU,z}|^{-1}}\right)
\end{equation}
uniformly on compact subsets of $\cU$.
\end{prop}
This will be accomplished by means of a partition of unity, using the following elementary special case of~\cite[Th.\,5.7]{Cle}.
\begin{lem}\label{lem:adapted}
There exists a family $((\cV_\alpha,z_\alpha))_{\alpha\in A}$ of adapted
coordinate charts, such that $(\cV_\alpha)_\alpha$ forms a locally finite
covering of $D$ and such that the strata $Y_\alpha$ of the $\cV_\alpha$ satisfy
\begin{equation}\label{equ:interadapted}
\bigcap_{\beta\in B} \cV_\beta\ne\emptyset\Longrightarrow\bigcap_{\beta\in B} Y_\beta\ne\emptyset
\end{equation}
for every finite $B\subset A$.
\end{lem}
\begin{proof}[Proof of Proposition~\ref{prop:globlog}]
Pick an open
cover $(\cV_\alpha)_\alpha$ as in Lemma~\ref{lem:adapted}, and denote by
$\Log_\alpha\colon \cV_\alpha\setminus D\to\sigma_\alpha$ the corresponding maps.
Set $\cV:=\bigcup_\alpha \cV_\alpha$, and pick a partition of unity $(\chi_\alpha)$
subordinate to $(\cV_\alpha)$.
We claim that for each $\xi\in \cV$ there exists an open neighborhood
$W$ of $\xi$
and a face $\sigma_W$ of $\Delta(D)$ such that
$$
W\cap\supp\chi_\alpha\ne\emptyset\Longrightarrow\sigma_\alpha\subset\sigma_W
$$
for any $\alpha\in A$. Indeed, using~\eqref{equ:interadapted} it is easy to see that
\begin{equation*}
W:=\bigcap_{\alpha\mid \xi\in \cU_\alpha} \cV_\alpha\setminus
\bigcup_{\alpha\mid\xi\notin\supp\chi_\beta}\supp\chi_\beta
\end{equation*}
satisfies this property. By convexity of $\sigma_W$, it follows that
$\Log_\cV:=\sum_\alpha\chi_\alpha\Log_{\cV_\alpha}$ is well-defined on
$W\setminus D$, and hence yields a continuous map
$\Log_\cV\colon \cV\setminus D\to\Delta(D)$.
The last property is a direct consequence of~\eqref{equ:Log'}.
\end{proof}
We extend the previous map as
$$
\Log_\cV\colon \cV^\hyb:=(\cV\setminus D)\cup\Delta(D)\to\Delta(D)
$$
by setting $\Log_\cV=\id$ on $\Delta(D)$.
\begin{defi}\label{defi:hybtop}
The \emph{hybrid topology} on $\cX^\hyb:=(\cX\setminus D)\cup\Delta(D)$
is defined as the coarsest topology such that:
\begin{itemize}
\item[(i)]
$\cX\setminus D\hto \cX^\hyb$ is an open embedding;
\item[(ii)]
For every open neighborhood $\cV$ of $D$ in $\cX$, the
set $(\cV\setminus D)\cup\Delta(\cX)$ is open in $\cX^\hyb$;
\item[(iii)]
$\Log_\cV\colon \cV^\hyb\to\Delta(D)$ is continuous.
\end{itemize}
\end{defi}
Using~\eqref{equ:globlog}, this definition is easily seen to be
independent of the choice of map $\Log_\cV$. If $D$ is compact and $K\subset
\cX$ is a compact neighborhood of $D$, then one easily checks that the
corresponding subset $K^\hyb=(K\setminus D)\cup\Delta(D)$ is compact
(Hausdorff).
When $D=b_0E_0$ has only one irreducible component,
$K^\hyb$ is simply the Tychonoff one-point
compactification of $K\setminus D$.
\begin{ex}\label{E301}
Set $\cX=\DD^2$ and $D=E_0+E_1$ the union of the coordinate
axes, with coordinates $(z_0,z_1)$. Then $\cU=\cX$ is itself an
adapted coordinate chart. In these coordinates,
$\Log_{\cU}\colon \cU\setminus D\to\sigma_\cU$ becomes the
map $(\DD^*)^2\to[0,1]$
sending $(z_0,z_1)$ to $\log|z_1|/\log|z_0z_1|$.
As a consequence, given $\zeta\in\RR_+^*$ and $0<\epsilon\ll1$,
the closure in $\cX^\hyb$ of the closed subset
\begin{equation*}
F_\epsilon:=\{0<|z_0|,\;|z_1|\le\epsilon,\; |z_0|^{\zeta+\epsilon}\le|z_1|\le|z_0|^{\zeta-\epsilon}\}\subset\DD^2
\end{equation*}
is given by $\bF_\epsilon=F_\epsilon\cup I_\epsilon$, where
\[
I_\epsilon:=\Bigl\{w\in[0,1]\Bigm| \frac{\zeta-\epsilon}{1+\zeta-\epsilon}\le w\le\frac{\zeta+\epsilon}{1+\zeta+\epsilon}\Bigr\}.
\]
Further, the sets $\bF_\epsilon$, for $0<\epsilon\ll1$ form a basis of closed neighborhoods of
the point $\sfrac{\zeta}{(1+\zeta)}\in\nobreak[0,1]$ in $\cX^\hyb$.
See Figure~\ref{F301}.
\end{ex}
\begin{figure}
\includegraphics[width=12cm]{boucksom-jonsson_fig}
\caption{The figure shows the closed subset $F_\epsilon$ in Example~\ref{E301}.}\label{F301}
\end{figure}
\section{Proof of Theorem~\ref{th:A}}\label{S314}
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.
\subsection{Residual measures}\label{sec:rescplex}
Let $\pi\colon\cX\to\DD$ be an snc degeneration, \ie a proper, surjective
holomorphic map from a connected complex manifold to the
unit disc in~$\CC$, whose restriction to $X:=\pi^{-1}(\DD^*)$ is a submersion
and such that $\cX_0:=\pi^{-1}(0)=\sum_{i\in I} b_i E_i$ has snc support.
Note that $X_t:=\pi^{-1}(t)$ is non-singular for $t\in\DD^*$.
The \emph{dual complex} $\Delta(\cX)$ is defined as that of $\cX_0$;
it is equipped with its natural $\ZZ$-PA structure.
The \emph{logarithmic canonical bundle} of $\cX$ is
$$
K^\lo_\cX:=K_\cX+\cX_{0,\red}.
$$
Setting $K^\lo_{\DD}:=K_\DD+[0]$, we define the \emph{relative logarithmic canonical bundle} as
$$
K^\lo_{\cX/\DD}:=K^\lo_\cX-\pi^*K^\lo_{\DD}=K_{\cX/\DD}+\cX_{0,\red}-\cX_0.
$$
Now suppose we are given a $\QQ$-line bundle $\cL$ on $\cX$
extending $K_{X/\DD^*}$. We then have a unique decomposition
\begin{equation*}
K^\lo_{\cX/\DD}=\cL+\sum_{i\in I}a_i E_i
\end{equation*}
with $a_i\in\QQ$. Set $\kappa_i:=a_i/b_i$ and $\kappa_{\min}:=\min_i\kappa_i$.
\begin{defi} We denote by $\Delta(\cL)$ the subcomplex of $\Delta(\cX)$ such that a face $\sigma$ of $\Delta(\cX)$ is in $\Delta(\cL)$ if and only if each vertex of $\sigma$ achieves $\min_i\kappa_i$.
\end{defi}
In general, $\Delta(\cL)$ is neither connected nor pure dimensional.
We say that a face of $\Delta(\cL)$ is \emph{maximal} if it is not
contained in a larger face of $\Delta(\cL)$.
\begin{lem}
Let $Y\subset\cX_0$ be a stratum corresponding to face $\sigma$ of $\Delta(\cX)$,
and denote by $J\subset I$ the set of irreducible
components $E_i$ cutting out $Y$. Then
\begin{equation*}
B^\cL_Y:=\sum_{i\notin J}(1-(a_i-\kappa_{\min}b_i))E_i|_Y
\end{equation*}
is a $\QQ$-divisor on $Y$ with snc support, and we have a canonical identification
\begin{equation*}
\cL|_Y=K_{(Y,B^\cL_Y)}:=K_Y+B^\cL_Y
\end{equation*}
as $\QQ$-line bundles. If we further assume that $\sigma$ is a
maximal face of $\Delta(\cL)$, then $B^\cL_Y$ has coefficients $<1$,
so the pair $(Y,B^\cL_Y)$ is subklt.
\end{lem}
\begin{proof}
The first point is a simple consequence of the triviality of the
normal bundle
$\cO_{\cX_0}(\cX_0)$ together with the adjunction formula
\begin{equation*}
K_Y=(K_\cX+\sum_{i\in J} E_i)|_Y,
\end{equation*}
canonically realized by Poincaré residues once an order on $J$ has
been chosen.
When~$\sigma$ is a maximal face of $\Delta(\cL)$, each $E_i$ meeting $Y$
properly satisfies $\kappa_i>\kappa_{\min}$, which implies that $B^\cL_Y$
has coefficients $<1$.
\end{proof}
If $\psi$ is a continuous metric on $\cL$, $\psi|_Y$ may thus be viewed
as a metric on $K_{(Y,B^\cL_Y)}$.
When $\sigma$ is a maximal face of $\Delta(\cL)$,
the pair $(Y,B^\cL_Y)$ is subklt,
and Lemma~\ref{lem:subklt} applies.
