For $x\in Y$ let $U\subset U^{\prime}$ be convex neighborhoods of $x$ such that $U$ is relatively compact in $U^{\prime}$. Let $l\in{\cal L}$. Then there is a natural embedding $f_{l}^{-1}(U)\to f_{l}^{-1}(U^{\prime})$.

In plain words a piece $L$ of $l$ is an equivalence class of a connected interval of $l\cap U$. Two connected intervals are equivalent if they are contained in a larger connected interval of $l\cap U^{\prime}$. The sole purpose of the introduction of the notion of a piece is to avoid some pathology. Namely, for any pair $(U,U^{\prime})$ as above, any $l\in{\cal L}$ and any $T\in{\bf R}_{>0}$, there is only a finite number of pieces of $l$ in $(U,U^{\prime})$ which have points with time parameter $t\in(0,T)$.

Let $L$ be a piece of $l$ defined by a pair $(U,U^{\prime})$. Then one can define an affine function $ord_{L}\in Aff_{{\bf Z},Y}(U^{\prime})$ in the following way. Let $t>0$ be such that $f_{l}(t)$ belongs to $L$. Since $U^{\prime}$ is convex, there is a unique continuation of $ord_{l}(t)$ to $U^{\prime}$. This is an affine function which does not depend on the choice of $t$. We will denote it by $ord_{L}$.

For any germ of a symplectomorphism $\varphi\in Symp_{p}$ at a point $p\in Y$ we define its convergence region as the maximal convex subset $\Omega(\varphi)\subset T_{p}Y$ such that the pullback $\exp_{p}^{\ast}(\varphi)$ extends to $\Omega(\varphi)$. Since the definition of $\varphi_{l}$ (and hence its convergence region) is covariant with respect to the affine connection we have the following result:

It is clear that one can define convergence regions for symplectomorphisms associated with pieces of lines, and a similar property holds for them.

Let us suppose that our collection of lines satisfies the following assumptions:

Assumption A1

There is a smooth metric $g=g_{B}$ and a collection of balls $D(s,r_{s})$ with centers at $s\in B^{sing}$ such that each ball $D(s,r_{s})$ contains exactly two lines $l_{\pm}\in{\cal L}_{in}$ outcoming of $s$.

Assumption A2

There exists $\varepsilon>0$ such that for any $p=f_{l}(t)\in Y^{\prime}:=B\setminus\cup_{s\in B^{sing}}D(s,r_{s})$ the distance in $T_{p}Y$ between $0\in T_{p}Y$ and the boundary of $P_{l,t}$ is greater or equal to $\varepsilon$.

We are going to show that such a collection does exist in Section 11.

Assumptions A1 and A2 are very artificial, they do not hold in physical picture which is the main motivation for the construction. It is quite possible that they can be weakened or even omitted. The main purpose of introducing them here is the possibility to define the sheaf of analytic functions by simple gluing. In complex geometry it is similar to the gluing of closed Riemann surfaces with boundaries by the mean of real-analytic identifications of the boundaries. It is well-known that one can replace real-analytic maps by smooth ones (or even by quasi-symmetric continuous maps). Maybe the rest of this section is unnecessary, and unpleasant technical arguments in Section 11.5 can be avoided.

Denote by $W_{\cal L}:=\cup_{l\in{\cal L}}f_{l}([0,+\infty))$ the set of all points of all lines. It has measure zero. Let $p$ be a point of $Y$. We consider two convex neighborhoods $U\subset U^{\prime}$ of $p$ such that $U$ is relatively compact in $U^{\prime}$.

For any two points $x,y$ belonging to $U\setminus W_{\cal L}$, and a path $\gamma$ joining $x$ and $y$ in $U$, we would like to define an infinite ordered product $i_{x,y}^{\gamma}$ of transformations $\varphi_{L}^{\pm 1}$, where factors correspond to the intersection points of $\gamma$ with all possible pieces $L$ relative to $(U,U^{\prime})$. Factors in the infinite product are ordered according to the time parameter of $\gamma$, the sign corresponds to the mutual position of orientations of $\gamma$ and a piece $L$ at the intersection point.

