Assume that we are given a non-compact manifold $B^{sm}$ with a ${\bf Z}$-affine structure. We would like to “compactify” it, i.e. to find a compact Hausdorff topological space $B$ such that $B^{sm}\subset B$ is an open dense subset. We do not require an extension of the ${\bf Z}$-affine structure to $B$. The question is: what kind of properties one should expect from such a compactification? We cannot give a complete list of such properties at the moment. Instead, we formulate two of them and illustrate the notion of compactification in PL case. Similarity between examples in Sections 3.2 and 4.2 suggests that the class of singularities which appear in integrable systems should be more or less the same as the class of singularities appearing in non-archimedean geometry.

Let $x\in B^{sing}:=B\setminus B^{sm}$ be a singular point of some compactification of $B^{sm}$. Then we require the following
Finiteness property. There is a fundamental system of neighborhoods $U\subset B$ of $x$ such that the number of connected components of $U\cap B^{sm}$ is finite.

Let $U\cap B^{sm}=\sqcup_{1\leq i\leq N}U_{i}$ be the disjoint union of the connected components. Let us pick a point $x_{i}\in U_{i}$ and consider a continuous path $\gamma:[0,1]\to B$ such that $\gamma(0)=x_{i},\gamma(1)=x,\gamma([0,1))\subset U_{i}$. Using the affine structure we can canonically lift this path to a path $\gamma^{\prime}:[0,1)\to T_{x_{i}}B$. We assume that the lifted path $\gamma^{\prime}$ extends to time $t=1$ and is analytic at $t=1$ (it is a technical assumption, helping to avoid some pathologies). Then we require the following
Independence property. Path $\gamma$ with the properties as above exists, and point $\gamma^{\prime}(1)\in T_{x_{i}}B$ does not depend on the choice of $\gamma$.

Independence property implies the existence of a fixed vector for the monodromy representation restricted to $\pi_{1}(U\cap B^{sm},x_{i})$ (this implies the Fixed Point property from Section 3.1).

Let $V$ be a finite set, $S\subset 2^{V}$ belongs to the set of $(n+1)$-element subsets of $V$. Then we have a $n$-dimensional simplicial complex $B=\cup_{Y\in S}\Delta^{Y}\subset\Delta^{V}$.

Let us choose a ${\bf Z}$-affine structure on $n$-dimensional faces of $B$ which is compatible with the standard affine structure, and consider all $(n-1)$-dimensional faces which enjoy the following property: they belong to exactly two $n$-dimensional faces. For any two such $n$-dimensional faces $\sigma$ and $\tau$ we choose a ${\bf Z}$-affine structure on $\sigma\cup\tau$ which is compatible with the already chosen ${\bf Z}$-affine structures on $\sigma$ and $\tau$ (such a choice is equivalent to a choice of ${\bf Z}$-affine structure in a neighborhood of the $(n-1)$-dimensional face $\sigma\cap\tau$). In this way we obtain a ${\bf Z}$-affine structure on the union $U$ of the interior points of all $n$-dimensional simplices and also the interior points of $(n-1)$-dimensional faces belonging to exactly two top-dimensional cells.

Proof. Let us proceed inductively by codimension of faces. The induction step reduces to the obvious remark that the extension of the standard ${\bf Z}$-affine structure on ${{\bf R}}^{n}\setminus L$ to a neighborhood of point $p\in L$ in ${\bf R}^{n}$ is unique in the case when $L\subset{\bf R}^{n}$ is an affine subspace, $\dim L\leq n-2\,\,$. $\blacksquare$

It is easy to see that $B^{sm}:=U_{\max}$ with ${\bf Z}$-affine structure on it, compactified by $B$ satisfies both Finiteness and Independence properties.

We introduce PL compactifications both as a “toy model”, and also (as we hope, see Conjecture 6 in Section 6.3) as a sufficiently representative class for applications. In this case we can try to formulate additional desired properties. One of goals is to find a good substitution for the algebro-geometric notion of a canonical singularity (which is, morally, a singularity of a non-collapsing limit of a family of Calabi-Yau manifolds with fixed Kähler class).

