We refer the reader to [Be1] for the general definition of an analytic space and more details. In this Appendix we restrict ourselves to analytic spaces associated with algebraic varieties (although we use the general definition in the paper as well).

Let $R=R/K$ be a commutative unital finitely generated $K$-algebra. The underlying set of the Berkovich spectrum $Spec^{an}(R):=Spec^{an}(R/K)$ can be defined in two ways. First one uses valuations (or, equivalently, multiplicative seminorms).

Having a valuation and a real number $q_{0}\in(0,1)$ one can define the multiplicative seminorm $|a|=q_{0}^{val_{K}(a)},a\in R$. In particular, in the previous definition one can take seminorms $|\cdot|_{x}$ instead of valuations $val_{x}(\cdot)$. The reader has noticed that in the main body of the paper, for $R=K$ we often took $|a|=e^{-val(a)}$. It is easy to translate the definition of Berkovich spectrum to the language of multiplicative seminorms. We use it freely in the paper.

The second way to define $X^{an}$ uses evaluations (characters).

The field $K_{x}$ is determined by $x\in X^{an}$ in a canonical way. We define for $r\in R$ and $x\in X^{an}$ the “value” $r(x)\in K_{x}$ as the image $eval_{x}(r)$.

In order to pass from the first description of $Spec^{an}(R/K)$ to the second, starting with a valuation $val_{x}$ one defines the field $K_{x}$ as the completion of the field of fractions of $R/I_{x}$, where $I_{x}=(val_{x})^{-1}(\{+\infty\})$.

An element $f\in R$ defines a function $f:Spec^{an}(R)\to K_{x}$, where $K_{x}$ is the non-archimedean valuation field, which is the completion of the field of fractions of the domain $R/\ker(val_{x})$. Since each $K_{x}$ carries a seminorm, we obtain a function $|f|:Spec^{an}\to{{\bf R}}_{\geq 0},x\mapsto|f(x)|$.

A fundamental system of neighborhoods $U=U_{x}\subset Spec^{an}(R)$ of a point $x$ is parametrized by the following data: a finite collections of functions

$$(f_{i})_{i\in I},\,\,(g_{j})_{j\in J}\,\,\,\in R$$

and numbers

$$\beta^{+}_{i},\beta^{-}_{i},\gamma_{j}\in{\bf R}_{>0}$$

such that $\beta_{i}^{-}<|{f_{i}(x)}|<\beta_{i}^{+},\,\,|{g_{j}(x)}|=0$, The corresponding neighborhood consists of points $x^{\prime}$ such that $\beta_{i}^{-}<|{f_{i}(x^{\prime})}|<\beta_{i}^{+},\,\,|{g_{j}(x^{\prime})}|<\gamma_{j}$ for all $i\in I,j\in J$ and $x^{\prime}\in U$.

Let us assume that elements $(f_{i})_{i\in I},\,(g_{j})_{j\in J}$ generate $R$, i.e.

$$R=K[(f_{i})_{i\in I},\,(g_{j})_{j\in J}]/I$$

where $I$ is an ideal. Let us consider the algebra of series

$$s=\sum_{n_{I}\in{\bf Z}^{I},\,m_{J}\in{\bf N}^{J}}c_{I,J}f_{I}^{n_{I}}g_{J}^{m_{J}}$$

with constants $c_{I,J}\in K$, absolutely convergent when variables $(f_{i})_{i\in I},\,(g_{j})_{j\in J}$ satisfy the above inequalities. The quotient of this algebra by the topological closure of ideal $I$ is the algebra ${\cal O}_{Spec^{an}(R/K)}(U)$.

As in the case of schemes we can glue $Spec^{an}(R/K)$ into ringed spaces called analytic spaces (or rigid analytic spaces). Moreover we get a functor

$$\begin{array}[]{ccc}(Schemes/K)&\to&(K-{\rm analytic}\,\,{\rm spaces})\\ X&\mapsto&(X^{an},{\cal O}_{X^{an}}).\end{array}$$

Define the cone over $X^{an}$ as

$$C_{X^{an}}({\bf R}):=X^{an}\times{\bf R}_{>0}\,\,.$$

We interpret a point ${\bf x}=(x,\lambda)$ of $C_{X^{an}}({\bf R})$ as a $K_{x}$-point of $X$, where $K_{x}\supset K$ is a complete field with the ${\bf R}$-valued valuation

$$val_{\bf x}:=\lambda\,val_{x}\,\,,$$

whose restriction to $K$ is proportional to $val_{K}$. The set of points ${\bf x}\in C_{X^{an}}({\bf R})$ such that the valuation $val_{\bf x}$ is ${\bf Z}$-valued is denoted by $C_{X^{an}}({\bf Z})$.

