Let us recall that an affine structure on manifold $Y$ (smooth, of dimension $n$) is given by a torsion-free flat connection $\nabla$ on the tangent bundle $TY$.

We will give below three equivalent definitions of the notion of an integral affine structure.

It is easy to see that if $Y$ carries a ${\bf Z}$-affine structure then for any point $y\in Y$ there exist small neighborhood $U$, local coordinate system $(x_{1},...,x_{n})$ in $U$ such that $\nabla=d$ in coordinates $(x_{1},...,x_{n})$, and the lattice $(T_{x}Y)^{{\bf Z}},x\in U$ is a free abelian group generated by the tangent vectors $\partial/\partial x_{i}\in T_{x}Y,1\leq i\leq n$. Let us call ${\bf Z}$-affine such a coordinate system in $U$ (sometimes we will call such $U$ a ${\bf Z}$-affine chart). For a covering of $Y$ by ${\bf Z}$-affine charts the transition functions belong (locally) to $GL(n,{\bf Z})\ltimes{\bf R}^{n}$. Explicitly, a change of coordinates is given by the formula

$$x_{i}^{\prime}=\sum_{1\leq j\leq n}a_{ij}x_{j}+b_{i}\,\,,$$

where $(a_{ij})\in GL(n,{{\bf Z}}),(b_{i})\in{{\bf R}}^{n}$.

Hence, Definition 1 is equivalent to the following

In the above definition $Y$ is just a topological manifold, $C^{\infty}$-structure on it can be reconstructed canonically from ${\bf Z}$-affine structure.

We can restate the notion of ${\bf Z}$-affine structure in the language of sheaves of affine functions.

We say that a real-valued function $f$ on ${{\bf R}}^{n}$ is ${\bf Z}$-affine if it has the form

$$f(x_{1},\dots,x_{n})=a_{1}x_{1}+\dots+a_{n}x_{n}+b\,\,,$$

where $a_{1},\dots,a_{n}\in{\bf Z}$ and $b\in{\bf R}$. We will denote by $Aff_{{\bf Z},{{\bf R}}^{n}}$ the sheaf of functions on ${{\bf R}}^{n}$ which are locally ${\bf Z}$-affine.

Equivalence of the last two definitions follows from the observation that a homeomorphism between two open domains in ${{\bf R}}^{n}$ preserving the sheaf $Aff_{{\bf Z},{{\bf R}}^{n}}$ is given by the same formula $x^{\prime}=A(x)+b,A\in GL(n,{{\bf Z}}),b\in{{\bf R}}^{n}$ as the change of coordinates between two ${\bf Z}$-affine coordinate systems.

With a given affine structure on $Y$ we can associate a flat affine connection $\nabla^{aff}$ (see [KN]). The corresponding parallel transport acts on tangent spaces by affine transformations. For a ${\bf Z}$-affine structure the monodromy of $\nabla^{aff}$ belongs to $GL(n,{\bf Z})\ltimes{\bf R}^{n}$, i.e. $\forall y\in Y$ we have a monodromy representation

$$\rho:\pi_{1}(Y,y)\to GL(n,{\bf Z})\ltimes{{\bf R}}^{n}\,\,.$$

Alternatively, we can define the monodromy representation by covering a loop in $Y$ by ${\bf Z}$-affine coordinate charts and composing the corresponding transition functions.

Notice that a ${\bf Z}$-affine structure on $Y$ gives rise to a class

$$[\rho]\in H^{1}(Y,T^{{\bf Z}}\otimes{{\bf R}})=H^{1}(Y,T_{Y}^{\nabla})\,\,\,,$$

where $T_{Y}^{\nabla}\subset T_{Y}$ is the subsheaf of $\nabla$-flat sections¹¹ 1 Here we slightly abuse notations because $Y$ is not necessarily connected.. De Rham representative of class $[\rho]$ is given by a differential $1$-form $\theta\in\Omega^{1}(Y,T_{Y})$ such that $\theta(v)=v$ for any tangent vector $v$. In affine coordinates one has $\theta=\sum_{i}\partial/\partial x_{i}\otimes dx_{i}$. Clearly $\nabla(\theta)=0$.

We will need later an explicit formula for the ${\bf R}$-valued pairing of $[\rho]$ with a closed singular 1-chain with coefficients in the local system $(T^{\ast})^{\bf Z}=(T^{*}Y)^{\bf Z}$, the dual covariant lattice in $T^{\ast}Y$. With any singular $1$-chain $c$ with values in $(T^{\ast})^{\bf Z}$ we associate a real number $j(c)$ in the following way. Suppose that $c$ is given by a continuous map $\gamma:[0,1]\to Y$ and a section $\alpha\in\Gamma([0,1],\gamma^{\ast}(T^{\ast})^{\bf Z})$. Parallel transport via the connection $\nabla^{aff}$ gives rise to a map $\overline{\gamma}:[0,1]\to T_{\gamma(0)}Y,\,\,\,\overline{\gamma}(0)=0$. Let $\alpha_{0}=\alpha(0)\in(T_{\gamma(0)}^{\ast})^{{\bf Z}}\subset T^{*}_{\gamma(0)}Y$. We define $j(c)=\langle\alpha_{0},\overline{\gamma}(1)\rangle$. We extend $j(c)$ to an arbitrary singular $1$-chain $c$ by additivity. Then the class $[\rho]$ can be calculated as $\langle[\rho],[c]\rangle=j(c)$ for any closed $1$-chain $c\in C_{1}(Y,(T^{\ast})^{{\bf Z}})$.