This leads to the following notion.
\begin{defi}
Let $Y$ be a stratum corresponding to a maximal face of
$\Delta(\cL)$. The \emph{residual measure} on $Y$ of a continuous metric
$\psi$ on $\cL$ is the (finite) positive measure on $Y$ defined by
\begin{equation*}
\Res_Y(\psi):=\exp\bigl(2(\psi|_Y-\phi_{B^\cL_Y})\bigr).
\end{equation*}
\end{defi}
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 $\cX_0$ that pass through $\xi$,
indexed so that $J=\{0,\dots,d\}$, where $0\le d\le p$,
and such that $t=\prod_{j=0}^pz_j^{b_j}$
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 $K^\lo_{\cX}$, hence induces a local trivialization
\hbox{$\Omega^\rel=\Omega\otimes(dt/t)^{-1}$} of $K^\lo_{\cX/\DD}$. We may then view $\tau:=\prod_{i=0}^p z_i^{a_i}\Omega^\rel$ as a local $\QQ$-generator of $\cL$. Under the identification $\cL|_Y=K_{(Y,B^\cL_Y)}$, we have
\begin{equation*}
\tau|_Y=\prod_{i=d+1}^p z_i^{a_i-\kappa_{\min} b_i}\Res_Y(\Omega)
\end{equation*}
with
\begin{equation*}
\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.
\end{equation*}
We infer
\begin{equation}\label{equ:rescplex}
\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}^ndz_i\bigg|^2.
\end{equation}
\subsection{Statement and first reductions}\label{S320}
It will be convenient to introduce the quantity
\begin{equation*}
\lambda(t):=(\log|t|^{-1})^{-1},
\end{equation*}
for $t\in\DD^*$. Note that $\lambda(t)\to0$ as $t\to 0$.
Let $\cX^\hyb:=X\coprod\Delta(\cX)$ be the locally compact
hybrid space constructed in~\S\ref{sec:hyb}. It comes with a
proper map $\pi\colon\cX^\hyb\to\Delta$ extending
$\pi\colon X\to\DD^*$ and such that $\Delta(\cX)=\pi^{-1}(0)$.
The next result implies Theorem A in the introduction.
\begin{thm}\label{thm:genconv}
Let $\pi\colon\cX\to\DD$ be an snc degeneration,
$\cL$ a $\QQ$-line bundle on $\cX$ extending $K_{X/\DD^*}$,
and $\psi$ a continuous metric on $\cL$.
Define $\kappa_{\min}$ as above, and set $d:=\dim\Delta(\cL)$.
Then, viewed as measures on $\cX^\hyb$,
\begin{equation*}
\mu_t:=\frac{\lambda(t)^d}{(2\pi)^d|t|^{2\kappa_{\min}}}\,e^{2\psi_t}
\end{equation*}
converges weakly to
\begin{equation*}
\mu_0:=\sum_{\sigma}\left(\int_{Y_\sigma}\Res_{Y_\sigma}(\psi)\right)b_\sigma^{-1}\lambda_\sigma,
\end{equation*}
where $\sigma$ ranges over the $d$-dimensional faces of $\Delta(\cL)$.
Here $\lambda_\sigma$ denotes normalized Lebesgue measure on $\sigma$
and $b_\sigma=\gcd_{i\in J}b_i$, where $\cX_0=\sum_i b_iE_i$
and $E_i$, $i\in J$ are the divisors defining $\sigma$.
\end{thm}
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
$\cO_\cX(\cX_0)$, and hence a smooth metric $\log|t|$, so we may replace
$\cL$ and $\psi$ with $\cL-\kappa_{\min}\cX_0$ 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 $a_i\ge 0$, with
equality if and only if $E_i$ corresponds to a vertex of $\Delta(\cL)$.
Next we reduce the assertion of Theorem~\ref{thm:genconv} to a local problem. Let $Y\subset\cX_0$ be the stratum of an arbitrary face $\sigma$ of $\Delta(\cX)$, and denote by $E_0,\dots,E_p$ the components of $\cX_0$ cutting out $Y$, ordered so that
$$
\kappa_0=\dots=\kappa_q<\kappa_{q+1}\le\dots\le\kappa_p.
$$
We can then make the identification
$$
\sigma=\bigl\{w\in\RR_+^{p+1}\bigm| b\cdot w=1\bigr\}
$$
with $b=(b_0,\dots,b_p)\in\ZZ_{>0}^{p+1}$.
Set $b'=(b_0,\dots,b_q)\in\ZZ_{>0}^{q+1}$ and
$$
\sigma':=\bigl\{w'\in\RR_+^{q+1}\bigm| b'\cdot w'=1\bigr\}.
$$
Then $\sigma'$ is a face of $\sigma$ under the embedding
$\RR_+^{q+1}\hto\RR_+^{p+1}$ given by $w'\to(w',0)$.
Let $Y'\supset Y$ be the corresponding stratum of $\cX_0$.
Note that $\sigma$ contains a face of $\Delta(\cL)$ if and only if $\kappa_0=0$; in that case, the face is unique, equal to $\sigma'$ (which then implies $q\le d$).
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 $0\le i\le p$ and
$$
t=\prod_{i=0}^p z_i^{b_i}.
$$
We may assume that $z$ is defined on a polydisc $\cU\simeq\DD(r)^{p+1}\times\DD^{n-p}$ with $00$ or $q0$
for $i>q$, it is easy to see that
\begin{equation*}
\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}).
\end{equation*}
By~\eqref{equ:normmeas}, it follows that
\begin{equation}\label{equ:massK}
\mu_t(U_t)=O(\lambda(t)^{d-q}|t|^{2\kappa_0}),
\end{equation}
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
\begin{equation}\label{equ:cvut}
\int_{U_t}(\varphi\circ\Log_{\cU})\chi\,\mu_t
\to\left(\int_{\sigma'}\varphi b_{\sigma'}^{-1}\lambda_{\sigma'}\right).
\left(\int_{Y'}\chi\Res_{Y'}(\psi)\right).
\end{equation}
Let us first express both sides of~\eqref{equ:cvut}
in logarithmic polar coordinates.
We start by the left-hand side.
Set $f:=\chi e^{2g}\in C^0(\cU)$.
By~\eqref{equ:normmeas} and Lemma~\ref{L301} we have
\begin{multline}\label{equ:normbis}
(2\pi)^{d-p}\int_{U_t}(\varphi\circ\Log_{\cU})\chi\,\mu_t\\
=\lambda(t)^{d-p}
\int\limits_{\sigma\times(Y\cap\cU)}
\varphi(w)
e^{-2\lambda(t)^{-1}a''\cdot w''}
b_\sigma^{-1}\lambda_\sigma(dw)\otimes|dy|^2
\int f\,\rho_{t,w,y}\\
=\int\limits_{\sigma'\times\RR_+^{p-d}\times(Y\cap\cU)}
H_t(w',x'')
b_{\sigma'}^{-1}\lambda_{\sigma'}(dw')\otimes|dx''|\otimes|dy|^2
\int f\,\rho_{t,w',x'',y},
\end{multline}
where
\begin{equation*}
H_t(w',x'')
=\mathbf{1}_{\sigma_t}
\varphi(Q_t(w',x''))e^{-2a''\cdot x''}(1-\lambda(t)b''\cdot x'')^d,
\end{equation*}
and $\rho_{t,w',x'',y}$ is the same measure as $\rho_{t,w,y}$
via the identification $Q_t(w',x'')=w$.
Note that $\lim_{t\to 0}Q_t(w',x'')=(w',0)$, so
\begin{equation*}
\lim_{t\to0}H_t(w',x'')
=\mathbf{1}_{\sigma'\times\RR_+^{p-d}}
\varphi(w',0)e^{-2a''\cdot x''}.
\end{equation*}
Consider the tropicalization map
\begin{equation*}
S\colon Y'\cap\cU\to\RR_+^{p-d}\times(Y\cap\cU)
\end{equation*}
given by $S=(-\log|z_{d+1}|,\dots,-\log|z_p|,y)$.
Each fiber $S^{-1}(x'',y)$ is a torsor for the
compact torus $(\RR/\ZZ)^{p-d}$ and hence carries a
unique invariant probability measure $\rho_{x'',y}$.
As $t\to0$, the probability measure $\rho_{t,w',x'',y}$
converges weakly to $\rho_{x'',y}$ for any $w'\in\sigma'$.
By dominated convergence it follows that
\begin{multline}\label{e303}
\lim_{t\to0}(2\pi)^{d-p}\int_{U_t}(\varphi\circ\Log_{\cU})\chi\,d\mu_t\\
=\int_{\sigma'\times\RR_+^{p-d}\times(Y\cap\cU)}
\varphi(w',0)e^{-2a''\cdot x''}b_{\sigma'}^{-1}\lambda_{\sigma'}(dw')\otimes|dx''|\otimes|dy|^2
\int f\,\rho_{x'',y}\\
=\left(\int_{\sigma'}\varphi b_{\sigma'}^{-1}\lambda_{\sigma'}\right)
\biggl(\int_{\RR_+^{p-d}}e^{-2a''\cdot x''}|dx''|\int_{Y\cap\cU}|dy|^2
\int f\,\rho_{x'',y}\biggr).
\end{multline}
It only remains to compare the second factor of~\eqref{e303}
to the second factor in~\eqref{equ:cvut}.