In order to give a precise meaning to the infinite product the neighborhood $U$ of $p$ should be sufficiently small. Then we will have an analytic continuation of symplectomorphisms $\varphi_{L}$ to $U$, and the convergence of the infinite product. We are also going to prove that the product is independent of the choice of path $\gamma$. In order to achieve these goals it suffices to assume:

C1

for any $l,t$ such $f_{l}(t)\in U$ the set $(\exp_{f_{l}(t)})^{-1}(U)$ is contained in $P_{l,t}\,$;

C2

for any $C\in{\bf R}$ there is only a finite number of pieces $L$ of lines in $U$ such that $\inf_{x\in U}ord_{L}(x)<C$.

Proof. Condition C1 implies that all transformations $\varphi_{L}$ admit an analytic continuation to $U$. Let us introduce a decreasing filtration by positive real numbers $Symp^{\geq r}(U),\,\,r\in{\bf R},r\geq 0$ on group $Symp(U)$ by the formula

$$\left\{g\in Symp(U)\,|\,\,\log|\xi^{\prime}/\xi-1|,\log|\eta^{\prime}/\eta-1|<-r\,\mbox{ where }(\xi^{\prime},\eta^{\prime})=g((\xi,\eta))\,\right\}$$

This is a complete filtration, and condition C2 implies that in any quotient $Symp(U)/Symp^{\geq r}(U)$ only a finite number of elements $\varphi_{L}$ are non-trivial. Therefore we can define the product in the quotient group.

In order to prove independence of $\gamma$, we consider the quotient group $Symp(U)/Symp^{\geq r}(U)$, and the finite $1$-dimensional CW-complex (graph) consisting of finitely many pieces $L$, such that $\varphi_{L}\neq 1$ in the quotient. For each vertex $v$ of the graph there is a natural cyclic order on the edges incident to $v$. The product $\varphi_{v}=\prod_{L}\varphi_{L}^{\pm 1}$ taken in the cyclic order over the set of edges incident to $v$ is equal to $id$ (this follows from the construction of $\varphi_{l}$ via factorizations). Since $U$ is simply-connected, we conclude that the image of $i_{x,y}^{\gamma}$ in $Symp(U)/Symp^{\geq r}(U)$ does not depend on $\gamma$. Using completeness of the filtration we see that $i_{x,y}:=i_{x,y}^{\gamma}$ does not depend on $\gamma$. Proof of the identity $i_{x,y}i_{y,z}=i_{x,z}$ is similar. $\blacksquare$

Proof. Assumption A1 implies that the result near any singular point $s\in B^{sing}\,$, as there are only two lines near $s$. If we are far from $B^{sing}$ then obviously A2 implies C1.

In order to check C2 we prove the following lemma

Proof: We proceed by induction in “complexity of the line”. Let $\delta\in{\bf R}_{>0}$ be the infimum of $ord_{l}(t)(f_{l}(t))$ where $l\in{\cal L}_{in}$ has a collision at time $t$. This number is strictly positive because the number of initial lines is finite, and by A1 there is no collisions at small times. Observe that the value of $ord_{l}(0)$ at the beginning of any composite line $l$ is greater or equal to the sum $ord_{l_{1}}(t_{1})+ord_{l_{2}}(t_{2})$. Therefore the inequality in the lemma implies that the number of collisions is bounded from above by $C/\delta$. Also we have an upper bound on integer coefficients $(n_{1},n_{2})$ in each collision (see Axiom $\bf 3b)$ in Section 9.2). Let us observe that the length of each edge of the ansector tree of $l$ is also bounded from above by $A\,ord_{l}$, for some absolute constant $A>0$. Hence we have only finitely many possibilities for intersections. $\blacksquare$

For point $p\in Y$ which is far from $B^{sing}$ we chose as $U$ a neighborhood of radius $\epsilon^{\prime}\ll\epsilon$ where $\epsilon>0$ is constant from Assumption A2. Then for any point of a line $f_{l}(t)\in U$ we will have the inclusion

$$U\subset\exp_{f_{l}(t)}\left(\frac{1}{2}P_{l,t}\right)\,\,.$$

This implies that $ord_{L}$ in $U$ for the corresponding piece $L$ is bounded below by

$$\frac{1}{2}ord_{l}(t)(f_{l}(t))\,\,.$$

Since (by the last lemma) there exists only a finite number of pieces $L$ intersecting such $U$, we obtain convergence condition C2. $\blacksquare$

For any point $p\in Y$ and a neighborhood $U$ satisfying conditions C1 and C2 we define the sheaf ${\cal O}^{modif}_{U}$ as the result of the identification of copies of the sheaf $\left({\cal O}^{can}_{Y}\right)_{|U}$ labeled by points $x\in U\setminus W_{\cal L}$, by isomorphisms $i_{x,y}$. It follows from formulas in Section 8 that near singular points one can identify canonically this sheaf with the restriction of the sheaf ${\cal O}^{model}_{{\bf R}^{2}}$ to a punctured neighborhood of $(0,0)\in{\bf R}^{2}$.

