First of all, one can introduce a small parameter $\hbar\in K,|\hbar|<1$ of noncommutativity in the picture, coordinates $\xi,\eta$ will not commute but instead satisfy the relation

$$\eta\xi=\xi\eta\exp(\hbar)\,\,.$$

For such a noncommutative analytic torus one can still define sheaf ${\cal O}^{can}_{\hbar}$ on ${\bf R}^{2}$ by the “same” formula as in the commutative case:

$${\cal O}^{can}_{\hbar}(U)=\left\{\sum_{n,m\in{\bf Z}}c_{n,m}\xi^{n}\eta^{m}\,|\,\forall(x,y)\in U\,\,\,\sup_{n,m}\left(\log|c_{n,m}|+nx+my\right)<\infty\right\}$$

where $U\subset{\bf R}^{2}$ is connected. Also one can construct a non-commutative deformation of the model sheaf near the singular point. All arguments with the groups work as well. In this way we will obtain a kind of quantized K3 surface over a non-archimeden field.

Secondly, we believe that one can generalize our construction to higher dimensions. Instead of lines there will be codimension one walls which should be flat hypersurfaces with respect to ${\bf Z}$-affine structure and carry foliations by parallel lines. Generically on the intersection of two such foliated hypersurfaces one can “separate” variables into the product of a purely 2-dimensional situation studied in the present paper, and $n-2$ dummy variables. Presumably everywhere except a countable union of codimension 2 subsets one can use 2-dimensional factorization and define gluing volume preserving maps. One can hope that by a kind of Hartogs principle the sheaf will have a canonical extension to the whole space $B$.