Let $(X,\omega)$ be a smooth symplectic manifold of dimension $2n$, $B_{0}$ a smooth manifold of dimension $n$, $\pi:X\to B_{0}$ a smooth map with compact fibers, such that $\{\pi^{\ast}(f),\pi^{\ast}(g)\}=0$ for any $f,g\in C^{\infty}(B_{0})$. Here $\{\cdot,\cdot\}$ denotes the Poisson bracket on $X$. We assume that $\pi$ is a submersion on an open dense subset $X^{\prime}\subset X$. Such a triple $(X,\pi,B_{0})$ is called an integrable system. In applications it is typically given by a collection of smooth functions $(H_{1},...,H_{n})$ on $X$ (these functions are called Hamiltonians) such that $\{H_{i},H_{j}\}=0,1\leq i,j\leq n$. Usually first Hamiltonian $H=H_{1}$ is identified with the energy of mechanical system.

Let us consider the case when $\pi$ is proper. It is a natural restriction, because in applications the energy $H_{1}$ is already a proper map $H_{1}:X\to{{\bf R}}$.

Let $x\in B_{0}$ be a point such that the restriction of $\pi$ to $\pi^{-1}(x)$ is a submersion. We call such points $\pi$-smooth. According to Sard theorem $\pi$-smooth points form an open dense subset of $B_{0}$. The fiber $\pi^{-1}(x)$ is a compact Lagrangian submanifold of $X$. The Liouville integrability theorem (see [Ar]) says that $\pi^{-1}(x)$ is a disjoint union of finitely many tori $T^{n}_{\alpha}$. Moreover, for each torus $T^{n}_{\alpha}$ there exists a local coordinate system $(\varphi_{1},...,\varphi_{n},I_{1},...,I_{n})$ in a neighborhood $W_{\alpha}$ of $T^{n}_{\alpha}$ such that $\varphi_{i}\in{{\bf R}}/2\pi{{\bf Z}},(I_{1},...,I_{n})\in{{\bf R}}^{n}$ and $\omega=\sum_{1\leq i\leq n}dI_{i}\wedge d\varphi_{i}$. These coordinates are called action-angle coordinates. The map $\pi$ in action-angle coordinates is given by the projection $(\varphi_{1},...,\varphi_{n},I_{1},...,I_{n})\mapsto(I_{1},...,I_{n})$. There is an ambiguity in the choice of action-angle coordinates. In particular action coordinates $I=(I_{1},...,I_{n})$ are defined up to a transformation $I^{\prime}=A(I)+b,A\in GL(n,{{\bf Z}}),b\in{{\bf R}}^{n}$. Indeed, the free abelian group generated by $1$-forms $dI_{i},1\leq i\leq n$ in each cotangent space $T_{x}^{\ast}B_{0}$ admits an invariant description. It is the free abelian group generated by the restrictions of $1$-forms $\int_{\gamma}\omega$ to $T_{x}^{\ast}B_{0}$, where $\gamma$ runs through closed singular $1$-chains in $\pi^{-1}(x)\cap W_{\alpha}$. In this way we obtain a ${\bf Z}$-affine structure on $\pi(W_{\alpha})$.

Let $B$ be the set of connected components of fibers of $\pi$. Endowed with the natural topology it becomes a locally compact Hausdorff space, projection from $X$ to $B$ will be denoted by the same letter $\pi$. The natural continuous map $B\to B_{0}$ is a kind of “ramified finite covering”. Let us define $B^{sm}\subset B$ as the set of connected components on which $\pi$ is a submersion (i.e. the set of all Liouville tori). Then $B^{sm}$ is an open dense subset in $B$. Hence it carries a ${\bf Z}$-affine structure given by the action coordinates.

The singular part $B^{sing}=B\setminus B^{sm}$ consists of projections of singular fibers. Typically the codimension of $B^{sing}$ is greater or equal to $1$. The codimension $1$ stratum consists of the boundary of the image of $\pi$ and of the ramification locus of the map $B\to B_{0}$. The structure of singularities of the integral affine structure in higher codimensions is less understood. It seems that the following property is always satisfied:
Fixed Point property . For any $x\in B^{sing}$ there is a small neighborhood $U$ such that the monodromy representation $\pi_{1}((U\setminus B^{sing})_{\alpha})\to GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$ for any connected component $(U\setminus B^{sing})_{\alpha}$ of $U\setminus B^{sing}$ has a fixed vector in ${{\bf R}}^{n}$ in the natural representation by affine transformations.

We will discuss this property in Section 6 devoted to compactifications.

We describe here few examples related to the rest of the paper.

In many examples an integrable system depends on parameters. It often happens that the parameter space ${\cal P}$ carries a natural foliation ${\cal F}$ such that the fundamental group $\pi_{1}({\cal F}_{p},p),p\in{\cal P}$ of any leaf acts on the base space $B_{p}$ of the corresponding torus fibration. This action is given by piecewise-linear homeomorphisms with integral linear parts.

Let us illustrate this phenomenon in the case of the family of integrable systems associated with a K3 surface discussed above.

Here the parameter space ${\cal P}$ has dimension $38$, which is twice of the complex dimension of the space of polynomials $F$ modulo unimodular linear transformations. On the other hand, the miniversal family of representations (up to a conjugation)

such that the monodromy around each puncture is conjugate to $\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right)$, has dimension $20$.

Thus, we obtain a foliation ${\cal F}$ of ${\cal P}$ of rank $18=38-20$. It is defined by the following property: if we continuously vary parameters $p\in{\cal P}$ along leaves of ${\cal F}$ then the conjugacy class of the monodromy representation $\rho$ remains unchanged.

Notice that in the local model described above we can move the position $(x_{0},0)$ at which we start the cut. Then we have on the sphere $S^{2}$ a set of $24$ “worms” (singular points, each of them can move in its preferred direction, which is the line invariant under the local monodromy). One can show easily that any continuous deformation of ${\bf Z}$-affine structure satisfying Fixed Point property (see Section 3.1) and preserving the conjugacy class of $\rho$, corresponds to a movement of worms. ²² 2 Notice that in our example $rk({\cal F})=18$ is less than $24$. This means that there are 6 constraints on moving worms.

Moving “worms”we get a canonical identification of manifolds with integral affine structures far enough from singular points. We will see later in Section 6.4 that we also have a canonical PL identification of manifolds near singular points. Therefore we obtain a local system along leaves of ${\cal F}$ with the fiber over $p\in{\cal P}$ being a manifold $B_{p}\simeq S^{2}$ with the above ${\bf Z}$-affine structure. In this way we get a homomorphism from $\pi_{1}({\cal F}_{p},p)$ to $Aut_{{\bf Z}PL}(S^{2})$, where ${\bf Z}PL$ denotes the group of integral PL transformations of $S^{2}$ equipped with the above ${\bf Z}$-affine structure. We will return to this action in Section 6.7 where it will be compared with another PL action on the same space.