For a large class of maximally degenerating Calabi-Yau manifolds there is a proposal by several authors (see [GS] and [HZh]) for a PL compactification $B$ conjecturally related to the Gromov-Hausdorff limit. Space $B$ is topologically a sphere $S^{n}$, it carries two dual cell decompositions. Each of these decompositions is identified with the boundary $\partial P_{1}$ or $\partial P_{2}$ of a convex $(n+1)$-dimensional polytope. Moreover, on each $n$-dimensional face of each polytope we have a ${\bf Z}$-affine structure compatible with the natural affine structure. The assumption is that for any two open $n$-cells $U,U^{\prime}$ from the first and the second cell decompositions, two induced ${\bf Z}$-affine structures on $U\cap U^{\prime}$ coincide. This gives a ${\bf Z}$-affine structure on $B\setminus\left(Sk_{n-1}\cap Sk_{n-1}^{\prime}\right)$ where $Sk_{n-1}$ and $Sk_{n-1}^{\prime}$ are $(n-1)$-skeletons of two CW-structures.

Our conjectures are in fact rather “wishes”, i.e. they are desired properties of $B^{sing}=B\setminus B^{sm}$. For simplicity we assume that $B^{sing}$ is a stratified set (say, CW complex) of dimension less or equal than $n-1$.

This conjecture gives a local model for a singular ${\bf Z}$-affine structure at a singular component of codimension one. Let us discuss the case of higher codimension. We start with the following definition.

One can think about closed set of “potential singularities” $B^{pre-sing}$ as containing the actual set of singularities $B^{sing}$).

Notice that in the case when the homotopy type of $B\setminus B_{t}^{pre-sing}$ remains unchanged the representation $\rho_{t}:\pi_{1}\left(B\setminus B_{t}^{pre-sing}\right)\to GL(n,{\bf Z})\ltimes{\bf R}^{n}$ stays the same.

We are going to give an example of a non-trivial path in the next subsection. We expect that singularities which appear in the collapse of Calabi-Yau manifolds satisfy the following

Let us remove the angle $\left\{(x,y)\in{\bf R}^{2}\,|\,\,0<x<y\,\right\}$ from ${{\bf R}}^{2}$. After that we identify sides of the angle by the affine transformation $(x,y)\mapsto(x+y,y)$. In this way we introduce a new ${\bf Z}$-affine structure on ${{\bf R}}^{2}\setminus\{(0,0)\}$ with the monodromy around $(0,0)$ given by the unipotent matrix (see Figure 3)

$$\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right)\,\,.$$

This ${\bf Z}$-affine structure does not admit a continuation to ${{\bf R}}^{2}$. Therefore we obtain a ${\bf Z}$-affine structure with singularities on ${{\bf R}}^{2}$. We will call standard the singularity at $(0,0)$.

Equivalently, we can describe this ${\bf Z}$-affine structure on ${{\bf R}}^{2}\setminus\{(0,0)\}$ by taking a cut along the ray $\{(x,0)\,|\,x>0\}$ in ${{\bf R}}^{2}$ and glue the standard ${\bf Z}$-affine structure above and below the cut by means of the affine transformation $(x,y)\mapsto(x+y,y)$ (see Figure 1 in Section 3.2.4). In this description it is clear that we can start the cut at arbitrary point $(x_{0},0)$ on the $x$-axes. The resulting singularity will be also called the standard one.

Standard singularity is called focus-focus singularity in the theory of integrable systems (see [Zu]). In non-archimedean geometry it appears as a singular value of some map $f:X^{an}\to{{\bf R}}^{2}$, where $X$ is an algebraic surface in $3$-dimensional affine space ${\bf A}_{K}^{3}$ (see Section 8).

Let us consider the Cartesian product of ${{\bf R}}^{2}\setminus\{(0,0)\}$ equipped with the above ${\bf Z}$-affine structure with the standard (non-singular) ${\bf Z}$-affine structure on ${{\bf R}}^{n-2}$. Let us choose a continuous function $f(z_{1},...,z_{n-2})$ and start the cuts at all points $(f(z_{1},...,z_{n-2}),0,z_{1},...,z_{n-2})$. This means that we introduce the standard non-singular ${\bf Z}$-affine structure in the region $y\neq 0$ as well as in the region $(y=0,x<f(z_{1},...,z_{n-2}))$. Near points $(y=0,x>f(z_{1},...,z_{n-2}))$ we introduce a modified ${\bf Z}$-affine structure by declaring functions

$$(y,x+\max(y,0),z_{1},\dots,z_{n-2})$$

to be ${\bf Z}$-affine coordinates. This gives an example of a “curved” singular set $B^{sing}$ of codimension 2. Since function $f$ can be approximated by PL functions, the above ${\bf Z}$-affine structure can be deformed to a PL one.

