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

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.