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).