Proof. [03W4]

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Proof. Here is the plan of the proof.

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.

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

\begin{array}[]{lll}U_{1}&=&\{(x,y)\in{{\bf R}}^{2}|x<\varepsilon|y|\,\}\\ U_{2}&=&\{(x,y)\in{{\bf R}}^{2}|x>0,y<\varepsilon x\,\}\\ U_{3}&=&\{(x,y)\in{{\bf R}}^{2}|x>0,y>0\}\end{array}

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

\pi_{i}(\xi_{i},\eta_{i})=\pi_{can}(\xi_{i},\eta_{i})=(\log|\xi_{i}|,\log|\eta_{i}|),\,\,\,i=1,3\,\,,

\pi_{2}(\xi_{2},\eta_{2})=\left\{\begin{array}[]{ll}(\log|\xi_{2}|,\log|\eta_{2}|)&\mbox{ if }|\eta_{2}|<1\\ (\log|\xi_{2}|-\log|\eta_{2}|,\log|\eta_{2}|)&\mbox{ if }|\eta_{2}|\geq 1\end{array}\right.\,\,.

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:

\begin{array}[]{lll}g_{1}(\xi_{1},\eta_{1})&=&({1\over{\xi_{1}}},\xi_{1}(1+\eta_{1}),{1\over{\eta_{1}}})\\ g_{2}(\xi_{2},\eta_{2})&=&({1+\eta_{2}\over{\xi_{2}}},\xi_{2},{1\over{\eta_{2}}})\\ g_{3}(\xi_{3},\eta_{3})&=&({1+\eta_{3}\over{\xi_{3}\eta_{3}}},\xi_{3}\eta_{3},{1\over{\eta_{3}}})\end{array}

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

\begin{array}[]{lll}f(X_{-})&=&\{\,(a,b,c)\in{{\bf R}}^{3}\,|\,c<0,a\geq 0,b\geq 0,\,ab(a+b+c)=0\,\}\\ f(X_{0})&=&\{\,(a,b,c)\in{{\bf R}}^{3}\,|\,c=0,a\geq 0,b\geq 0,\,ab=0\,\}\\ f(X_{+})&=&\{\,(a,b,c)\in{{\bf R}}^{3}\,|\,c>0,a\geq 0,b\geq 0,\,ab=0\,\}\end{array}

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

j(x,y)=\left\{\begin{array}[]{lll}(-x\,,\,\max(x+y,0)\,,\,-y\,)&\mbox{ if }&x\leq 0\\ (\,0\,,\,x+\max(y,0)\,,\,-y\,)&\mbox{ if }&x\geq 0\end{array}\right.

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. $\blacksquare$