Proof. Existence of $\Omega$ follows from the fact that all modifications associated with lines are symplectomorphisms. In order to finish the proof it suffices to check that the modification associated with a line does not change the $K$-affine structure on $Y$. In local coordinates we may assume that $\Omega={d\xi\over\xi}\wedge{d\eta\over\eta}$ and the modification is of the form $\varphi(\xi,\eta)=(\xi f(\eta^{-1}),\eta)$, where $f(z)=1+\sum_{n\geq 1}c_{n}z^{n}\in K[[z]]$ is convergent in an appropriate domain. We need to check that the automorphism $\varphi$ acts trivially on the quotient sheaf $\pi_{\ast}({\cal O}_{X}^{\times})/\ker\,p_{\Omega}$ (see Section 7.2 for the notation). This check reduces to the calculation of

$$p_{\Omega}\left({\xi f(\eta)\over\xi}\right)=\exp\left({Res(\Omega\log(\xi f(\eta)/\xi)))\over Res(\Omega)}\right)\,\,.$$

The latter is equal to $\exp\left(Res(\Omega\log(f(\eta)))\right)=1$ because $\log(f(\eta^{-1}))$ belongs to $\eta^{-1}K[[\eta^{-1}]]$ and therefore has no constant term. $\blacksquare$

Thus, we have a solution of the Lifting Problem under Assumptions A1 and A2.

We would like to show that there exists a smooth metric $g$ and a collection of lines satisfying the Assumptions A1 and A2.

Let $g_{0}$ be an arbitrary smooth metric, flat near singular points. We define germs of lines $l\in{\cal L}_{in}$ in such a way that for each $s\in B^{sing}$ in local coordinates these lines are given by $\{(0,y)|y>0\}$ and $\{(0,y)|y<0\}$. The metric $g$ will coincide with $g_{0}$ in a sufficiently small neighborhood $U=\cup_{s\in B^{sing}}D(s,r_{s})$ of the singular set. Hence Assumption A1 will be satisfied.

In order to construct the whole family of lines we introduce a $3$-dimensional manifold ${\cal M}$ consisting of pairs $(x,P)$ where $x\in B\setminus\overline{U}_{1}$ and $P$ is a half-plane in $T_{x}B$ whose boundary contains zero. Here $U_{1}:=\cup_{s\in B^{sing}}D(s,2r_{s})$ is a larger neighborhood of $B^{sing}$.

We would like to construct a smooth section $v:(x,P)\mapsto v_{(x,P)}\in T_{x}B$ of the pull-back to ${\cal M}$ of the tangent bundle $TB$ satisfying the following conditions:

1. 1.

   for any $(x,P)\in{\cal M}$ one has $v_{(x,P)}\in int(P)\,\,$;

2. 2.

   for any $x\in B\setminus\overline{U}_{1}$ the map $(x,P)\mapsto{{\bf R}}_{>0}^{\times}\cdot v_{(x,P)}$ is an orientation-preserving diffeomorphism

   $$S^{1}\simeq(T_{x}B^{\ast}\setminus\{0\})/{{\bf R}}_{>0}^{\times}\to S^{1}\simeq(T_{x}B\setminus\{0\})/{{\bf R}}_{>0}^{\times}\,\,\,;$$

3. 3.

   for every $l\in{\cal L}_{in}$ there exists a smooth extension of the piece of $l$ in $U_{1}$ to a larger piece intersecting $\partial U_{1}$ such that

   $$\dot{f}_{l}(t)\in{{\bf R}}_{>0}^{\times}\cdot v_{(f_{l}(t),P_{l,t})}\,\,,$$

   for such $t>0$ that $f_{l}(t)\in B\setminus\overline{U}_{1}\,\,$;

Let us associate with the section $v$ a nowhere vanishing vector field $\hat{v}$ on $T^{\ast}(B\setminus\overline{U}_{1})\setminus(Zero\,\,\,Section)$ in the following way:

• •

  For each $(x,\alpha)\in T_{x}^{\ast}B$ the vector $\hat{v}(x,\alpha)$ is tangent to the horizontal distribution associated with the flat connection $\nabla$ (the one which defines the affine structure on $B\setminus B^{sing}$).