To this end, we again use logarithmic polar coordinates.
We have
\begin{equation}\label{equ:rescplexbis}
\chi\Res_{Y'}(\psi)
=f\prod_{i=d+1}^p|z_i|^{2a_i-2}|dz''|^2\otimes|dy|^2.
\end{equation}
For $d0}$.
We can realize the simplex $\sigma$ (\resp $\sigma'$)
as the subset $\{\sum_{i=0}^pb_iw_i=1\}\subset\RR_+^{p+1}$
(\resp $\{\sum_{j=0}^{p'}b'_jw'_j=1\}\subset\RR_+^{p'+1}$),
where $b_i$ (\resp $b'_j$) is the multiplicity of $E_i$ in $\cX_0$
(\resp of $E'_j$ in $\cX'_0$).
The restriction of $r_{\cX\cX'}$ to $\sigma'$ is then given by\vspace*{-5pt}\enlargethispage{.5\baselineskip}%
\begin{equation}\label{e305}
w_i=\sum_{j=0}^{p'} a_{ij}w'_j.\vspace*{-3pt}
\end{equation}
for $0\le i\le p$.
It is clear that $r_{\cX\cX'}$ defines a continuous, integral affine
map from $\Delta(\cX')$ to $\Delta(\cX)$. Further, if $\cX$, $\cX'$ and
$\cX''$ are snc models with~$\cX''$ dominating~$\cX'$, and
$\cX'$ dominating $\cX$, then $r_{\cX\cX'}\circ r_{\cX'\cX''}=r_{\cX\cX''}$.
In general, it may happen that $\rho(Y')$ is a strict subvariety of $Y$,
and the linear map defining $r_{\cX\cX'}|_{\sigma'}$ could fail to be injective or surjective.
\begin{defi}
With notation as above, we say that
$\sigma'$ is \emph{active} for $r_{\cX\cX'}$
if the restriction $\rho|_{Y'}\colon Y'\to Y$ is a bimeromorphic morphism
and the $\QQ$-linear map defining $r_{\cX\cX'}|_{\sigma'}$ is an isomorphism.
In this case, $\sigma'$ and $\sigma$ have the same dimension, and
$r_{\cX\cX'}$ maps $\sigma'$ homeomorphically
onto a $\ZZ$-subsimplex of $\sigma$ of the same dimension.
\end{defi}
Denote by $A_{\cX\cX'}$ the union of all simplices in
$\Delta(\cX')$ that are active for $r_{\cX\cX'}$.
Our goal in this subsection is to prove the following result.
\begin{prop}\label{P301}
Let $\cX$ and $\cX'$ be snc models, with $\cX'$ dominating $\cX$.
Then $r_{\cX\cX'}$ maps $A_{\cX\cX'}$ homeomorphically onto $\Delta(\cX)$.
\end{prop}
\begin{cor}\label{C401}
The images under $r_{\cX\cX'}$ of the active simplices in $\Delta(\cX')$
form a simplicial $\ZZ$-subdivision of $\Delta(\cX)$.
As a consequence, there exists a unique, $\ZZ$-PA map
$i_{\cX'\cX}\colon\Delta(\cX)\to\Delta(\cX')$ such that $i_{\cX'\cX}(\Delta(\cX))=A_{\cX\cX'}$ and
$r_{\cX\cX'}\circ i_{\cX'\cX}=\id$.
\end{cor}
When $\pi$, $\pi'$ and $\rho$ are projective, one can prove Proposition~\ref{P301}
using the algebraic tool of valuations. Here we follow an ad hoc approach,
based on Lemma~\ref{L303}.
\begin{lem}\label{L401}
Suppose $\cX$, $\cX'$ and $\cX''$ are snc models, with
$\cX'$ dominating $\cX$ and $\cX''$ dominating $\cX'$.
Let $\sigma''$ be a simplex of $\Delta(\cX'')$, and let $\sigma'$ be the
smallest simplex of $\Delta(\cX')$ containing $r_{\cX'\cX''}(\sigma'')$.
Then $\sigma''$ is active for $r_{\cX\cX''}$ iff $\sigma''$ is active for
$r_{\cX'\cX''}$ and $\sigma'$ is active for $r_{\cX\cX'}$.
As a consequence,
\[
A_{\cX\cX''}=A_{\cX'\cX''}\cap r_{\cX'\cX''}^{-1}(A_{\cX\cX'}).
\]
\end{lem}
\begin{proof}
To ease notation, set $r':=r_{\cX'\cX''}$ and $r:=r_{\cX\cX'}$.
Let $\sigma$ be the smallest simplex of $\Delta(\cX)$ containing $r(\sigma')$.
Write $Y$, $Y'$ and $Y''$ for the strata of $\cX_0$, $\cX'_0$ and~$\cX''_0$
corresponding to $\sigma$, $\sigma'$ and $\sigma''$, respectively.
The restrictions $r'|_{\sigma''}\colon\sigma''\to\sigma'$
and $r_{\sigma'}\colon\sigma'\to\sigma$ are given by $\QQ$-linear maps,
and we have induced morphisms $Y''\to Y'$ and $Y'\to Y$.
First suppose that $\sigma''$ is active for $r'$ and $\sigma'$
is active for $r$. Then $r'|_{\sigma''}$ and $r|_{\sigma'}$ are given
by $\QQ$-linear isomorphisms; hence so is the composition
$r_{\cX\cX''}|_{\sigma''}$.
Similarly, the maps $Y''\to Y'$ and $Y'\to Y$ are bimeromorphic morphisms;
hence so is the composition $Y''\to Y$.
It follows that $\sigma''$ is active for $r_{\cX\cX''}$.
Conversely, suppose $\sigma''$ is active for $r_{\cX\cX''}$.
Since the map $Y''\to Y$ is a bimeromorphic morphism,
the map $Y''\to Y'$ (\resp $Y'\to Y$) must be injective (\resp surjective).
In particular, $\dim Y''\le\dim Y'$ and $\dim Y\le\dim Y'$.
Similarly, since the $\QQ$-linear map defining
$r_{\cX\cX''}|_{\sigma''}=r|_{\sigma'}\circ r'|_{\sigma''}$ is an isomorphism,
the $\QQ$-linear map defining $r'|_{\sigma''}$ (\resp $r|_{\sigma'}$) must be
injective (\resp surjective).
In particular, $\dim\sigma''\le\dim\sigma'$ and $\dim\sigma\le\dim\sigma'$.
Now
\begin{equation*}
\dim Y''+\dim\sigma''=\dim Y'+\dim\sigma'=\dim Y+\dim\sigma=n-1,
\end{equation*}
so we infer that
$\dim Y''=\dim Y'=\dim Y$ and $\dim\sigma=\dim\sigma'=\dim\sigma''$.
This further implies that the maps $Y''\to Y'$ and $Y'\to Y$ are
bimeromorphic morphisms, and that the $\QQ$-linear maps
defining $r'|_{\sigma''}$ and $r|_{\sigma'}$ are isomorphisms.
Hence $\sigma''$ and $\sigma'$ are active for $r'$ and $r$, respectively.
\end{proof}
\begin{lem}\label{L304}
Suppose $\cX$, $\cX'$ and $\cX''$ are snc models, with
$\cX'$ dominating $\cX$ and~$\cX''$ dominating $\cX'$.
\begin{itemize}
\item[(a)]
If $r_{\cX\cX''}\colon A_{\cX\cX''}\to\Delta(\cX)$ is surjective, then
so is $r_{\cX\cX'}\colon A_{\cX\cX'}\to\Delta(\cX)$.
\item[(b)]
If $r_{\cX\cX''}\colon A_{\cX\cX''}\to\Delta(\cX)$ is injective and
$r_{\cX'\cX''}\colon A_{\cX'\cX''}\to\Delta(\cX')$ is surjective, then
$r_{\cX\cX'}\colon A_{\cX\cX'}\to\Delta(\cX)$ is injective.
\item[(c)]
If $r_{\cX\cX'}\colon A_{\cX\cX'}\to\Delta(\cX)$ and
$r_{\cX'\cX''}\colon A_{\cX'\cX''}\to\Delta(\cX')$ are both surjective, then
so is $r_{\cX\cX''}\colon A_{\cX\cX''}\to\Delta(\cX)$.
\item[(d)]
If $r_{\cX\cX'}\colon A_{\cX\cX'}\to\Delta(\cX)$ and
$r_{\cX'\cX''}\colon A_{\cX'\cX''}\to\Delta(\cX')$ are both injective, then
so is $r_{\cX\cX''}\colon A_{\cX\cX''}\to\Delta(\cX)$.
\end{itemize}
\end{lem}
\begin{proof}
This is formal consequence of the relations
$r_{\cX\cX''}=r_{\cX\cX'}\circ r_{\cX'\cX''}$ and
$A_{\cX\cX''}=A_{\cX'\cX''}\cap r_{\cX'\cX''}^{-1}(A_{\cX\cX'})$.
For example, let us prove~(a).
Pick any point $w\in\nobreak\Delta(\cX)$. The assumption implies that we can find
$w''\in A_{\cX\cX''}$ with \hbox{$r_{\cX\cX''}(w'')=w$}. Then $w':=r_{\cX'\cX''}(w'')\in A_{\cX\cX'}$
and $r_{\cX\cX'}(w')=w$.
Thus~(a) holds. The proofs of~(b)--(d) are similar and left to the reader.