Here we will describe explicitly the main example for our paper. Let $X={\bf G}_{m}^{n}=Spec(K[z_{i}^{\pm 1}]),1\leq i\leq n$ be an algebraic torus. and $X^{an}=({\bf G}_{m}^{an})^{n}$ the corresponding analytic space.

Firstly, we define an embedding $i_{can}:{\bf R}^{n}\hookrightarrow X^{an}$. For real vector $(x_{i})_{1\leq i\leq n}\in{\bf R}^{b}$ the corresponding point $p:=i_{can}(x_{1},\dots,x_{n})\in X^{an}$ will be described in terms of valuations.

For every Laurent polynomial $f=\sum_{I\in{{\bf Z}}^{n}}c_{I}z^{I},\,\,c_{I}\in K$ we set

$$val_{p}(f):=\min_{I\in{\bf Z}^{n}}\left(val(c_{I})-\sum_{i=1}^{n}x_{i}I_{i}\right)\,\,.$$

Secondly, we define a projection $\pi_{can}:X^{an}\to{\bf R}^{n}$ by formula

$$\pi_{can}(y)=\left(-val_{y}(z_{1}),\dots,-val_{y}(z_{n})\right)=\left(\log|z_{1}|_{y},\dots,\log|z_{n}|_{y}\right)\,\,.$$

The fiber over a point $(x_{1},\dots,x_{n})\in{{\bf R}}^{n}$ can be identified with the set of such seminorms $|\cdot|_{y}$ that $|z_{i}|_{y}=\exp(x_{i}),1\leq i\leq n$. We see $\pi_{can}$ is a kind of torus fibration⁷⁷ 7 This is the origin of the term “analytic torus fibration” introduced in Section 4.1.. Moreover, $\pi_{can}\circ i_{can}=id_{\,{\bf R}^{n}}$.

For any open connected $U\in({{\bf R}})^{n}$ the $K$-algebra of analytic functions on $\pi_{can}^{-1}(U)$ consists of series $f=\sum_{I\in{{\bf Z}}^{n}}c_{I}z^{I}$ with coefficients $c_{I}\in K$ such that for any $p=(x_{1},\dots,x_{n})\in U$ we have $\log|c_{I}|+\sum_{i=1}^{n}x_{i}I_{i}\to+\infty$ when $|l|\to+\infty$. It is easy to see that $\pi_{can}^{-1}(U)=\pi_{can}^{-1}(Conv(U))$ where $Conv(U)$ is the convex hull of $U$.

The sheaf $(\pi_{can})_{\ast}({\cal O}_{X^{an}}):={\cal O}^{can}_{{{\bf R}}^{n}}$ (canonical sheaf) plays an important role in the paper (see Sections 4.1, 7.3, 8).

Let $X$ be a smooth proper scheme over the non-archimedean field $K$. We assume that $K$ carries a discrete valuation $val$ such that $val(K^{\times})={{\bf Z}}$.

A model has no nontrivial automorphisms. Thus, the stack of equivalence classes of models is in fact a set, which we denote by $Mod_{X}$. It carries a natural partial order. Namely, we say that ${\cal X}_{1}\geq{\cal X}_{2}$ if there exists a map ${\cal X}_{1}\to{\cal X}_{2}$ over $Spec({{\cal O}}_{K})$. Such a map is automatically unique.

By the resolution of singularities, in the case $char\,k=0$ we know that every model is dominated by a model with normal crossings.

The set of equivalence classes of snc models will be denoted by $Mod_{X}^{snc}$. It is a filtered partially ordered set. The order is given by dominating maps of models which give the identity automorphism on the generic fiber.

It is easy to show that starting with any model with normal crossings and applying blow-ups centered at certain self-intersection loci of the special fiber we can get a snc model. In what follows we use snc models only. This choice is dictated by convenience and not by necessity. Working with snc models has the advantage that all definitions and calculations can be made very transparent. The reader can consult [Be2] for the approach in the general case, without the use of the resolution of singularities.