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)

$$\left\{\,\rho:\pi_{1}(S^{2}-\{24\mbox{ points}\})\rightarrow SL(2,{\bf Z})\ltimes{\bf R}^{2}\,\right\}$$

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.

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.

A $K$-analytic space $X$ is called Stein if the natural map

$$X\to Spec^{an}(\Gamma(X,{\cal O}_{X}))$$

is a homeomorphism. Here $\Gamma(X,{\cal O}_{X})$ is considered as a topological $K$-algebra. This definition is equivalent to the standard one. Let us call the projection $\pi:X\to B$ Stein if for any $b\in B$ there exists a fundamental systems of neighborhoods $U_{i}$ of $x$ such that $\pi^{-1}(U_{i})\subset X$ is a Stein domain. If $\pi$ is Stein then we can reconstruct $(X,{\cal O}_{X})$ and $\pi$ from the space $B$ endowed with the sheaf $\pi_{*}({\cal O}_{X})$ of topological $K$-algebras.

Proof.³³ 3 We thank to Ofer Gabber for suggesting the proof below It suffices to prove that $n$-dimensional cells are dense in $B$, where $n=\dim X$. For any open $U\subset B,\,\,U\neq\emptyset$ we have $H_{c}^{n}(U,\pi_{\ast}(\Omega^{n}_{X}))\simeq H_{c}^{n}(\pi^{-1}(U),\Omega^{n}_{X})$.

The last group is nontrivial, because for any non-empty open $V\subset X^{an}$ the integration map $\int:H_{c}^{n}(V,\Omega^{n}_{X})\to K$ is onto. Therefore $\dim(U)\geq n$. $\blacksquare$

All the examples in Sections 4.2.1–4.2.5 (except Section 4.2.3) have Stein property.

We recall that a Calabi-Yau metric on a complex manifold $X$ is a Kähler metric with vanishing Ricci curvature. If such a metric exists then $c_{1}({TX})=0\in H^{2}(X,{{\bf R}})$ and hence the class of the canonical bundle $\bigwedge^{\dim X}(T^{*}X)$ is torsion in $Pic(X)$. According to the famous Yau theorem, for any compact Kähler manifold $X$ such that $c_{1}({TX})=0\in H^{2}(X,{{\bf R}})$, and any Kähler class $[\omega]\in H^{2}(X,{{\bf R}})$ there exists a unique Calabi-Yau metric $g_{CY}$ with the class $[\omega]$⁴⁴ 4 Notice that there is a discrepancy in terminology. In algebraic situation one usually calls Calabi-Yau a projective variety with the trivial canonical class in $Pic(X)$, and the polarization is not considered as a part of data.. Up to now, there is no explicitly known non-flat Calabi-Yau metric on a compact manifold.

In Mirror Symmetry one studies the limiting behavior of $g_{CY}$ as the complex structure on $X$ approaches a “cusp” in the moduli space of complex structures (“maximal degeneration”). Well-known conjecture of Strominger, Yau and Zaslow (see [SYZ]) claims a torus fibration structure of Calabi-Yau manifolds near the cusp. A metric approach to the maximal degeneration (see [GW], [KoSo]) explains the structure of such Calabi-Yau manifolds in terms of their Gromov-Hausdorff limits. We recall this picture below following [KoSo].

We start with the definition of a maximally degenerating family of algebraic Calabi-Yau manifolds.

Let ${{\bf C}}_{t}^{mer}=\{f=\sum_{n\geq n_{0}}a_{n}t^{n}\}$ be the field of germs at $t=0$ of meromorphic functions in one complex variable, and ${X}_{mer}$ be an algebraic $n$-dimensional Calabi-Yau manifold over ${{\bf C}}_{t}^{mer}$ (i.e. ${X}_{mer}$ is a smooth projective manifold over ${{\bf C}}_{t}^{mer}$ with the trivial canonical class: $K_{{X}_{mer}}=0$). We fix an algebraic non-vanishing volume element $\Omega\in\Gamma({X}_{mer},K_{{X}_{mer}})$. The pair $({X}_{mer},\Omega)$ defines a 1-parameter analytic family of complex Calabi-Yau manifolds $(X_{t},\Omega_{t}),0<|t|<\epsilon$, for some $\epsilon>0$.

Let $[\omega]\in H^{2}_{DR}({X}_{mer})$ be a cohomology class in the ample cone. Then for every $t$, such that $0<|t|<\epsilon$ it defines a Kähler class $\omega_{t}$ on $X_{t}$. We denote by $g_{X_{t}}$ the unique Calabi-Yau metric on $X_{t}$ with the Kähler class $[\omega_{t}]$.

It follows from the resolution of singularities, that as $t\to 0$ one has

$$\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}=C(\log|t|)^{m}|t|^{2k}(1+o(1))$$

for some $C\in{{\bf C}}^{\times},k\in{{\bf Z}},0\leq m\leq n=\dim\,({X}_{mer})$.