• •

  Projection of $\hat{v}(x,\alpha)$ to $B$ coincides with $v_{(x,P_{\alpha})}$, where $P_{\alpha}=\{\gamma|(\alpha,\gamma)>0\}$.

Clearly these conditions determines $\hat{v}$ uniquely. Now we formulate last condition:

1. 4.

   there exits $r_{s}^{\prime}>2r_{s}$ such that for almost all (in the sense of Baire category) initial values $(x_{0},P_{0})\in{\cal M}$ the integral curve of $\hat{v}$ starting at $(x_{0},P_{0})$ reaches the pullback of $B\setminus\cup_{s\in B^{sing}}D(s,r_{s}^{\prime})$ in finite time.

Using the vector field $\hat{v}$ we will construct (under certain genericity assumptions) a set ${\cal L}$ of lines satisfying Assumption A1. Namely, the data consisting of a line $l$ and an integer-valued $1$-form $\alpha_{l}$ (see Section 9) will be an integral line of $\hat{v}$.

We are going to construct lines by induction by the number of collisions. Lines $l\in{\cal L}_{in}$ will be constructed using condition 3. The genericity assumption mentioned after the condition 4 is the assumption that no more than two lines collide and that initial values for newborn lines will be sufficiently generic. Conditions 1 and 4 plus genericity imply that one can parametrize any line $l\in{\cal L}$ by the new “time” $t>0$ such that the Axiom 2 is satisfied. Axiom 6 follows from the condition 2. Other axioms and the Assumption A1 will be satisfied automatically.

Now we would like to discuss Assumption A2.

Proof. In order to satisfy Assumption A2 it suffices to find such $\varepsilon>0$ that for any $x\in B\setminus\overline{U}_{1}$ and any half-plane $P_{x}\subset T_{x}B,\,\,0\in int(P_{x})$ with the distance $dist_{g_{x}}(0,\partial P_{x})=\varepsilon$, and another half-plane $P_{x}^{\prime}\subset T_{x}B$ parallel to $P_{x}$ such that $0\in\partial P_{x}^{\prime}$, one has the following property: if $P_{x+\delta tv_{(x,P_{x}^{\prime})}}$ is the half-plane obtained from $P_{x}$ by a small covariant (with respect to the affine connection $\nabla^{aff}$) shift $\delta t$ in the direction of $v_{(x,P_{x}^{\prime})}$, then

$$dist_{g_{x+\delta tv_{(x,P_{x}^{\prime})}}}(0,P_{x+\delta tv_{(x,P_{x}^{\prime})}})\geq dist_{g_{x}}(0,\partial P_{x})\,\,.$$

Here $g_{x}$ etc. denotes the induced flat metric on the tangent space $T_{x}B$. This property guarantees that the condition $dist_{g_{x}}(0,\partial P_{l,t})\geq\varepsilon$ will propagate along the line. For a new line obtained as a result of collision of $l_{1}$ and $l_{2}$ at the times $t_{1}$ and $t_{2}$ respectively one has

$$dist_{g_{x}}(0,\partial P_{l,0})\geq\min\{dist_{g_{x}}(0,\partial P_{l_{1},t_{1}}),dist_{g_{x}}(0,\partial P_{l_{2},t_{2}})\}$$

since $\partial P_{l,0}$ contains the intersection point $\partial P_{l_{1},t_{1}}\cap\partial P_{l_{2},t_{2}}$, see Figure 7.

One can easily see that the infinitesimal inequality from above is equivalent to

$$\delta tg_{x}(v_{(x,P_{x}^{\prime})},n_{P_{x}^{\prime}})+\varepsilon/2(g_{x+\delta tv_{(x,P_{x}^{\prime})}}-g_{x})(n_{P_{x}^{\prime}},n_{P_{x}^{\prime}})\geq 0$$

(the change of the distance consists of two summands: one corresponds to the shift along $\delta tv_{(x,P_{x})}$ with the fixed metric, and the other one corresponds to the change of the metric). Taking the limit $\delta t\to 0$ we arrive to the inequality for the covariant derivative of the metric with $C=2/\varepsilon$. $\blacksquare$

Now our goal is to construct the field of directions $v$ and the metric $g$ satisfying the conditions 1–4 and the inequality from the last Propostion. This will conclude the construction of the set ${\cal L}$ of lines satisfying the Assumptions A1 and A2.