\end{proof}
\begin{lem}\label{L302}
The assertions of Proposition~\ref{P301} hold when
$\rho$ is a simple blowup.
\end{lem}
\begin{proof}
This is well known (see~\eg \cite[p.\,381]{KS06}) but we supply a proof
for the convenience of the reader.
To simplify notation, we set $r:=r_{\cX\cX'}$,
$A:=A_{\cX\cX'}$, $\Delta:=\Delta(\cX)$ and $\Delta':=\Delta(\cX')$.
Let $W$ be the center of the blowup $\rho$, and
$Z$ the smallest stratum of $\cX_0$ containing~$W$.
Let $E_i$, $i\in I$ be the irreducible components of $\cX_0$,
$J\subset I$ the subset such that $Z$ is an component of $E_J$,
and $\sigma_Z$ the simplex defined by $Z$.
Let $E'_i$, $i\in I$ be the strict transform of $E_i$ to $\cX'$.
Finally, let $E'$ be the exceptional divisor
of $\rho$. It corresponds to a vertex $v'=v'_{E'}$ of $\Delta'$.
First assume $W\subsetneq Z$.
In this case, $\Delta'$ 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 $\cX_0$.
By the definition of a simple blowup, $W$ meets every irreducible component
of $\cX_0$ 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 $Y'$ of $Y$ defines a stratum of $\cX'_0$
as well as a simplex $\sigma'$ of $\Delta'$, whose vertices
correspond to the strict transforms of the vertices of $\sigma$.
In this case, $r$ maps $\sigma'$ onto $\sigma$,
and $\rho\colon Y'\to Y$ is a bimeromorphic morphism,
so $\sigma'$ is active for $r$.
This proves that $r\colon A\to\Delta$ is surjective.
To prove injectivity, consider a stratum~$Y'$ of $\cX'_0$,
with corresponding simplex $\sigma'$ of $\Delta'$.
If $Y'$ is not contained in $E'$,
then $\rho$ is a biholomorphism at the general point of $Y'$,
$Y:=\rho(Y')$ is a stratum of $\cX_0$ of the same dimension
as $Y'$, and $Y'$ is the strict transform of $Y$. Thus we are in the
situation above.
On the other hand, if $Y'$ is contained in $E'$, then there exist
irreducible components $E_i$, $i\in J$ of $\cX_0$, having
strict transforms $E'_i$, $i\in J$, such that $\sigma'$ has
$v'$ and $v'_i$, $i\in J$ as vertices.
Since $W$ is not a stratum of $\cX_0$, the smallest stratum $Y$
containing $\rho(Y')$ is cut out by $E_i$, $i\in J$.
It follows that $r$ maps the simplex $\sigma'$ onto the lower-dimensional simplex $\sigma$,
so $\sigma'$ is not active for $r$.
Hence $r\colon A\to\Delta$ is injective.
Now assume $W=Z$ is stratum of $\cX_0$, defining a
simplex $\sigma$ with vertices $v_i$, $i\in J$.
In this case, $\Delta'$ 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 $\cX_0$ that is not contained in $W$, and $Y'$
is the strict transform, then the simplex $\sigma'_{Y'}$
is active for $r$ and $r(\sigma'_{Y'})=\sigma_Y$.
Further, $\sigma'_{Y'}$ is the unique simplex in~$\cX'_0$ that is active for $r$ and whose
image under $r$ meets the interior of $\sigma_Y$.
It remains to consider strata of $\cX_0$ 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
$Y'_i$, $i\in J$, each cut out by $E'$ and $E'_j$, $j\in K\setminus\{i\}$.
The restriction $\rho|_{Y'_i}\colon Y'_i\to Y$ is a bimeromorphic morphism,
and the corresponding simplex $\sigma'_i$ is active for $r$ and maps
homeomorphically onto a simplex contained in $\sigma_Y$.
Further, these simplices $r(\sigma'_i)$ have disjoint interiors and
cover $\sigma_Y$.
Finally, if $Y'$ is a stratum of $\cX'_0$ contained in
$E=\rho^{-1}(Z)$, then $Y=\rho(Y')$ is a stratum contained in $Z$,
hence $Y'=Y'_i$ is one of the strata above.
This completes the proof.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{P301}]
Since $r_{\cX\cX'}$ is continuous, $A_{\cX\cX'}$ is compact, and
$\Delta(\cX)$ is Hausdorff, it suffices to prove that
$r_{\cX\cX'}\colon A_{\cX\cX'}\to\Delta(\cX)$ is bijective.
Using Lemma~\ref{L304}~(c)--(d) and Lemma~\ref{L302}, one proves by
induction on the
number of blowups that $r_{\cX\cX'}\colon A_{\cX\cX'}\to\Delta(\cX)$ is bijective when
$\cX'\to\cX$ is a composition of simple blowups.
Now consider the general case.
Using Lemma~\ref{L303} we find an snc model $\cX''$
dominating both $\cX$ and $\cX'$ and such that the morphism
$\cX''\to\cX$ is a composition of simple blowups.
Thus $r_{\cX\cX''}\colon A_{\cX\cX''}\to\Delta(\cX)$ is bijective.
By Lemma~\ref{L304}~(a), it follows that
$r_{\cX\cX'}\colon A_{\cX\cX''}\to\Delta(\cX)$ is surjective.
Since $\cX$ and $\cX'$ were arbitrary snc models with $\cX'$
dominating $\cX$, it follows that $r_{\cX'\cX''}\colon A_{\cX'\cX''}\to\Delta(\cX)$
is also surjective. It now follows from Lemma~\ref{L304}~(b) that
$r_{\cX\cX'}\colon A_{\cX\cX''}\to\Delta(\cX)$ is injective, which completes the proof.
\end{proof}
\subsection{Induced maps between hybrid spaces}\label{S304}
To any snc model $\cX$ of $X$ we associated in~\S\ref{sec:hyb}
a hybrid space $\cX^\hyb=X\coprod\Delta(\cX)$. Let us briefly recall
the topology on $\cX^\hyb$ in the present context.
Extend $\pi\colon X\to\DD^*$ to a map
\begin{equation*}
\pi\colon\cX^\hyb\to\DD
\end{equation*}
by declaring $\pi=0$ on $\Delta(\cX)$.
For $0\kappa_{\min}$.
As a result, $D|_Y$ contains each lc center $E_i\cap Y$ of $(Y,B_Y)$,
which yields the last assertion.
\end{proof}
\subsection{Skeleta and base change}
Now we study how skeleta of snc models and of metrics behave under
base change.
For $m\in\ZZ_{>0}$ consider the Galois extension $K':=k\lau{\unipar^{1/m}}$ of
$K=k\lau{\unipar}$, with Galois group $G=\ZZ/m\ZZ$, and set $X'=X_{K'}$.
Then $G$ acts on $X'^{\an}$ and the canonical map
$p\colon X'^{\an}\to X^{\an}$ induces a homeomorphism
\begin{equation*}
X'^{\an}/G\simto X^{\an}.
\end{equation*}
If $\cX$ is a model of $X$, then its normalized base change yields a model
$\cX'$ of $X'$ with a finite morphism $\rho\colon\cX'\to\cX$.
If $D$ is a $\QQ$-divisor on $\cX$ defining a model function $\phi_D$ on $X^\an$, then
\begin{equation}\label{equ:pullmodel}
\phi_{\rho^*D}=m p^*\phi_D.
\end{equation}
When $\cX$ is an snc model, $\cX'$ is toroidal,
by~\cite[pp.\,98--102]{KKMS}.
The following rather detailed description will be useful later on.
\begin{lem}\label{lem:base}
We have $p^{-1}(\Sk(\cX))=\Sk(\cX')$.
Further, for each face $\sigma$ of $\Delta(\cX)$, there exist positive
integers $e_\sigma$, $f_\sigma$ and $g_\sigma$ satisfying
\begin{equation*}
e_\sigma=\frac{m}{\gcd(m,b_\sigma)}
\qand
f_\sigma g_\sigma=\gcd(m,b_\sigma)
\end{equation*}
and such that the following properties hold:
$p^{-1}(\sigma)$ is a union of $g_\sigma$ faces $\sigma'_\alpha$ of $\Delta(\cX')$,
and these are permuted by $G$. For each $\alpha$:
\begin{itemize}
\item[(a)]
$p$ induces a $\QQ$-affine isomorphism $\sigma'_\alpha\simto\sigma$;
\item[(b)]
$p$ induces a generically finite map $Y_{\sigma'_\alpha}\to Y_\sigma$,
of degree $f_\sigma$;
\item[(c)]
$mp^*M_\sigma\subset M_{\sigma'_\alpha}$, and
$[M_{\sigma'_\alpha}:mp^*M_\sigma]=e_\sigma$.
\end{itemize}
\pagebreak[2]
Furthermore, we have:
\begin{itemize}
\item[(i)]
$M_{\sigma'_\alpha}=p^*\left(mM_\sigma+\ZZ 1_{\sigma}\right)$;
\item[(ii)]
$\vol(\sigma'_\alpha)=m^{\dim\sigma}\vol(\sigma)$;
\item[(iii)]
$b_{\sigma'_\alpha}=b_\sigma/\gcd(m,b_\sigma)$.
\end{itemize}
\end{lem}
\begin{proof}
The proof uses the toroidal theory of~\cite{KKMS} together with
elementary ramification theory of valuations~\cite{ZS}.
