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 $\Gamma_{v}$.

Let $X=(X_{t})_{t\to 0}$ be a maximally degenerate K3 surface over the field ${{\bf C}}_{t}^{mer}$ (see Section 5.1). We denote by $\Lambda_{X}$ the quotient group $[\gamma_{0}]^{\perp}/{{\bf Z}}[\gamma_{0}]$ where $[\gamma_{0}]\in H_{2}(X_{t},{\bf Z})$ is the vanishing cycle. Then $\Lambda_{X}\simeq\Lambda_{2,18}$. Let us assume that the monodromy acts trivially on $\Lambda_{X}$.

We define a natural homomorphism $\rho_{X}:\Lambda_{X}\to({{\bf C}}_{t}^{mer})^{\times}$ by the formula

$$\rho_{X}([\gamma])=\exp\left(2\pi i{\int_{\gamma}\Omega_{t}\over{\int_{\gamma_{0}}\Omega_{t}}}\right),\,\,\,[\gamma]\in[\gamma_{0}]^{\perp}\,\,.$$

One can give a more abstract definition of $\rho_{X}$ in terms of the variation of Hodge structure. It is easy to see that $\left(val_{{{\bf C}}_{t}^{mer}}\circ\rho_{X}\right)([\gamma])=(v_{X},[\gamma])$ where $v_{X}\in\Lambda_{X}$ is a vector such that $(v_{X},v_{X})>0$, and $val_{{{\bf C}}_{t}^{mer}}$ is the standard valuation on the field ${{\bf C}}_{t}^{mer}\subset{{\bf C}}((t))$.

Let $X^{an}$ be the corresponding analytic K3 surface over the field $K={{\bf C}}((t))$. We have an analytic torus fibration over $S^{2}\setminus\{x_{1},...,x_{24}\}$ which can be extended to a continuous map $X^{an}\to S^{2}$. Let us call such an extension a singular analytic torus fibration with standard singularities.

Notice that for K3 surfaces any birational automorphism is biregular. Hence the group of birational automorphisms $Aut^{brt}(X)$ acts by a ZPL-transformations of the sphere $S^{2}$ which is equipped with a singular ${\bf Z}$-affine structure (see Section 6.6), i.e. we have a homomorphism

$$Aut^{brt}(X)=Aut(X)\to Aut_{{{\bf Z}}PL,v_{X}}(Sk(X^{an},\Omega))\simeq Aut_{{{\bf Z}}PL,v_{X}}(S^{2})\,\,.$$

Let us consider the special case when vector $v$ is a lattice vector, i.e. $v\in\Lambda_{2,18}$. In A-model picture it corresponds to the integrality of the class $[\omega]$ of symplectic 2-form. In B-model this means that the non-archimedean field $K$ has valuation in ${\bf Z}\subset{\bf R}$. In terms of ${\bf Z}$-affine structures it means that the monodromy of the affine connection is reduced to $SL(2,{\bf Z})\ltimes{\bf Z}^{2}$. Group $\Gamma_{v}$ is a subgroup (and also a quotient group) of an arithmetic subgroup in the Lie group $SO(1,18)$. Also in this case there is a $\Gamma_{v}$-invariant notion of a point with integer coordinates on $B\simeq S^{2}$, as well of points with coordinates in $\frac{1}{N}{\bf Z}$ for any integer $N\geq 1$. The number $M_{v,N}$ of such points is finite. It is not hard to see that $M_{v,N}=Area_{v}+2={{(v,v)}\over{2}}N^{2}+2$ where $Area_{v}$ is the area of $B$ with a ${\bf Z}$-PL structure corresponding to $v$. This is analogous to the Riemann-Roch formula $rk\,\left(\Gamma(X_{\bf C},L^{\otimes N})\right)=\int_{X}{{c_{1}(L)^{2}}\over{2}}+2$ for an ample line bundle $L$ on a complex K3 surface $X_{\bf C}$.

The action of $\Gamma_{v}$ on $S^{2}$ gives rise to a homomorphism $\Gamma_{v}\to S_{M_{v,n}}$ where $S_{M_{v,n}}$ is the symmetric group. Also the action gives a homomorphism from $\Gamma_{v}$ to the mapping class group $\pi_{1}({\cal{M}}_{0,M_{v,N}}^{unord})$, the fundamental group of the moduli space of genus zero complex curves with $M_{v,N}$ unordered distinct marked points. The last group is closely related to the braid group. The conclusion is that we have constructed homomorphisms from arithmetic groups to a tower of braid groups.

One can deduce from Torelli theorem an interpretation of $\Gamma_{v}$ as a quotient group of the fundamental group of a neighborhood $U$ of a cusp in 19-dimensional moduli space of polarized complex algebraic K3-surfaces, where vector $v$ corresponds to the polarization. Therefore the homomorphism $\Gamma_{v}\to S_{M_{v,n}}$ gives a finite covering $U^{\prime}$ of $U$. One may wonder whether there exists a line bundle over $U^{\prime}$ whose direct image to $U$ coinsides with the direct image of the sheaf $L^{\otimes N}$ from the universal family of K3 surfaces (this question is in spirit of some ideas of Andrey Tyurin, see e.g. [Tyu]).