Since $\partial U_{1}$ is a boundary of the convex set, we can locally model it by the graph of function $y=f(x)$ such that $f^{\prime\prime}(x)>0$, $f^{\prime}(x_{0})=0$. We may assume that $P=P_{0}$ is the upper half-plane. Then we take

$$v_{\left((x,y),P\right)}=\partial/\partial y+{(f(x)-f(x_{0}))/f^{\prime}(x)\over{f(x)-f(x_{0})+f(x)-y}}\,\,\partial/\partial x\,\,.$$

We extend this local model of $v$ near $\partial U_{1}$ to $B\setminus\overline{U}_{1}$ in such a way that conditions 1 and 2 are satisfied. It is clear that we can satisfy conditions 3,4 as well by taking a small perturbation of $v$. On Figure 8 there is a picture of the field $(x,y)\mapsto v_{\left((x,y),P_{0}\right)}$.

For an arbitrary choice of the metric $g$ we have $g(n_{P},v_{z,P})>0$ for all $(z,P)\in{\cal M}$. The problem with inequality

$$(\nabla_{v_{(z,P)}}\,g)(n_{P},n_{P})\leq Cg(n_{P},v_{(z,P)})$$

arises only as the point $z$ approaches $\partial U_{1}$. Indeed, in this case the vector $v_{(z,P)}$ can be very close to the tangent vector to $\partial P_{z}\subset T_{z}B$.

Proof. We need to check that the ratio

$${(\nabla_{v_{(z,P)}}\,g)(n_{P},n_{P})}\over{g(n_{P},v_{(z,P)})}$$

is bounded for $(z,P)\in{\cal M}$.

It suffices to prove the Lemma assuming that $U_{1}$ is the parabolic domain $\{(x,y)\in{{\bf R}}^{2}|y>x^{2}\}$ and $P$ is the upper half-plane. The vector field $v_{(z,P)}$ is given for $z=(x,y)$ by the formulas

$$v_{(z,P)}=\partial/\partial y+{x\over{4x^{2}-2y}}\,\partial/\partial x\,\,.$$

The denominator is equal to $g(n_{P},v_{(z,P)})=\langle dy,v_{(z,P)}\rangle\cdot\sqrt{g(\partial/\partial y,\partial/\partial y)}=\sqrt{g(\partial/\partial y,\partial/\partial y)}=\exp(O(1))$ near $(0,0)$.

The numerator is equal to

$${x\over{4x^{2}-2y}}f_{1}(x,y)+f_{2}(x,y)\,\,,$$

where $f_{1}(x,y)=(\nabla_{\partial/\partial x}\,g)(n_{P},n_{P})$ and $f_{2}(x,y)=(\nabla_{\partial/\partial y}\,g)(n_{P},n_{P})$ are two $C^{\infty}$-functions.

By assumption of the Lemma we have $f_{1}(0,0)=0$. Therefore $|f_{1}(x,y)|\leq const\,\max\{|x|,|w|\}$ where $w=x^{2}-y$ is a convenient local coordinate near the point $(0,0)$. Notice also that $f_{2}(x,y)=O(1)$.

Now we can estimate first summand of the numerator assuming that $|x|$ and $|w|$ are sufficiently small. As we have seen, it is bounded by

$$I:={x\over{x^{2}+w}}O(\max\{|x|,|w|\}\,\,.$$

There are three cases which we need to consider.

a) If $0<w<x^{2}$ then $I={x\over{x^{2}}}O(|x|)=O(1)$.

b) if $x^{2}\leq w<x$ then $I={x\over w}O(|x|)=O(1)$.

c) If $x\leq w\leq 1$ the $I={x\over w}O(|w|)=O(1)$.

We see that the numerator is bounded. This concludes the proof of Lemma. $\blacksquare$

Finally, we have the following result.