Let $\sigma$ be the face of $\Delta(\cX)$ corresponding to an irreducible component
$Y$ of \hbox{$E_0\cap\dots\cap E_p$}. Set $b_i=\ord_{E_i}(\unipar)$.
With the identification
\begin{equation*}
\sigma=\{w\in\RR_+^{p+1}\mid\ts\sum_i b_i w_i=1\},
\end{equation*}
the integral affine structure $M_\sigma$ is given
by the lattice $\ZZ^{p+1}$. 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 $\widehat\cO_{\cX,\xi}\simeq k\cro{z_0,\dots,z_n}$
such that $\unipar=\prod_{i=0}^p z_i^{b_i}$.
A toric computation (\cf \cite[pp.\,98--102]{KKMS})
shows that $\xi$ has $\gcd(m,b_\sigma)$ preimages $\xi'_\alpha$ in
$\cX'_0$, with~$\cX'$ formally isomorphic, at each $\xi'_\alpha$,
to the product of $\AA_k^{n-p}$ with the affine toric $k$-variety
corresponding to the cone $\RR_+^{p+1}\subset\RR^{p+1}$ with lattice%\vspace*{-2pt}
$$
M':=\ZZ^{p+1}+\ZZ\left(\sfrac{b_0}{m},\dots,\sfrac{b_p}{m}\right).
$$
It follows that $p^{-1}(\sigma)$ is the union of the corresponding
faces $\sigma'_\alpha$ of $\Delta(\cX')$, each isomorphic to%\vspace*{-5pt}
$$
\sigma'=\bigl\{w'\in\RR_+^{p+1}\bigm|\ts\sum_i b_i w'_i=m\bigr\},
$$
with integral affine structure induced by $M'$.
Now $p$ restricts to a homeomorphism
$\sigma'_\alpha\simto\sigma$ given by $w=w'/m$.
Thus $M_{\sigma'_\alpha}=m p^*M_\sigma+\ZZ 1_{\sigma'_\alpha}$.
This implies~(i), and \hbox{(ii)--(iii)} easily follow.
Now note that%\vspace*{-5pt}
\begin{multline*}
[M_{\sigma'_\alpha}':mp^*M_\sigma]
=[mp^*M_\sigma+\ZZ 1_{\sigma'_\alpha}:mp^*M_\sigma]\\[-2pt]
=[\ZZ^{p+1}+\ZZ(\sfrac{b_0}{m},\dots,\sfrac{b_p}{m}):\ZZ^{p+1}]
=\frac{m}{\gcd(m,b_\sigma)}
=:e_\sigma.
\end{multline*}
It remains to analyze the degree $f_\sigma$ of the restriction $Y_{\sigma'_\alpha}\to Y_\sigma$.
For this we use ramification theory.
The function field $F(X')=F(X)(\unipar^{1/m})$ is a Galois extension of $F(X)$ of degree~$m$,
with Galois group $G$. For any valuation $v'\in X'^\val$, we have $v'|_{F(X)}=m p(v')$.
Let $v\in X^\an$ be a valuation corresponding to a point $w\in\sigma$.
Assume $w$ is ``general'' in the sense that $\dim_\QQ\sum_{i=0}^p\QQ w_i=p$.
The point $w$ has $g_\sigma$ preimages $w'_\alpha$ under $p$, one in each $\sigma'_\alpha$,
and the valuations $v'_\alpha:=m^{-1}w'_\alpha$ are all the extensions of $v$ to $F(X')$.
Let us compute the residue degree and ramification index of these extensions.
The residue fields of $v$ and $v'_\alpha$ are exactly the function fields of $Y$ and $Y'_\alpha$,
respectively, so the residue degree of the extension $v'_\alpha$ of $v$ is equal to $f_\sigma$.
The value group $\Gamma_v=v(F(X))$ of $v$ is given by $\Gamma_v=\sum_{i=0}^p\ZZ w_i$.
Similarly, the value group of $v'_\alpha$ is given by
$\Gamma_{v'_\alpha}=\frac1m\ZZ+\frac1m\sum_{i=0}^p\ZZ w'_i=\frac1m\ZZ+\sum_{i=0}^p\ZZ w_i$.
It follows that the ramification index of the extension $v'_\alpha$ of $v$ is given by
\begin{equation*}
\ts
[\Gamma_{v'_\alpha}:\Gamma_v]
=\bigl[\frac1m\ZZ+\sum_{i=0}^p\ZZ w_i:\sum_{i=0}^p\ZZ w_i\bigr]
=\gcd(\ZZ\cap m\sum_{i=0}^p\ZZ w_i)
=\dfrac{m}{\gcd(m,b_\sigma)}
=e_\sigma.
\end{equation*}
By~\cite[p.\,77]{ZS} we now have $e_\sigma f_\sigma g_\sigma=m$,
which completes the proof.
\end{proof}
Next we study skeleta of metrics. Generalizing~\cite[Lem.\,4.1.9]{NX13}, we prove:
\begin{lem}\label{lem:skelmetrbase}
Let $\psi$ be a continuous metric on $K_X^\an$,
$\psi'$ the metric on $K_{X'}^\an\simeq p^*K_X^\an$
corresponding to $p^*\psi$, and set $\kappa':=A_{X'}-\psi'$.
Then $\kappa'=m p^*\kappa$.
As a consequence, $\Sk(\psi')=p^{-1}\Sk(\psi)$ and
$\kappa'_{\min}=m\kappa_{\min}$.
\end{lem}
\begin{proof}
By~\cite[Th.\,7.12]{Gub98} (see also~\cite[Cor.\,2.3]{siminag}),
model metrics are dense in the set of continuous metrics
on $K_X^\an$.
Hence we may assume $\psi$ is a model
metric. Using~\eqref{equ:Asup}, it is enough to show that
$\kappa'(v')=m\kappa(p(v'))$ for a divisorial valuation $v'\in
X'^\div$. Let $\cX$ be an snc model with $p(v')\in\Sk(\cX)$, and
such that $\psi=\phi_\cL$ for a model $\cL$ of $K_X$ on $\cX$. Since
the normalized base change $\cX'$ of $\cX$ is toroidal, we can
choose a toroidal modification $\cX''\to\cX'$ with $\cX''$ snc. The
induced morphism $\rho\colon\cX''\to\cX$ is toroidal;
hence it satisfies the log ramification formula
$$
mK^\lo_{\cX''/S'}=\rho^*K^\lo_{\cX/S}.
$$
By~\eqref{equ:pullmodel}, we infer
$\phi_{K^\lo_{\cX''/S'}}-\psi'=p^*(\phi_{K^\lo_{\cX/S}}-\psi)$,
which gives the desired result since $v'\in\Sk(\cX'')$,
$p(v')\in\Sk(\cX)$ imply $A_{\cX''}(v')=A_{\cX}(p(v'))=0$.
\end{proof}
\section{Skeletal measures}\label{sec:skeletal}
From now on, we assume that $k=\CC$, and that
$X$ is a smooth, projective, geometrically connected
variety over the non-Archimedean field $K=\CC\lau{\unipar}$.
Our goal is to construct measures of the types appearing
in Theorem~\ref{th:A} and Corollary~\ref{cor:B}.
\subsection{Residually metrized models}
As explained above, to any model $\cL$ of a line bundle $L$ on $X$,
defined on a proper dlt model $\cX$ of $X$,
we can associate a skeleton $\Sk(\cL)\subset\Sk(\cX)\subset X^\an$.
To produce a measure on $\Sk(\cL)$ we need additional data.
\begin{defi}
Let $L$ be a line bundle on $X$. A \emph{residually metrized model}
of~$L$ is a pair $\cL^{\#}=(\cL,\psi_0)$ where $\cL$ is a model of
$L$, determined on a proper dlt model~$\cX$ of~$X$, and $\psi_0$ is a
continuous Hermitian metric on $\cL_0:=\cL|_{\cX_0}$, viewed as a
holomorphic line bundle over the complex space $\cX_0$. A
\emph{residually metrized model metric} $\psi^{\#}$ on $L$
is an equivalence class of such pairs, modulo pull-back to a higher model.
\end{defi}
\begin{ex} 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 $\cX$ by $\cL=\cO_{\cX}$
and $\psi_0$ the trivial metric on $\cO_{\cX_0}$.
\end{ex}
\subsection{Residual measures}\label{S309}
Let $\cL^{\#}=(\cL,\psi_0)$ be a residually metrized model of~$K_X$,
determined on a proper dlt model $\cX$. If $Y$ is a stratum
corresponding to a top-dimensional face of $\Delta(\cL)$,
Lemma~\ref{lem:boundaries} shows that the restriction of $\psi_0$ to $\cL|_Y$
induces a Hermitian metric $\psi_Y$ on $K_{(Y,B^\cL_Y)}:=K_Y+B^\cL_Y$,
with $(Y,B^\cL_Y)$ subklt.
By Lemma~\ref{lem:subklt}, we may thus introduce:
\begin{defi}
Let $Y$ be a stratum corresponding to a top-dimensional face of
$\Delta(\cL)$.
The \emph{residual measure} of $\cL^{\#}$ on $Y$ is the (finite) positive measure
$$
\Res_Y(\cL^{\#}):=\exp\bigl(2(\psi_Y-\phi_{B^\cL_Y})\bigr).
$$
\end{defi}
This definition is of course compatible with one
in~\S\ref{sec:rescplex}, and can be more explicitly described as
follows.