Let $B$ be a connected compact oriented topological surface, $B^{sing}\subset B$ a finite set. Assume that $B^{sm}=B\setminus B^{sing}$ carries a ${\bf Z}$-affine structure such that for any $x\in B^{sing}$ there exists a small neighborhood $U$ such that $U=\cup_{i\in I}U_{i}$ where $I$ is a finite set and each $U_{i}$ is affine equivalent to a germ of an angle in ${{\bf R}}^{2}$, with $x$ being the apex of each angle.

The aim of this section is to define a map $i_{loc}:B^{sing}\to{\frac{1}{12}}{{\bf Z}}$ (which depends only on ${\bf Z}$-affine structure near $B^{sing}$) and prove the following result (a kind of Gauss-Bonnet theorem).

We start with the construction of $i_{loc}$. Let us denote by $\widetilde{SL(2,{{\bf Z}})}$ the pre-image of $SL(2,{{\bf Z}})$ in the universal covering $\widetilde{SL(2,{{\bf R}})}$ of the group $SL(2,{{\bf R}})$. The group $\widetilde{SL(2,{{\bf R}})}$ contains $\pi_{1}(SL(2,{{\bf R}}))\simeq{{\bf Z}}$. Let $u$ be a generator of the latter (it belongs also to $\widetilde{SL(2,{{\bf Z}})}$).

We have an exact sequence of groups

$$1\to{{\bf Z}}\to\widetilde{SL(2,{{\bf Z}})}\to PSL(2,{{\bf Z}})\to 1\,\,.$$

Notice that $PSL(2,{{\bf Z}})$ is a free product ${{\bf Z}}/2*{{\bf Z}}/3$. Moreover, in the above exact sequence ${{\bf Z}}$ is embedded into the center of $\widetilde{SL(2,{{\bf Z}})}$. Notice that $u$ is the image of $2\in{{\bf Z}}$. One can choose representatives $a_{2},a_{3}$ of the standard generators of $PSL(2,{{\bf Z}})$ in such a way that $\widetilde{SL(2,{{\bf Z}})}$ is generated by $u,a_{2},a_{3}$ subject to the relations $a_{2}^{2}=a_{3}^{3},a_{2}^{4}=a_{3}^{6}=u$. This gives a homomorphism of groups $\phi:\widetilde{SL(2,{{\bf Z}})}\to{{\bf Z}}$ such that $\phi(a_{2})=3,\phi(a_{3})=2,\phi(u)=12$. Dividing by $12$ we obtain a homomorphism $i:\widetilde{SL(2,{{\bf Z}})}\to{\frac{1}{12}}{{\bf Z}}$ such that $i(u)=1$.

Let us consider a topological $S^{1}$-bundle $E$ over $B$, such that the fiber over $x\in B$ is the union of all affine rays outcoming of $x$. Then the restriction of $E$ to $B^{sm}$ is just the spherical bundle. Let us pick $x_{0}\in B^{sm}$ and remove it from $B$ together with small neighborhoods of all points $B^{sing}$. We denote by $B_{1}$ the topological space obtained in this way. We can trivialize the tangent bundle over $B_{1}$ (we choose a $C^{\infty}$-trivialization, compatible with $SL(2,{{\bf R}})$-structure) in such a way that it extends to a continuous trivialization of $S^{1}$-bundle $E$ over $B\setminus\{x_{0}\}$. Let $\alpha\in\Omega^{1}(B_{1})\otimes sl(2,{{\bf R}})$ be a $1$-form defined by means of the affine structure $\nabla$ on $B_{1}$. Then $d\alpha+{\frac{1}{2}}[\alpha,\alpha]=0$ and $\alpha$ defines a flat connection on a trivial $\widetilde{SL(2,{{\bf R}})}$-bundle on $B_{1}$. It gives a a monodromy representation $\pi_{1}(B_{1})\to\widetilde{SL(2,{{\bf Z}})}$ defined up to a conjugation. Composing it with the homomorphism $i$ we obtain a homomorphism $\pi_{1}(B_{1})\to{\frac{1}{12}}{{\bf Z}}$. Since ${\frac{1}{12}}{{\bf Z}}$ is an abelian group, the latter homomorphism is a composition $\pi_{1}(B_{1})\to H_{1}(B_{1},{\bf Z})\to{\frac{1}{12}}{{\bf Z}}$. Let us pick small circles $[\gamma_{x}]\in H_{1}(B,{\bf Z})$ for each $x\in B^{sing}$. Then the above homomorphism gives us a number denoted by $i_{loc}(x)\in{\frac{1}{12}}{{\bf Z}}$.