Proof: First of all, the condition on $g$ from Lemma 6 is the condition on a loop $g_{|T_{z}B}$ of scalar products on 2-dimensional spaces, here $z\in\partial U_{1}\simeq S^{1}$. We can write $g=\exp(\psi)g_{0}$ where $\det(g_{0})=1$ and $\psi$ is a smooth function. Then we have

$$\nabla_{e_{z}}(\exp{\psi}g_{0})=\exp({\psi})\nabla_{e_{z}}g_{0}+\exp(\psi)\partial_{e_{z}}(\psi)\,g_{0}\,\,.$$

The equation of Lemma 6 gives $\partial_{e_{z}}\psi=-(\nabla_{e_{z}}g_{0})(n_{z},n_{z})/g_{0}(n_{z},n_{z})$. The RHS of this expression is known as long as we know $g_{0}$. Hence we can say that $d\psi=\beta_{g_{0}}$, where $\beta_{g_{0}}$ is a 1-form depending on the restriction $(g_{0})_{|\partial U_{1}}$. We see that it suffices to find such $g_{0}$ that $\int_{\partial U_{1}}\beta_{g_{0}}=0$ (then $\psi$ and hence $g$ does exist).

Let us consider the functional $I(g_{0})=\int_{S^{1}}\beta_{g_{0}}$. We can interpret a metric $g_{0}$ as a point in the Lobachevsky plane ${\cal H}=SL(2,{\bf R})/SO(2)$. More precisely, let us consider the space $S$ of pairs $(g_{0},P)$ where $g_{0}$ is a positive quadratic form on ${\bf R}^{2}$ such that $\det(g_{0})=1$ and $P$ is a half-plane in ${\bf R}^{2}$ (the meaning of $P$ is the inward oriented tangent half-plane to $\partial U_{1}$ at point $z\in\partial U_{1}$). This space is naturally diffeomorphic to $S^{\ast}({\bf R}^{2})\times{\cal H}$. The latter manifold can be identified in $SL(2,{\bf R})$-equivariant way with the manifold consisting of pairs $(x,y)$, where $x\in{\cal H}$ and $y$ belongs to the absolute. Hence $(g_{0})_{|\partial U_{1}}$ is (locally) a non-parametrized path in $S$ (it would be a global path, if the bundle over $S^{1}$ given by the all metrics on $S^{1}$ with the determinant $1$ was trivial).

Next we observe that the variation $\delta I(g_{0})=\int_{N}\omega$, where $N$ is a $2$-dimensional surface bounded by the paths defined by $g_{0}$ and $g_{0}+\delta g_{0}$, and $\omega$ is a canonical $SL(2,{\bf R})$-invariant $2$-form on $S$. One can show that even by a small variation of the path defined by $g_{0}$ we can make $I(g_{0})$ an arbitrary real number. In particular, we can find $g_{0}$ such that $I(g_{0})=0$. This concludes the proof of Lemma 6. $\blacksquare$

Summarizing, we have constructed a set of lines satisfying the Assumptions A1 and A2. This concludes the proof of Theorem 5. Thus we have obtained a solution of the Lifting Problem, which is a $K$-analytic K3 surface.

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.

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).

Our construction of $(X^{an},\Omega)$ works even without the assumption $char\,k=0$ where $k$ is the residue field of $K$. This can be explained from the point of view of factorization theorem (see Section 10.4). It turns out that symplectomorphisms which appear in the infinite product in the RHS of the factorization theorem are infinite series whose coefficients are integer polynomials in the coefficients of the “parent” symplectomorphisms.

For example, let $f_{0}(z)=1+\sum_{n\geq 1}c_{n}z^{n}$ and $f_{\infty}(z)=1+\sum_{n\geq 1}d_{n}z^{n}$ be two power series convergent when $|z|<1$. Let us consider two symplectomorphisms: $F_{0}(\xi,\eta)=(\xi,\eta f_{0}(\xi^{-1}))$ and $F_{\infty}(\xi,\eta)=(\xi f_{\infty}(\eta^{-1}),\eta)$ and decompose $F_{\infty}\circ F_{0}$ into the infinite ordered product $\prod_{\to}(F_{\lambda})$. Here

$$F_{p/q}(\xi,\eta)=(\xi f_{p/q}(\xi^{-p}\eta^{-q})^{q},\eta f_{p/q}(\xi^{-p}\eta^{-q})^{-p})$$

where $f_{p/q}(z)=1+\sum_{n\geq 1}c_{n}^{p/q}z^{n}$. Then one can check that for any coprime $p,q\in{\bf Z}_{>0}$ and any $n\geq 1$ one has

$$c_{n}^{p/q}\in{{\bf Z}}[c_{1},c_{2},\dots,d_{1},d_{2},\dots]\,\,.$$

This implies that our construction works when one replaces $K$ by arbitrary commutative ring $R$ endowed with a complete non-trivial valuation $val:R\to(-\infty,+\infty]$.

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 $B$.