11.6 Independence and uniqueness [03X5]

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11.6 Independence and uniqueness

It is natural to ask how the above construction of the $K$ -analytic K3 surface $(X^{an},\Omega)$ depends on the choice of the set ${\cal L}$ of lines. We know that the “periods” of $\Omega$ (they are encoded in the initial $K$ -affine structure) do not depend on ${\cal L}$ (see Sections 7.3, 10.4). In the light of Torelli theorem (see Appendix B) it is natural to formulate the following conjecture.

Conjecture 11

The isomorphism class of the pair $(X^{an},\Omega)$ does not depend on the choice of the set ${\cal L}$ of lines.

More precisely, the change of ${\cal L}$ corresponds to the change of the projection $\pi:=\pi_{\cal L}:X^{an}\to B$ (see Section 7.3).

Remark 4

For $B=S^{2}$ and $B^{sing}=\{x_{1},\dots,x_{24}\}$ with the standard singular ${\bf Z}$ -affine structure we have constructed a $K$ -analytic K3 surface depending on $20$ parameters in $K^{\times}$ . More precisely, we have a $20$ -dimensional $K$ -analytic space of conjugacy classes of representations

\pi_{1}(S^{2}\setminus B^{sing})\to SL(2,{{\bf Z}})\ltimes(K^{\times})^{2}

such that the monodromy around each singular point is conjugate to the pair $(A,(1,1))$ where $A\in SL(2,{{\bf Z}})$ is equal to

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

(compare with Section 3.3).