Let $\xi$ be a (closed) point of $Y\cap\cX_\snc$, index the
irreducible components $E_0,\dots,E_p$ passing through $\xi$ so that $Y$ is a component of $\bigcap_{0\le i\le d} E_i$ with $d=\dim\Delta(\cL)\le p$. In the notation of Example~\ref{ex:log}, the Poincaré residue
\begin{equation*}
\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
\end{equation*}
is a generator of $K_{(Y,B_Y)}=K^\lo_{\cX/S}\big|_Y$.
Setting $a_i:=\kappa(v_{E_i})b_i\in\QQ$, we have
\begin{equation*}
K^\lo_{\cX/S}=\cL+\sum_i a_i E_i,
\end{equation*}
and we may thus view
\begin{equation*}
\tau
:=\unipar^{\kappa_{\min}}\prod_{i=0}^p z_i^{a_i-\kappa_{\min}b_i}\Omega^\rel
=\unipar^{\kappa_{\min}}\prod_{i=d+1}^p z_i^{a_i-\kappa_{\min}b_i}\Omega^\rel
\end{equation*}
as a local $\QQ$-generator of $\cL$.
Further, $B^\cL_Y=\sum_{i=d+1}^p(1-(a_i-\kappa_{\min}b_i))E_i|_Y$,
and $\tau|_Y$ corresponds to
\begin{equation*}
\prod_{i=d+1}^p z_i^{a_i-\kappa_{\min} b_i}\Res_Y(\Omega)
\end{equation*}
under the identification $\cL|_Y=K_{(Y,B^\cL_Y)}$. We arrive at
\begin{align}\label{equ:res}
\Res_Y(\cL^{\#})
&=\frac{\prod_{i=d+1}^p|z_i|^{2(a_i-\kappa_{\min} b_i)}}
{\left|\unipar^{\kappa_{\min}}\prod_{i=d+1}^p z_i^{a_i-\kappa_{\min}b_i}\Omega^\rel\right|^2_{\psi_0}}\,
\left|\Res_Y(\Omega)\right|^2\notag\\
&=\frac{\prod_{i=d+1}^p|z_i|^{2(a_i-\kappa_{\min} b_i-1)}}
{\left|\unipar^{\kappa_{\min}}\prod_{i=d+1}^p z_i^{a_i-\kappa_{\min}b_i}\Omega^\rel\right|^2_{\psi_0}}\,
|dz_{d+1}\wedge\dots\wedge dz_n|^2.
\end{align}
\subsection{Measures on dual complexes}
We now define measures associated to residually metrized model metrics.
\begin{defi}
Let $\cL^{\#}$ be a residually metrized model of $K_X$,
determined on a proper dlt model $\cX$ of $X$.
To $\cL^{\#}$ we associate a positive measure $\mu_{\cL^{\#}}$ on
\hbox{$\Delta(\cL)\subset\Delta(\cX)$} defined by
\begin{equation*}
\mu_{\cL^{\#}}
=\sum_\sigma\left(\int_{Y_\sigma}\Res_{Y_\sigma}(\cL^{\#})\right)
b_\sigma^{-1}\lambda_\sigma,
\end{equation*}
where $\sigma$ runs over the top-dimensional faces of $\Delta(\cL)$.
\end{defi}
By Lemma~\ref{lem:integral}, we have
$$
\mu_{\cL^{\#}}(\sigma)=\frac{\int_{Y_\sigma}\Res_{Y_\sigma}(\cL^{\#})}{d!\prod_{i\in J} b_i}
$$
for each face $\sigma$ corresponding to a component of some $E_J$.
\subsection{Skeletal measures on Berkovich spaces}
Now consider a residually metrized model metric $\psi^{\#}$ on $K_X$.
Pick any representative $\cL^{\#}=(\cL,\psi_0)$ for $\psi^{\#}$,
where $\cL$ is a model of $K_X$ determined on a proper dlt model
$\cX$ of $X$, and where $\psi_0$ is a continuous metric on $\cL_0:=\cL|_{\cX_0}$.
\begin{defi}
The \emph{skeletal measure} $\mu_{\psi^{\#}}$ is the image of the measure
$\mu_{\cL^{\#}}$ under the embedding
$\Delta(\cL)\hto X^\an$.
We view it as a positive measure on $X^\an$, supported on the
skeleton $\Sk(\psi^{\#}):=\Sk(\phi_\cL)$.
\end{defi}
This definition makes sense, in view of the following result.
\begin{lem}\label{L305}
The skeletal measure $\mu_{\cL^{\#}}$ is independent of the choice
of representative $\cL^{\#}$ for $\psi^{\#}$.
\end{lem}
\begin{proof}
Let $\cX$, $\cX'$ be proper dlt models of $X$, with $\cX'$
dominating $\cX$ via a proper birational morphism
$\rho\colon\cX'\to\cX$.
Let $\cL^{\#}=(\cL,\psi_0)$ be a residually metrized model of $K_X$
consisting of a model $\cL$ of $K_X$ determined on $\cX$
and a continuous metric~$\psi_0$ on $\cL_0$.
Set $\cL'=\rho^*\cL$, $\psi'_0=\rho^*\psi_0$ and
$\cL'^{\#}=(\cL',\psi'_0)$.
We must prove that $\mu_{\cL'^{\#}}=\mu_{\cL^{\#}}$.
Let $\sigma'$ be a top-dimensional face of $\Delta(\cL')$,
$Y'$ the associated stratum of $\cX'_0$, $Y$ the
minimal stratum of $\cX_0$ containing $\rho(Y')$
and $\sigma=\sigma_Y$ the associated simplex of $\Delta(\cX)$.
Then $\sigma$ and $\sigma'$ have the same dimension, and
if we (somewhat abusively) identify $\sigma$ and $\sigma'$ with their images
in $\Sk(\phi_\cL)\subset X^\an$, then $\sigma'$ is a
rational subsimplex of $\sigma$.
It suffices to prove that $\mu_{\cL'^{\#}}(\sigma')=\mu_{\cL^{\#}}(\sigma')$.
Now $\rho$ restricts to a birational morphism of
$Y'\!\to\!Y$, so since $\lambda_{\sigma}|_{\sigma'}\!=\!\lambda_{\sigma'}$ and
$b_{\sigma}\!=\!b_{\sigma'}$,
it suffices to prove that $\Res_{Y'}(\cL'^{\#})=\rho^*\Res_Y(\cL^{\#})$.
But this is formal. Indeed, we have
$(\rho|_Y)_*(B_{Y'}^{\cL'})=B_Y^\cL$ and we can identify
$(\rho|_Y)^*K_{(Y,B_Y^\cL)}$ with $K_{(Y',B_{Y'}^{\cL'})}$
in such a way that the restriction of $\psi'_0$ to
$\cL'|_{Y'}=K_{(Y',B_{Y'}^{\cL'})}$ coincides with the pullback
under $\rho|_{Y'}$ of the restriction of $\psi_0$ to $\cL|_Y=K_{(Y,B_Y^\cL)}$.
\end{proof}
\subsection{Behavior under base change}
Fix $m\in\ZZ_{>0}$.
As before, denote by $X'$ the base change of $X$ to $K'=\CC\lau{\unipar^{1/m}}$,
with induced map $p\colon X'^\an\to X^\an$.
\begin{thm}\label{thm:skelmesfinite}
Let $\psi^{\#}$ be a residually metrized model metric on $K_X$,
and let $\psi'^{\#}$ be its pull-back to $X'$. Then
\begin{equation*}
p_*\mu_{\psi'^{\#}}=m^d\mu_{\psi^{\#}}
\end{equation*}
with $d=\dim\Sk(\psi^{\#})$.
\end{thm}
\begin{proof}
Pick a representative $\cL^{\#}=(\cL,\psi_0)$ of $\psi^{\#}$
such that $\cL$ is defined on a proper snc model $\cX$.
Let $\cX'$ be the
normalized base change by $\unipar=\unipar'^m$.
Let $\sigma$ be a $d$-dimensional face of $\Delta(\cL)$.
By Lemma~\ref{lem:base}, $p^{-1}(\sigma)$ is the union of
$g_\sigma$ distinct isomorphic faces $\sigma'_\alpha$ of $\Delta(\cX)$ such that
\begin{align}\label{equ:bprime}
b_{\sigma'_\alpha}&=b_\sigma/\gcd(m,b_\sigma)
\\
\label{equ:volprime}
\vol(\sigma'_\alpha)&=m^d\vol(\sigma).
\end{align}
Further, the induced map $Y'_{\sigma'_\alpha}\to Y$ 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 $\cX''\to\cX'$ with~$\cX''$ snc,
denote by $\rho\colon\cX''\to\cX$ the composition, and set
$\cL'':=\rho^*\cL$.
Each face $\sigma'_\alpha$ above is subdivided into
simplices $\sigma''_{\alpha\beta}$ of $\Delta(\cL'')$ of dimension $d$,
each corresponding to a stratum $Y''_{\alpha\beta}$ of $\cX''_0$,
and $\rho|_{Y''_{\alpha\beta}}\colon Y''_{\alpha\beta}\to Y$ is generically finite,
of degree $f_\sigma$.
Further,~\eqref{equ:bprime} implies that
\begin{equation}\label{e309}
b_{\sigma''_{\alpha\beta}}=b_{\sigma'_\alpha}=b_\sigma/\gcd(m,b_\sigma)
\quad\text{for all $\alpha,\beta$}.