Let ${\cal X}$ be an snc model and $I=I_{{\cal X}}$ the set of irreducible components of ${\cal X}^{0}_{red}$. Denote by $D_{i}\subset{\cal X}$ the divisor corresponding to $i\in I$. For any finite non-empty subset $J\subset I$ put

$$D_{J}:=\bigcap_{j\in J}D_{j}\,\,.$$

By the snc property the set $D_{J}$ is either empty or is a smooth connected proper variety over $k$ of dimension $\dim(D_{J})=(n-|J|+1)$. For a divisor $D_{i}\subset{\cal X}^{0}$ we denote by $d_{i}\in{\bf Z}_{>0}$ the order of vanishing of $u$ at $D_{i}$, where $u\in K$ is an uniformizing element, $val_{K}(u)=1$. Equivalently, $d_{i}$ is the multiplicity of $D_{i}$ in ${\cal X}^{0}$.

Clearly, $S_{\cal X}$ is a nonempty connected CW-complex. We will also consider the cone over $S_{\cal X}$:

$$C_{\cal X}({\bf R}):=\left\{\sum_{i\in I}a_{i}\langle D_{i}\rangle|\,a_{i}\in{{\bf R}}_{\geq 0},\,\,\,\bigcap_{i:\,a_{i}>0}D_{i}\neq\emptyset\right\}\setminus\{0\}\subset{\bf R}^{I}\,\,.$$

Analogously, we can define $C_{\cal X}({\bf Z}),\,$.

We identify $S_{\cal X}$ with the following subset of $C_{\cal X}({\bf R})$:

$$\left\{\sum_{i\in I}a_{i}\langle D_{i}\rangle\in C_{\cal X}({\bf R})|\,\sum_{i}a_{i}d_{i}=1\right\}\,\,.$$

Obviously, we can also describe $S_{\cal X}$ as a quotient of $C_{\cal X}({\bf R})$:

$$S_{\cal X}=C_{\cal X}({\bf R})/{\bf R}^{\times}_{+}\,\,.$$

Let ${\cal X}$ be an snc model, $J\subset I_{\cal X}$ a non-empty subset and $Y\subset D_{J}$ a smooth irreducible variety of dimension less or equal than $n$. Let us assume that $Y$ intersects transversally (in $D_{J}$) all subvarieties $D_{J^{\prime}}$ of $D_{J}$ (for $J^{\prime}\supset J$), and that all intersections $Y\cap D_{J}$ are either empty or irreducible. It is obvious that the blow-up ${\cal X}^{\prime}:=Bl_{Y}({\cal X})$ of ${\cal X}$ with the center at $Y$ is again a snc model.

Let us describe the behavior of $S_{\cal X}$ under simple blow-ups. To the set of vertices we add a new vertex corresponding to the divisor $\widetilde{Y}$ obtained from $Y$:

$$I_{{\cal X}^{\prime}}=I_{\cal X}\sqcup\{new\},\,\,\,D_{new}:=\widetilde{Y}\,\,.$$

The degree of the new divisor is (for both the first and the second type)

$$d_{new}:=\sum_{i\in J}d_{j}\,\,.$$

For blow-ups of the first type we have automatically $\#J>1$. Here is the list of faces of $S_{\cal{X}^{\prime}}$:

1) $I^{\prime}$ for $I^{\prime}\in Faces(S_{\cal X}),\,I^{\prime}\not\subset J$;

2) $I^{\prime}\sqcup\{new\}$ for $I^{\prime}\in Faces(S_{\cal X}),\,I^{\prime}\neq J,\,I^{\prime}\cup J\in Faces(S_{\cal X})$;

3) the vertex $\{new\}$.
For blow-ups of the second type the list of faces of $S_{{\cal X}^{\prime}}$ is

1) $I^{\prime}$ for $I^{\prime}\in Faces(S_{\cal X})$;

2) $I^{\prime}\sqcup\{new\}$ for $I^{\prime}\in Faces(S_{\cal X}),\,I^{\prime}\supset J,\,Y\cap D_{I^{\prime}}\neq\emptyset$;

3) the vertex $\{new\}$.

