Proof. Let us write in local coordinates $\Omega=\varphi(z_{1},...z_{n})\bigwedge_{1\leq i\leq n}{dz_{i}\over z_{i}}$. Define residue $Res(\Omega)\in K$ as the constant term $\varphi_{0}$ in the Laurent expansion $\varphi(z_{1},...z_{n})=\sum_{I\in{\bf Z}^{n}}\varphi_{I}z^{I}$. It is easy to see that $Res(\Omega)$ does not depend (up to a sign) on the choice of local coordinates. For non-vanishing everywhere $\Omega$ satisfying Constant Norm Assumption we have $\exp(-Val(\varphi))=|\varphi|=|\varphi_{0}|$. Therefore we have $Res(\Omega)\neq 0$.

Let us return to the proof of the Theorem. Let $F$ be the sheaf of abelian groups $F\subset\pi_{\ast}({\cal O}_{X}^{\times})$ consisting of $f$ such that $Val(f)=0$. Then we have an exact sequence of sheaves

where ${\cal O}_{K}$ denotes the constant sheaf with the fiber being the ring of integers of $K$. Indeed we embed $K^{\times}/{\cal O}_{K}^{\times}$ into $\pi_{\ast}({\cal O}_{X}^{\times})/F$ as constant functions. The projection $\pi_{\ast}({\cal O}_{X}^{\times})/F\to(T_{X}^{\ast})^{{\bf Z}}$ assigns to the function $f$ the linear part of the corresponding ${\bf Z}$-affine function $Val(f)$.

Notice that if $U\subset B^{sm}$ is a connected domain then any $f\in\Gamma(U,F)$ can be written (non-canonically) as $f=a(1+r)$, where $a\in{\cal O}_{K}^{\times}$ and $r=o(1)$ in $\pi^{-1}(U)$.

We define an epimorphism of sheaves $p_{\Omega}:F\twoheadrightarrow{\cal O}_{K}^{\times}$ by formula

Here $\exp$ and $\log$ are understood as infinite convergent series (in order to make sense of them we use Zero Characteristic Assumption).

It is easy to see that $p_{\Omega}$ is well-defined. Then the exact sequence of sheaves

defines a $K$-affine structure on $B^{sm}$ compatible with $Aff_{{{\bf Z}},B^{sm}}^{can}$. This concludes proof of the Theorem. $\blacksquare$