\end{equation}
We shall need the following result:
\begin{lem}\label{lem:resfinite}
With notation as above, we have, for all $\alpha$, $\beta$:
\begin{equation*}
\Res_{Y''_{\alpha\beta}}(\cL''^{\#})=\gcd(m,b_\sigma)^{-2}\rho^*\Res_Y(\cL^{\#}).
\end{equation*}
\end{lem}
Grant this result for the moment.
Lemma~\ref{lem:resfinite} implies
\begin{equation*}
\int_{Y''_{\alpha\beta}}\Res_{Y''_{\alpha\beta}}(\cL''^{\#})=f_\sigma\gcd(m,b_\sigma)^{-2}\int_Y\Res_Y(\cL^{\#}),
\end{equation*}
and hence
\begin{multline*}
(p_*\mu')(\sigma)
=\sum_{\alpha,\beta}\mu'(\sigma''_{\alpha\beta})
=\sum_{\alpha,\beta}\biggl(\int_{Y''_{\alpha\beta}}\Res_{Y''_{\alpha\beta}}(\cL''^{\#})\biggr)
b_{\sigma''_{\alpha\beta}}^{-1}\vol(\sigma''_{\alpha\beta})\\
=f_\sigma\gcd(m,b_\sigma)^{-2}\left(\int_Y\Res_Y(\cL^{\#})\right)b_\sigma^{-1}
\gcd(m,b_\sigma) \sum_\alpha \vol(\sigma'_\alpha)\\
=m^d\left(\int_Y\Res_Y(\cL^{\#})\right)b_\sigma^{-1}\vol(\sigma)=m^d\mu(\sigma),
\end{multline*}
thanks to~\eqref{equ:volprime} and~\eqref{e309}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:resfinite}]
Pick a closed point $\xi''\in\mathring{Y''}$ and set
$\xi=\rho(\xi'')\in\mathring{Y}$.
We use the notation at the end of~\S\ref{S309} with $p=d$.
Namely, pick local coordinates $(z_i)_{0\le i\le n}$ at $\xi$ and
$(z''_j)_{0\le j\le n}$ at $\xi''$ such that $E_i=\{z_i=0\}$ for $0\le i\le d$
and $E''_j=\{z_j''=0\}$ for $0\le j\le d$.
We have $\rho^*z_i=u_i\prod_{j=0}^d(z''_j)^{c_{ij}}$ for $0\le i\le d$,
where $c_{ij}\in\ZZ_{\ge0}$ and $u_i\in\cO_{\cX'',\xi''}$ is a unit.
Further, by Lemma~\ref{lem:base},
the matrix $(c_{ij})$ has determinant~$\pm e_\sigma$,
where $e_\sigma=m/\gcd(m,b_\sigma)$.
Set
\begin{equation*}
\Omega_1:=\frac{dz_0}{z_0}\wedge\dots\wedge\frac{dz_d}{z_d}
\qand
\Omega_2:=dz_{d+1}\wedge\dots\wedge dz_n,
\end{equation*}
and define $\Omega_1''$, $\Omega''_2$ similarly.
Then $\Omega:=\Omega_1\wedge\Omega_2$
and $\Omega'':=\Omega''_1\wedge\Omega_2''$ are
local $\QQ$\nobreakdash-gen\-era\-tors of $K^\lo_\cX$ and $K^\lo_{\cX''}$
at $\xi$ and $\xi''$, respectively.
Further,
\begin{equation*}
\Res_Y(\Omega)=\Omega_2|_Y
\qand
\Res_{Y''}(\Omega'')=\Omega''_2|_{Y''}.
\end{equation*}
Now
\begin{equation*}
\rho^*\Omega_1
=\pm e_\sigma\Omega''_1
+\frac1{z_0''\dots z''_d}\,\tilde\Omega''_1,
\end{equation*}
where $\tilde\Omega''_1$ is a regular $(d+1)$-form vanishing at
$\xi''$,
and
\begin{equation*}
\rho^*\Omega_2
=q\Omega''_2+\tilde\Omega''_2,
\end{equation*}
where $q\in\cO_{\cX,\xi''}$ and $\tilde\Omega''_2$ is a regular
$(n-d)$-form at $\xi''$ satisfying $\Omega''_1\wedge\tilde\Omega''_2=0$.
On the one hand, this leads to
\begin{equation*}
(\rho|_{Y''})^*\Res_Y(\Omega)
=(\rho|_{Y''})^*(\Omega_2|_Y)
=q\Omega''_2|_{Y''}
=q\Res_{Y''}(\Omega'').
\end{equation*}
On the other hand, we also get
\begin{equation*}
\rho^*\Omega=\pm qe_\sigma(1+h)\Omega'',
\end{equation*}
with $q$ as above and $h\in\cO_{\cX'',\xi''}$ vanishing along $Y''$.
Define $\Omega^\rel$ and $\Omega^{''\rel}$ by
$\frac{d\unipar}{\unipar}\otimes\Omega^\rel=\Omega$ and
$\frac{d\unipar'}{\unipar'}\otimes\Omega^{''\rel}=\Omega''$,
respectively.
Then
\begin{equation*}
m\frac{d\unipar'}{\unipar'}\otimes\rho^*\Omega^\rel
=\rho^*\Bigl(\frac{d\unipar}{\unipar}\Bigr)\otimes\rho^*\Omega^\rel
=\rho^*\Omega
=\pm qe_\sigma(1+h)\Omega''
=\pm qe_\sigma(1+h)\frac{d\unipar'}{\unipar'}\otimes\Omega^{''\rel},
\end{equation*}
so that
\begin{equation*}
\rho^*\Omega^\rel=\pm\frac{e_\sigma}{m}\,q(1+h)\Omega^{''\rel}.
\end{equation*}
As a consequence,
\begin{equation*}
\rho^*|\unipar^{\kappa_{\min}}\Omega^\rel|_{\psi_0}
=\frac{e_\sigma}{m}\,|q|\,|(1+h)|\,|(\unipar')^{\kappa'_{\min}}\Omega^{''\rel}|_{\psi'_0}.
\end{equation*}
Since $h$ vanishes along $Y''$, this finally leads to
\begin{multline*}
(\rho|_{Y''})^*\Res_Y(\cL^\#)
=\frac{(\rho|_{Y''})^*|\Res_Y(\Omega)|^2}
{(\rho|_{Y''})^*|\unipar^{\kappa_{\min}}\Omega^\rel|^2_{\psi_0}}
=\frac{|(\rho|_{Y''})^*(\Omega_2|_Y)|^2}
{(\sfrac{e^2_\sigma}{m^2})|q|^2|(\unipar')^{\kappa'_{\min}}\Omega^{''\rel}|_{\psi'_0}^2}\\
=\Bigl(\frac{m}{e_\sigma}\Bigr)^2
\frac{|\Res_{Y''}(\Omega'')|^2}
{|(\unipar')^{\kappa'_{\min}}\Omega^{''\rel}|_{\psi'_0}^2}
=\Bigl(\frac{m}{e_\sigma}\Bigr)^2\Res_{Y''}(\cL^{''\#}),
\end{multline*}
which completes the proof since $e_\sigma=\sfrac{m}{\gcd(b_\sigma,m)}$.
\end{proof}
\section{The Calabi--Yau case}\label{S317}
As in~\S\ref{sec:skeletal}, we assume that $X$ is a smooth, projective,
geometrically connected variety over $\CC\lau{\unipar}$.
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 $K_X^\an$,
determined on any model $\cX$ by $\cL=\cO_\cX$, with $\eta$ providing the identification $\cL|_X\simeq K_X$. Denote also by $\log|\eta|^{\#}$ the residually metrized model metric induced by the trivial Hermitian metric $\psi_0=0$ on $\cO_{\cX_0}$.
The function $\kappa:=A_X-\log|\eta|=-\log|\eta|_{A_X}$
coincides with the weight function of~\cite{MN,NX13}.
By definition, the \emph{Kontsevich--Soibelman skeleton} of $X$ is
$\Sk(X):=\Sk(\log|\eta|^{\#})$. It is indeed independent of the choice of
$\eta$, since any other trivializing section of $K_X$ is of the form
$\eta'=f\eta$ with $f\in\CC\lau{\unipar}^*$, and hence $\kappa'=\kappa+\ord_0(f)$.
\subsection{Topology of the skeleton}
By~\cite[Th.\,4.2.4]{NX13}, the $\ZZ$-PA-space $\Sk(X)$ is connected, of pure dimension $d$, and is a deformation retract of $X^\an$. Further,
$\Sk(X)$ is a \emph{pseudomanifold with boundary}, \ie for some (or, equivalently, any) triangulation~$\Delta$ of $\Sk(X)$, we have:
\begin{itemize}
\item[(a)] Non-branching property: every $(d-1)$-simplex of $\Delta$ is contained in at most two $d$-simplices
\item[(b)] Strong connectedness: every pair of $n$-simplices $\sigma$, $\sigma'$ is joined by a chain of $n$-simplices $\sigma=\sigma_1,\dots,\sigma_N=\sigma'$ with $\sigma_i$ and $\sigma_{i+1}$ sharing a common $(n-1)$-face.
\end{itemize}
In the maximally degenerate case $d=n$, if $X$ has semistable reduction, then
$\Sk(X)$ is even a \emph{pseudomanifold}, \ie~(a) is replaced by
\begin{itemize}
\item[(a')]
every $(n-1)$-simplex of $\Delta$ is contained in exactly two $n$-simplices.
