Recall that in Section 3.3 we constructed a $38$-dimensional space ${\cal P}$ parameterizing integrable systems $(X,\omega)\to B$ with $B\simeq S^{2}$. The space ${\cal P}$ carries a codimension $20$ foliation ${\cal F}$ corresponding to small deformations of integrable systems which do not change the invariant $[\rho]$ of the local system $\rho:\pi_{1}(B^{sm})\to SL(2,{{\bf Z}})\ltimes{{\bf R}}^{2}$. We explained that the fundamental group of a leaf of ${\cal F}$ acts by PL homeomorphisms of $S^{2}$. Here we are going to give a (partial) description of ${\cal P}$ and ${\cal F}$ in cohomological terms using Torelli theorem (see Appendix B).

An algebraic polarized K3 surface $X/{{\bf C}}$ elliptically fibered over ${{\bf C}}P^{1}$, equipped with a holomorphic volume form $\Omega$ can be encoded by the data $(\Lambda,(\cdot,\cdot),[\omega],[\Omega],[\gamma],{\cal K}_{X})$, where

1. 1.

   $(\Lambda,(\cdot,\cdot),{\bf C}[\Omega],{\cal K}_{X})$ is a K3 period data;

2. 2.

   $[\omega],[\gamma]\in\Lambda$, $\Omega\in\Lambda\otimes{{\bf C}}\,\,$;

3. 3.

   $[\omega]\in{\cal K}_{X},\,\,\,\gamma\in\partial{\cal K}_{X},\,\,\,([\omega],[\Omega])=([\gamma],[\Omega])=([\gamma],[\gamma])=0\,\,$;

4. 4.

   $\gamma$ is a non-zero primitive lattice vector.

Here $[\omega]$ is the class of polarization (projective embedding) of $X$, $[\gamma]$ is dual to the class of generic fiber of the elliptic fibration $\pi:X\to{{\bf C}}P^{1}$.

Perhaps one can express in cohomological terms the fact that $\pi$ has exactly $24$ critical values. The latter is an open condition.

Let $L\subset H_{2}(X,{{\bf Z}})$ be a subgroup consisting of homology classes which can be represented by cycles which are projected into graphs in $B^{sm}$ (such cycles are circle fibrations over graphs). When we move along a leaf of ${\cal F}$ then the pairing of $Re([\Omega])$ with $L$ remains unchanged (see Section 3.1.1). Clearly $L\subset[\gamma]^{\perp}$, and moreover, one can check that $L=[\gamma]^{\perp}\simeq{{\bf Z}}^{21}$. The pairing with $Re([\Omega])$ gives a map $\Lambda_{2,18}:=[\gamma]^{\perp}/{{\bf Z}}[\gamma]\to{{\bf R}}$, where $\Lambda_{2,18}$ is the following even unimodular lattice of signature $(2,18)$:

$$\Lambda_{2,18}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\oplus\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\oplus\left(-E_{8}\right)\oplus\left(-E_{8}\right)\,\,,$$

where $-E_{8}$ is the Cartan matrix for Dynkin diagram $E_{8}$ taken with the minus sign.

The functional $(Re[\Omega],\cdot)$ on $\Lambda_{2,18}$ can be represented as $(v_{Re[\Omega]},\cdot)$ where $v_{Re[\Omega]}\in\Lambda_{2,18}\otimes{{\bf R}}$ is a vector with the strictly positive square norm. One can show that the (non-Hausdorff) space of leaves of ${\cal F}$ is canonically identified with the set $\{v\in\Lambda_{2,18}\otimes{{\bf R}}|(v,v)>0\}/Aut(\Lambda_{2,18})$.

The fundamental group of the leaf ${\cal F}_{v}$ corresponding to a vector $v\in\Lambda_{2,18}\otimes{{\bf R}}$ maps onto the group $\Gamma_{v}\subset Aut(\Lambda_{2,18})$. This group is (up to a conjugation) the stabilizer in $(Aut(\Lambda_{2,18}),(\cdot,\cdot)_{2,18},v)$ of the cone $K_{v}$, which is a connected component of the set

$$\{w\in\Lambda_{2,18}\otimes{{\bf R}}|(w,v)=0,(w,w)>0\}\setminus\bigcup_{\gamma\in\Lambda_{2,18},(\gamma,\gamma)=-2,(\gamma,v)=0}H_{\gamma}$$

and $H_{\gamma}\in\Lambda_{2,18}\otimes{{\bf R}}$ is the hyperplane orthogonal to $\gamma$ (cf. Appendix B). Let us denote by $Aut_{{{\bf Z}}PL,v}(S^{2})$ the group of piecewise-linear transformations of $S^{2}$ with integer linear part. Index $v$ signifies the dependence of ${\bf Z}PL$-structure on $S^{2}$ on $v$.

One can consider the whole moduli space ${\cal M}_{44}$ of ${\bf Z}$-affine structures on $S^{2}$ with $24$ standard singularities. This space is a Hausdorff orbifold (with a natural ${\bf Z}$-affine structure!) of dimension $44$, and it carries a foliation of codimension $20$ as before. It seems that using our main result (Theorem 5 in Part III) together with certain natural assumption (see Conjecture 11 in Section 11.6) one can show that the action by ${\bf Z}PL$ transformations of $S^{2}$ of the fundamental group of leaves of the foliation on the larger space ${\cal M}_{44}$ is again reduced to the action of $\Gamma_{v}$.