Here we will construct an analytic torus fibration corresponding to standard singularity (see Sections 3.2.4 and 6.4).

Let $X\subset{\bf A}^{3}$ be the algebraic surface given by equation $(\alpha\beta-1)\gamma=1$ in coordinates $(\alpha,\beta,\gamma)$, and $X^{an}$ be the corresponding analytic space. We define a continuous map $f:X^{an}\to{{\bf R}}^{3}$ by the formula $f(\alpha,\beta,\gamma)=(a,b,c)$ where $a=\max(0,\log|\alpha|_{p}),b=\max(0,\log|\beta|_{p}),c=\log|\gamma|_{p}=-\log|\alpha\beta-1|_{p}$. Here $|\cdot|_{p}=\exp(-val_{p}(\cdot))$ denotes the multiplicative seminorm corresponding to the point $p\in X^{an}$ (see Appendix A).

Proof. Here is the plan of the proof.

1. 1.

   We define three open domains $T_{i},\,\,i=1,2,3$ in three copies of the standard two-dimensional analytic torus $({\bf G}_{m}^{an})^{2}$, and continuous maps $\pi_{i}:T_{i}\to{{\bf R}}^{2}$ such that all points of the image $U_{i}=\pi_{i}(T_{i})$ are $\pi_{i}$-smooth (i.e. each $\pi_{i}$ is an analytic torus fibration). Domains $U_{i}$ cover ${\bf R}^{2}\setminus\{(0,0)\}$.

2. 2.

   For each $i,1\leq i\leq 3$ we construct an open embedding $g_{i}:T_{i}\hookrightarrow X^{an}$.

3. 3.

   We construct an embedding $j:{{\bf R}}^{2}\hookrightarrow{{\bf R}}^{3}$ such that each open set $U_{i}$ is homeomorphically identified with $f(g_{i}(T_{i}))$ and $j((0,0))=(0,0,0)$. Moreover, $\pi_{i}$-smooth points are mapped into $f$-smooth points.

The Proposition will follow from 1)-3).

Let us describe the constructions and formulas. We start with open sets $U_{i},1\leq i\leq 3$. Let us fix a number $0<\varepsilon<1$ and define

Clearly ${{\bf R}}^{2}\setminus\{(0,0)\}=U_{1}\cup U_{2}\cup U_{3}$. We define also a slightly modified domain $U_{2}^{\prime}$ as $\{(x,y)\in{{\bf R}}^{2}|x>0,y<\frac{\varepsilon}{1+\varepsilon}x\,\}$.

We define $T_{i}:=\pi_{can}^{-1}(U_{i})\subset({\bf G}_{m}^{an})^{2},i=1,3$ and $T_{2}:=\pi_{can}^{-1}(U_{2}^{\prime})\subset({\bf G}_{m}^{an})^{2}$. Then the projections $\pi_{i}:T_{l}\to U_{l}$ are given by the formulas

In these formulas $(\xi_{i},\eta_{i})$ are coordinates on $T_{i},1\leq i\leq 3$.

We define inclusion $g_{i}:T_{i}\hookrightarrow X,1\leq i\leq 3$ by the following formulas:

Let us decompose $X^{an}=X_{-}\cup X_{0}\cup X_{+}$ according to the sign of $\log|\gamma|_{p}$ where $p\in X^{an}$ is a point. It is easy to see that

From this explicit description we see that $f$ is proper and the image of $f$ is homeomorphic to ${{\bf R}}^{2}$.

Let us consider the embedding $j:{{\bf R}}^{2}\to{{\bf R}}^{3}$ given by formula

One can easily check that the image of $j$ coincides with the image of $f$, $j\circ\pi_{i}=f\circ g_{i}$ and $f^{-1}(j(U_{i}))=g_{i}(T_{i})$ for all $1\leq i\leq 3$. This concludes the proof of Proposition. $\blacksquare$

We can derive more from explicit formulas given in the proof.

Let us denote by $\pi:X^{an}\to{{\bf R}}^{2}$ the map $j^{(-1)}\circ f$. It is an analytic torus fibration outside of point $(0,0)$. The induced ${\bf Z}$-affine structure on ${{\bf R}}^{2}\setminus\{(0,0)\}$ is in fact the standard singular ${\bf Z}$-affine structure described in Sections 3.2.4 and 6.4, as follows immediately from formulas for projections $\pi_{i},\,i=1,2,3$.

Let us introduce another sheaf ${\cal O}^{can}$ on ${{\bf R}}^{2}\setminus\{(0,0)\}$. It is defined as $(\pi_{i})_{*}\left({\cal O}_{T_{i}}\right)$ in each domain $U_{i}$, with identifications

Let us consider the direct image sheaf $\pi_{\ast}({\cal O}_{X^{an}})$. It is easy to see that on the sets $U_{1}$ and $U_{2}\cup U_{3}$ this sheaf is canonically isomorphic to ${\cal O}^{can}$. The isomorphism is given by the identification of coordinates $(\xi_{1},\eta_{1})$ on $U_{1}$, and of coordinates $(\xi_{2},\eta_{2})$ and $(\xi_{3},\eta_{3})$ on $U_{2}\cup U_{3}$. Therefore on the intersection $U_{1}\cap(U_{2}\cup U_{3})$ we identify two copies of the canonical sheaf by certain automorphism $\varphi$ of ${\cal O}^{can}$ which preserves one coordinate (namely, the coordinate $\eta$). We will develop the theory of such transformations and their analytic continuations in Section 11. The explicit formulas for $\varphi$ is

We would like to say now few words about analytic volume forms. Notice that each $T_{i}\subset({\bf G}_{m}^{an})^{2}$ carries a nowhere vanishing top degree analytic form given by the formula $\Omega_{i}={d\xi_{i}\wedge d\eta_{i}\over{\xi_{i}\eta_{i}}}$. Then a straightforward calculation shows that $\Omega^{T_{i}}$ is the pullback under $g_{i}$ of nowhere vanishing on $X^{an}$ analytic top degree form

Form $\Omega$ satisfies Constant Norm Assumption, hence it gives a $K$-affine structure on ${\bf R}^{2}\setminus\{(0,0)\}$. On the other hand, the sheaf ${\cal O}^{can}$ of algebras is also endowed with top-degree form $\Omega^{can}$, equal to ${d\xi_{i}\wedge d\eta_{i}\over{\xi_{i}\eta_{i}}}$ in local coordinates.

Proof: Using definitions from Section 7.2 one sees immediately that the statement of the Lemma follows from the equality $p_{\Omega^{can}}(1+\eta)=1$, which is straightoforward: $p_{\Omega^{can}}(1+\eta)=exp\left(Res(\Omega^{can}\,\log(1+\eta))\right)=1\in{\cal O}_{K}^{\times}$. $\,\blacksquare$

In all the definitions and formulas in this section on can shift domains $U_{i},_{,}i=1,2,3$ by vector $(x_{0},0)\in{\bf R}^{2}$ for arbitrary $x_{0}\in{\bf R}$, thus giving a map $X^{an}\to{\bf R}^{2}$ with singularity at the point $(x_{0},0)$.

Finally, we denote $\pi_{\ast}({\cal O}_{X^{an}})$ by ${\cal O}^{model}_{{\bf R}^{2}}$. This will be our model for the sheaf ${\cal O}_{B}$ near each point of the singular set $B^{sing}$.