\end{itemize}
See also~\cite{KX} for even more precise results on the structure of $\Sk(X)$.
For example, $\Sk(X)$ is homeomorphic to a sphere if $n\le 3$
(still in the maximally degenerate case and $X$ having semistable reduction).
\subsection{The skeletal measure}
Consider the skeletal measure $\mu_{\log|\eta|^{\#}}$ on $\Sk(X)$. Choose an snc model $\cX$, and write as usual $\cX_0=\sum_{i\in I} b_i E_i$. The form $\eta$ defines an identification $K^\lo_{\cX/S}=\sum_{i\in I} a_i E_i$, and Proposition~\ref{prop:skelmetr} yields
\begin{equation}\label{equ:kappamin}
\kappa_{\min}=\min_i\frac{a_i}{b_i}.
\end{equation}
If $\kappa_{\min}\in\ZZ$, then
$$
\omega:=\frac{d\unipar}{\unipar^{\kappa_{\min}+1}}\wedge\eta
$$
is a logarithmic form on $\cX$. For each face $\sigma$ of $\Delta(\cL)$,
ordering the set $J\subset I$ of components cutting out the stratum $Y=Y_\sigma$
yields a well-defined Poincaré residue
$\Res_Y(\omega)$.
By Lemma~\ref{lem:boundaries}, $\Res_Y(\omega)$ is a rational section of $K_Y$, with divisor
$$
-B^\cL_Y=\sum_{i\notin J}(a_i-\kappa_{\min}b_i-1)E_i|_Y.
$$
When $\sigma$ is a maximal face, $\Res_Y(\omega)$ is thus a holomorphic
form on $Y$; using the formulas in~\S\ref{S309},
it is easy to see that the residual measure on $Y$ is given by
$$
\Res_Y(\log|\eta|^{\#})=|\Res_Y(\omega)|^2.
$$
The following result corresponds to Theorem~\ref{th:C} in the introduction.
\begin{thm}\label{thm:resCY}
Assume that $X$ is maximally degenerate, \ie $\dim\Sk(X)=n$, and that
$X$ has semistable reduction,
Then the skeletal measure $\mu_{\log|\eta|^{\#}}$ is a multiple of the integral Lebesgue
measure of $\Sk(X)$.
\end{thm}
\begin{proof} Let $\cX$ be a semistable model, \ie $\cX$ is snc with $\cX_0$ reduced. By~\eqref{equ:kappamin}, we have $\kappa_{\min}\in\ZZ$. Since some non-empty $E_J$ might have several components,
the dual complex $\Delta(\cX)$ is possibly not a triangulation of $\Sk(X)$. However, the barycentric subdivision $\Delta'$ of $\Delta(\cX)$ is a triangulation; the corresponding toroidal modification~$\cX'$ is snc, with $\cX'_0$ is possibly non-reduced, but $b_{\sigma}=1$ for each $n$-simplex $\sigma$ of $\Delta'$. Applying the above discussion to $\cX'$, we infer
$$
\mu_{\log|\eta|^{\#}}=\sum_{\sigma}|\Res_{y_\sigma}(\omega)|^2\lambda_\sigma,
$$
with $\sigma$ ranging over the $n$-dimensional faces of $\Delta'$, with corresponding strata $y_\sigma\in\cX'_0$ reduced to single points. It will thus be enough to show that $|\Res_{y_\sigma}(\omega)|$ is independent of $\sigma$.
By the strong connectedness property, any two $n$-simplices $\sigma$, $\sigma'$ of $\Delta'$ can be joined by a chain of $n$-simplices $\sigma=\sigma_1,\dots,\sigma_N=\sigma'$ 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 $\cX'$, we thus have $y_i,y_{i+1}\in Y_i$. Further, the Poincaré residue $\Res_{Y_i}(\omega)$ has poles precisely at $y_i,y_{i+1}$, since any other pole would correspond to an $n$-simplex of~$\Delta'$ containing~$\tau_i$, contradicting the non-branching property. Since $\Res_{y_i}\Res_Y(\omega)=\Res_{y_i}(\omega)$, the residue theorem applied to the Riemann surface $Y_i$ yields $\Res_{y_i}(\omega)+\Res_{y_{i+1}}(\omega)=0$, and hence $|\Res_{y_1}(\omega)|=\dots=|\Res_{y_N}(\omega)|$.
\end{proof}
\begin{rmk} Theorem~\ref{thm:resCY} fails in general when $X$ does not
have semistable reduction. Indeed, the semistable reduction theorem~\cite{KKMS} shows that the base change $p\colon X'\to X$ to $\CC\lau{\unipar^{1/m}}$ has semistable reduction for some $m$ divisible enough. By Lemma~\ref{lem:skelmetrbase}, $\dim\Sk(X')=n$, and $\mu_{\log|\eta'|^{\#}}$ is thus a multiple of the integral Lebesgue measure $\lambda'$ of $\Sk(X')$, by Theorem~\ref{thm:resCY}. By Theorem~\ref{thm:skelmesfinite}, $\mu_{\log|\eta|^{\#}}=m^{-n}p_*\lambda'$. However, $p_*\lambda'$ is not proportional to the integral Lebesgue measure $\lambda$ of $\Sk(X)$ in general. Indeed, for each $n$-simplex $\sigma$ of $\Delta(\cL)$, Lemma~\ref{lem:base} shows that $(p_*\lambda')_{\sigma}=m^n b_\sigma\lambda_{\sigma}$, and $b_\sigma$ is in general not independent of $\sigma$.
\end{rmk}
\enlargethispage{-3\baselineskip}%
\pagebreak[2]
\section{Extensions}\label{S312}
In this section we extend the main results in various directions.
\subsection{A singular version of Theorem A}
Let $\pi\colon\cX\to\DD$ be a projective, flat holomorphic map
of a normal complex space onto the disc, with $X:=\pi^{-1}(\DD^*)$
smooth over $\DD^*$.
Since $\pi$ is projective, it defines a smooth projective
variety $X_{\CC\lau{\unipar}}$ over $\CC\lau{\unipar}$, as well as a model
$\cX_{\CC\cro{\unipar}}$.
Let $\cL$ be a $\QQ$-line bundle on
$\cX$ extending $K_{X/\DD^*}$, and $\psi$ a continuous Hermitian metric on $\cL$.
This data induces a continuous Hermitian metric $\psi_t$
on $K_{X_t}$ for $t\in\DD^*$, as well as a residually metrized model $\cL^{\#}$ of
$K_{X_{\CC\lau{\unipar}}}$, the model given by $\cL_{\CC\cro{\unipar}}$ and the metric
by the restriction of $\psi$ to $\cL_0=\cL|_{\cX_0}$.
Thus we obtain a skeletal measure $\mu_{\cL^{\#}}$ on $X^\an_{\CC\lau{\unipar}}$.
Denote by $\cL'$ (\resp $\psi'$) the pull-back of $\cL$ (\resp $\psi$) to a log resolution \hbox{$\cX'\!\to\!\cX$}. By invariance of skeletal measures under pull-back, we have $\mu_{\cL'^{\#}}=\mu_{\cL^{\#}}$, and Theorem~\ref{thm:genconv} therefore implies:
\begin{thm}\label{thm:gencvsing} The rescaled measures
$$
\mu_t:=\frac{e^{2\psi_t}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^d},
$$
viewed as measures on $\cX'^\hyb$, converge weakly to $\mu_{\cL^{\#}}$.
\end{thm}
\begin{cor}\label{cor:mass} If $\cX$ (\ie the pair $(\cX,\cX_{0,\red})$) is dlt, then
$$
\lim_{t\to 0}\frac{\int_{\cX_t} e^{2\psi_t}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^d}=\sum_{\sigma}\left(\int_{Y_\sigma}\Res_{Y_\sigma}(\cL^{\#})\right) b_\sigma^{-1}\vol(\sigma),
$$
where $\sigma$ runs over the $d$-dimensional faces of $\Delta(\cL)$.
\end{cor}
When $d=0$, this implies the following slight generalization of~\cite[Lem.\,1]{Li}.
\begin{cor} Assume that $\cX_0$ has klt singularities (and hence $\cX$ is dlt by inversion of adjunction). Let $\psi$ be a continuous metric on $K_{\cX/\DD}$. Then $t\mto\int_{\cX_t} e^{2\psi_t}$ is continuous at $t=0$.
\end{cor}
\subsection{Corollary~\ref{cor:B} for pairs}
Suppose $(X,B)$ is a projective subklt pair over $\DD^*$ that is meromorphic
at $0\in\DD$.
By Bertini's theorem (see~\cite[4.8]{KollarPairs} and also below),
the pair $(X_t,B_t)$ is subklt for all $t\in\DD^*$ outside a discrete
subset $Z$.
Let $\psi$ be a continuous metric on $K_{{(X,B)}/\DD^*}$.
As explained in~\S\ref{S310}, $\psi$ induces a finite positive measure
$e^{2(\psi_t-\phi_{B_t})}$ on $X_t$ for $t\in\DD^*\setminus Z$.
Assume that $\psi$ has analytic singularities in the sense that
there exists a flat projective map $\cX\to\DD$ extending $X\to\DD^*$,
with $\cX$ normal, and a $\QQ$-line bundle $\cL$ on~$\cX$ extending
$K_{(X,B)/\DD^*}$ such that $\psi$ extends continuously to $\cL$.
Our assumptions imply that $X$ is defined over the Banach ring $A_r$
described in the appendix for $0