Here we are going to define an analog of the notion of integrable system in the framework of rigid analytic geometry. Roughly speaking, it is a triple $(X,\pi,B)$, where $X$ is a variety defined over a non-archimedean field (see [Be1] and Appendix A), $B$ a CW complex and $\pi:X\to B$ a continuous map. More precisely, let $K$ be a field with non-trivial valuation, $X$ an irreducible algebraic variety over $K$ of dimension $n$, $f=(f_{1},...,f_{N})$ a collection of non-zero rational functions on $X$. Then we have a multivalued map

$$X(\overline{K})\to[-\infty,+\infty]^{N},\,\,x\mapsto val_{\overline{K}}(f(x)):=\left(val_{\overline{K}}\left(f_{1}(x)),\dots,val_{\overline{K}}(f_{N}(x)\right)\right)\,.$$

Here $\overline{K}$ is the algebraic closure of $K$, and $val_{\overline{K}}$ denotes valuation on $\overline{K}$.

Let $\psi:[-\infty,+\infty]^{N}\to B$ be a continuous map such that the composition $\pi=\psi\circ val(f)$ is single-valued. Our map $\pi$ will always be of this form. More generally, we can take $X$ to be a (not necessarily algebraic) compact smooth $K$-analytic space, and $\pi:X\to B$ be a continuous map which factorizes as the composition of the projection $p_{\cal X}:X\to S_{\cal X}$ to the Clemens polytope $S_{\cal X}$ of some model ${\cal X}$ of $X$ and a continuous map $\pi^{\prime}:S_{\cal X}\to B$ (see Section 4.2.3).

Now we would like to be more precise. Let $K$ be a complete non-archimedean field, with valuation $val$ and the corresponding norm $|x|:=\exp(-val(x))\in{\bf R}_{\geq 0}$. Before giving next definition we observe that there is a canonical continuous map $\pi_{can}:({\bf G}_{m}^{an})^{n}\to{{\bf R}}^{n}$ (see Section A.2 in Appendix A). Here ${\bf G}_{m}^{an}$ is a multiplicative group (considered as an analytic space over $K$) and the restriction of $\pi_{can}$ to $(\overline{K}^{\times})^{n}$ is given by the formula

$$\pi_{can}(z_{1},...,z_{n})=(\log|z_{1}|,\dots,\log|z_{n}|)\,\,.$$

The sheaf ${\cal O}^{can}_{{{\bf R}}^{n}}:=(\pi_{can})_{\ast}({\cal O}_{({\bf G}_{m}^{an})^{n}})$ of $K$-algebras is called the canonical sheaf.

Let $X$ be a smooth $K$-analytic space of dimension $n$, $\pi:X\to B$ a continuous map of $X$ into a Hausdorff topological space $B$.

Let $B^{sm}$ denotes the set of smooth points of $B$. It is a topological subspace of $B$ (in fact a topological manifold of dimension $n$).

Proof. We start with the following Lemma.

In order to prove Lemma we observe that any analytic function $\psi\in{\cal O}^{\times}_{{({\bf G}_{m}^{an})}^{n}}(\pi_{can}^{-1}(V))$ can be decomposed into Laurent series:

$$\psi=\sum_{I=(i_{1},\dots,i_{n})\in{{\bf Z}}^{n}}c_{I}z^{I},\,\,\,c_{I}\in K$$

satisfying certain convergence conditions (see Section A.2).

Then for a non-zero analytic function $\psi$ on $\pi_{can}^{-1}(V)$ we introduce a real-valued function $Val(\psi)(x):=\inf_{I\in{{\bf Z}}^{n}}(val(c_{I})-\langle I,x\rangle),x\in V$. It is a concave, locally piecewise-linear function on $V$. It is easy to see that

a)

there is a dense open subset $V_{1}\subset V$ such that for any $x\in V_{1}$ the infimum in the definition of $Val(\psi)$ is achieved for a single multi-index $I$;

b)

$Val(\psi_{1}\psi_{2})=Val(\psi_{1})+Val(\psi_{2})$.

For an invertible function $\varphi$ we have $0=Val(1)=Val(\varphi\varphi^{-1})=Val(\varphi)+Val(\varphi^{-1})$. Since both $Val(\varphi)$ and $Val(\varphi^{-1})$ are concave, their sum can be equal to zero iff they are both affine. Moreover they are both ${\bf Z}$-affine since the linear part of $Val(\varphi)$ is given by the integer vector $I$ for some single multi-index $I$. Finally, observe that $val_{x}(\varphi(x))\geq\pi_{can}^{*}(Val(\varphi))(x),x\in\pi_{can}^{-1}(V)$. Therefore $val_{x}(\varphi(x))=Val(\varphi)(\pi_{can}(x))$ for invertible $\varphi$. $\blacksquare$

Now we can finish the proof of the Theorem. The above formula gives us a coordinate-free description of $\pi_{can}^{*}(Val(\varphi))$. It is easy to see that any ${\bf Z}$-affine function on $V$ is of the form $Val(\varphi)+c,c\in{{\bf R}}$ for some invertible $\varphi$ (in the case of ${{\bf R}}^{n}$ it suffices to take monomials as $\varphi$). We can identify $\pi^{-1}(U)\to U$ with $\pi_{can}^{-1}(V)\to V$ for some small open $U\subset X$ and $V\subset{{\bf R}}^{n}$. Then we can define $Val(\varphi)$ for any invertible $\varphi\in{\cal O}_{X}(\pi^{-1}(U))$ by the above formula. Finally we define a sheaf of ${\bf Z}$-affine functions on $B^{sm}$ by taking all functions of the form $Val(\varphi)+c,c\in{{\bf R}}$. It follows from the above discussion that in this way we obtain a ${\bf Z}$-affine structure on $B^{sm}$, which is locally isomorphic to the standard one on ${{\bf R}}^{n}$. $\blacksquare$

We will denote by $Aff^{can}_{{{\bf Z}},B^{sm}}$ the sheaf of ${\bf Z}$-affine functions constructed in the proof.