On can deduce from results [AKMW] the following

Let ${\cal X}$ be a snc model of $X$. We define a map

$$i_{\cal X}:C_{\cal X}({\bf R})\to C_{X^{an}}({\bf R})$$

such as follows. For $J=\{j_{1},\dots,j_{k}\}\subset I_{\cal X}$ such that $D_{J}\neq\emptyset$ let us consider a point $x\in C_{\cal X}({\bf R})$

$$x=\sum_{i=1}^{k}a_{i}\langle D_{j_{i}}\rangle,\,\,\,a_{i}\in{{\bf R}}_{>0}\,\,\forall i\in\{1,\dots,k\}$$

and an affine Zariski open subset $U\subset{\cal X}$ containing the generic point of $D_{J}$. One can embed ${\cal O}(U)$ into the algebra of formal series $K_{J}[[z_{1},\dots,z_{k}]]$ where $K_{J}$ is the field of rational functions on $D_{J}$ and $z_{i}=0$ are equations of divisors $D_{j_{i}},\,\,i=1,\dots,k\,$. We define a valuation $v_{x}$ of ${\cal O}(U)$ by the formula

$$v_{x}\left(\sum_{n_{1},\dots,n_{k}\geq 0}c_{n_{1},\dots,n_{k}}\prod_{i=1}^{k}z_{i}^{n_{i}}\right)=\inf\left\{\sum a_{i}n_{i}\,|\,c_{n_{1},\dots,n_{k}}\neq 0\right\}\,\,.$$

We define $i_{\cal X}(x)$ to be the image of the point $v_{x}\in Spec^{an}({\cal O}(U)/K)$ in $X^{an}$. It is easy to check that the element $i_{\cal X}(x)$ does not depend on the choice of the open subset $U$.

The following proposition is obvious:

We will denote also by $i_{\cal X}$ the induced embedding $S_{\cal X}\hookrightarrow X^{an}$.

For a model ${\cal X}$ we can interpret elements of $C_{X^{an}}({\bf Z})$ as paths in ${\cal X}$, i.e. equivalence classes of maps

$$\phi:Spec({{\cal O}}_{L})\to{\cal X},$$

where ${{\cal O}}_{L}$ is the ring of integers in a field $L$ with discrete valuation in ${\bf Z}$, such that the image of $\phi$ does not lie in ${\cal X}$. We define the map

$$p_{\cal X}^{\bf Z}\,:\,C_{X^{an}}({\bf Z})\to C_{\cal X}({\bf Z})$$

as

$$p_{\cal X}^{\bf Z}([\phi]):=\sum_{i}a_{i}\langle D_{i}\rangle,$$

where $a_{i}\in{\bf Z}_{\geq 0}$ is the multiplicity of the intersection of the path $\phi$ with the divisor $D_{i},\,\,i\in I_{\cal X}$.

The following proposition can be derived from [Be1].

We denote by $p_{\cal X}:X^{an}\to S_{\cal X}$ the map induced by $p_{\cal X}^{\bf R}$.

Let $f:{\cal X}^{\prime}\to{\cal X}$ be a dominating map of models. Let us denote by $m_{i,i^{\prime}}\in{{\bf Z}}_{\geq 0}$ the multiplicity of a divisor $D_{i^{\prime}},i^{\prime}\in I_{{\cal X}^{\prime}}$ in the proper pull-back of $D_{i},i\in I_{\cal X}$. We define $p_{{\cal X}^{\prime},{\cal X}}^{{\bf Z}}:C_{X^{an}}({\bf Z})\to C_{\cal X}({\bf Z})$ by the formulas $\sum_{i^{\prime}}a_{i^{\prime}}\langle D_{i^{\prime}}\rangle\mapsto\sum_{i}m_{i,i^{\prime}}a_{i^{\prime}}\langle D_{i}\rangle$. Let $p_{{\cal X}^{\prime},{\cal X}}:S_{X^{an}}({\bf R})\to S_{\cal X}({\bf R})$ be the corresponding by map of Clemens polytopes.

Then we have the following result, which is easy to prove.

With any meromorphic at $t=0$ family of smooth complex projective varieties $X_{t},\,\,\,0<|t|<\epsilon$ one can associate a variety $X$ over the field ${\bf C}((t))$. It is easy to see that for any snc model $\cal X$ one can canonically complete the family $X_{t}$ by adding $S_{\cal X}$ as the fiber over $t=0$. The total space is not a complex manifold by just a Hausdorff locally compact space which maps properly to the dick $\{t\in{\bf C}\,|\,|t|<\epsilon\}$. Passing to the projective limit we see that one can compactify the family $X_{t}$ at $t=0$ by $X^{an}$.