Proof of the Theorem. Let us pick up a small circle $[\gamma_{x_{0}}]\in H_{1}(B_{1},{\bf Z})$ around $x_{0}$. Then $\sum_{x\in B^{sing}}[\gamma_{x}]+[\gamma_{x_{0}}]=0$. The monodromy around $x_{0}$ can be easily computed via the winding number of the induced vector field (section of $E_{|\gamma_{x_{0}}}$) and is equal to $-\chi(B)u$. Applying homomorphism $i$ we obtain the result. $\blacksquare$

Proof. It is easy to see that for each point $x\in B^{sing}$ one has $i_{loc}(x)={\frac{1}{12}}$. Then from Gauss-Bonnet theorem one deduces that $\chi(B)=2-2g$, where $g$ is the genus of the Riemann surface $B$. Then we have

$${\frac{|B^{sing}|}{12}}=2-2g\,\,.$$

Since LHS is non-negative we conclude that either $g=1$ or $g=0$. In the first case $|B^{sing}|=0$ and we have a ${\bf Z}$-affine structure on a torus. In the second case we have $|B^{sing}|=24$ and $g=0$. $\blacksquare$

Similarly, for the affine structure with the monodromy at each point conjugate to

$$\left(\begin{array}[]{cc}1&4\\ 0&1\end{array}\right)$$

one has $i_{loc}(x)=\frac{1}{3}$, and we have six singular points on $S^{2}$ (see Section 4.2.5).

Let $X=X/K$ be a smooth proper algebraic variety over a non-archimedean field $K$, $\dim\,X=n$ and $\Omega\in\Gamma(X,\Omega_{X}^{n})$ be a non-zero top degree form on $X$. We will associate canonically with the pair $(X,\Omega)$ a piecewise-linear compact space $Sk(X,\Omega)\subset X^{an}$ such that $\dim_{{\bf R}}Sk(X,\Omega)\leq n$.

Let us assume for simplicity that $K={{\bf C}}((t))$ and $X$ is defined over ${{\bf C}}_{t}^{mer}\subset K$. Analytic space $X^{an}$ contains a dense subset $X_{Div}$ of divisorial points corresponding to irreducible components of special fibers of all snc models ${\cal X}$ of $X$ (see Appendix A):

$$X_{Div}=\cup_{\cal X}i_{\cal X}(V_{S_{\cal X}})\,\,\,,$$

where $V_{S_{\cal X}}$ is the set of vertices of the Clemens polytope $S_{\cal X}$.

Top degree form $\Omega$ gives rise to a map $\psi_{\Omega}:X_{Div}\to{\bf Q}$. Namely, if $p:{\cal X}\to Spec({{\bf C}}_{t}^{mer})$ is a snc model and $D\subset{\cal X}_{0}$ is an irreducible divisor of the special fiber then we define

$$\psi_{\Omega}(D)={{ord_{D}(\Omega\wedge dt/t)}\over{{ord_{D}(p^{\ast}(t))}}}\,\,.$$

Here $\Omega\wedge dt/t$ is a meromorphic top degree form on ${\cal X}/{{\bf C}}$.

It is easy to show that $\psi_{\Omega}(D)$ depends only on the point $i_{\cal X}(D)\in X^{an}$. Function $\psi_{\Omega}$ is (globally) bounded from below.

Let ${\cal X}$ be a snc model. We will explain how to describe $Sk(X,\Omega)$ in terms of ${\cal X}$ and $\Omega$. In fact it is a nonempty simplicial subcomplex of $i_{\cal X}(S_{\cal X})$.

Let us call ${\cal X}$-essential a divisor $D_{i}\subset{\cal X}_{0}$ such that

$$\psi_{\Omega}(D_{i})=\min_{D_{j}\in{\cal X}_{0}}\psi_{\Omega}(D_{j})\,\,.$$

A nonempty collection $D_{i_{1}},...,D_{i_{l}}$ of divisors in ${\cal X}_{0}$ is called ${\cal X}$-essential if all $D_{i_{k}}$ are ${\cal X}$-essential, the intersection $D_{i_{1}}\cap D_{i_{2}}\cap...\cap D_{i_{l}}$ is non-empty and does not belong to the closure of the divisor of zeros of $\Omega$ is ${\cal X}\setminus{\cal X}_{0}$.

Sketch of the proof. Notice that for any snc model ${\cal X}$ the subset $S_{\cal X}({\bf Q})\subset S_{\cal X}$ consisting of points with rational barycentric coordinates is mapped by $i_{\cal X}$ into $X_{Div}$. Namely, we can modify ${\cal X}$ by blowing up at nonempty intersections of irreducible components of the special fiber and then continue this process indefinitely. Divisorial points obtained in this way exhaust all points of $i_{\cal X}(S_{\cal X}({\bf Q}))$.

We will prove that the set of essential points in $X^{an}$ coincides with $i_{\cal X}(S_{\cal X}({\bf Q})\cap Sk({\cal X},\Omega))$. First of all, a direct computation shows that $\psi_{\Omega}$ being restricted to $i_{\cal X}(S_{\cal X}({\bf Q}))$ achieves its absolute minimum on $i_{\cal X}(S_{\cal X}({\bf Q})\cap Sk({\cal X},\Omega))$. Secondly, another straightforward computation shows that the latter set does not change under blow-ups of first and second type (see Section A.5 in Appendix A). This concludes the proof. $\blacksquare$

For Calabi-Yau manifold $X$ we will denote $Sk(X,\Omega)$ simply by $Sk(X)$, as there exists only one (up to a scalar) non-zero top-degree form $\Omega$ on $X$ and $Sk(X,\lambda\Omega)=Sk(X,\Omega)\,\,\,\forall\lambda\in K^{\times}$.

One can prove that the PL space $Sk(X,\Omega)$ is in fact a birational invariant. Moreover the group $Aut^{brt}(X)$ of birational automorphisms of $X$ acts on the skeleton by ${{\bf Z}}PL$ transformations. In order to obtain non-trivial examples of such actions we need Calabi-Yau manifolds with large groups of birational automorphisms. An example of a ${{\bf Z}}PL$-action is considered in the next subsection.

There are many families of Calabi-Yau varieties with huge groups of birational automorphisms. The following example we learned from D.Panov and D.Zvonkine. For any real numbers $l_{1},\dots,l_{n}>0$ we can consider the space of planar $n$-gons with the length of edges equal to $l_{1},\dots,l_{n}$, modulo the group of orientation-preserving motions. This space can be identified with the space of solutions of the following system of equations

$$\sum l_{i}z_{i}=0,\,\,\,\sum l_{i}z_{i}^{-1}=0$$

where $(z_{1}:\dots:z_{n})\in{\bf C}P^{n-1}$ is a point satisfying the reality condition $|z_{i}|=1,\,\,i=1,\dots,n$. Hence we obtain a singular subvariety of ${\bf C}P^{n-1}$ of codimension $2$, depending on parameters $l_{1},\dots,l_{n}$. One can check that this variety is birationally isomorphic to a non-singular Calabi-Yau variety. For any proper set $I\subset\{1,\dots,n\}$, $2\leq|I|\leq n-2$ we have a birational involution $\sigma_{I}$ defined by the formula

$$\sigma_{I}^{*}(z_{i})=\left\{\begin{array}[]{ll}c/z_{i}&\mbox{ if $i\in I$}\\ z_{i}&\mbox{ if $i\notin I$}\end{array}\right.$$

where $c:=\frac{\sum_{i\in I}l_{i}z_{i}}{\sum_{i\in I}l_{i}/z_{i}}$.

We do not know at the moment the structure of the group $G_{n}$ generated by involutions $\sigma_{I}$. One can obtain easily explicit formulas for the action of $G_{n}$ by piecewise-linear homemorphisms of $S^{n-3}$. Length parameters $l_{i}$ should be replaced by elements of a non-archimedean field $K$ with “generic” norms $\lambda_{i}=val_{K}(l_{i})\in{\bf R}$. Denote by $\zeta_{i},\,\,i=1,\dots,n$ real variables which have the meaning of valuations of variables $z_{i}\in K$. Sphere $S^{n-3}$ is obtained in the following way. In ${\bf R}^{n}$ we consider the intersection of two subsets:

$$\{(\zeta_{1},\dots\zeta_{n})\,|\,\,\,\min_{i}(\lambda_{i}+\zeta_{i})\mbox{ is achieved at least twice}\}$$

and

$$\{(\zeta_{1},\dots\zeta_{n})\,|\,\,\,\min_{i}(\lambda_{i}-\zeta_{i})\mbox{ is achieved at least twice}\}$$

and then take the quotient by the action of ${\bf R}$:

$$(\zeta_{1},\dots\zeta_{n})\rightarrow(\zeta_{1}+c,\dots\zeta_{n}+c)$$

corresponding to the projectivization. For appropriately chosen $(\lambda_{1},\dots,\lambda_{n})$ we obtain a set which is the union of $S^{n-3}$ with several “wings” going to infinity. The action of the involution $\sigma_{I}$ is obtained from algebraic formulas from above, in which one replace non-archimedean variables by real ones, addition by minimum and multiplication (division) by addition (subtraction).