For the case of collapse with singular fibers, the rigid analytic space $Y^{an}$ constructed as above, seems to be a “wrong” one. First of all, it is not compact because $Y$ is not compact. But there is also a more fundamental problem. It seems that $Y^{an}$ can not be embedded into a compact analytic space associated with a projective algebraic variety. There are several indications that there exists another sheaf of algebras ${\cal O}^{\prime}_{Y}$ which is (locally on $Y$) isomorphic to ${\cal O}_{Y}$, and the rigid analytic space $(Y^{an})^{\prime}$ associated with $(Y,{\cal O}^{\prime}_{Y})$ admits an algebraic compactification. In general, sheaves ${\cal O}^{\prime}_{Y}$ which are twisted versions of ${\cal O}_{Y}$ are classified by the first non-abelian cohomology $H^{1}(Y,{\underline{Aut}}({\cal O}_{Y}))$ where ${\underline{Aut}}({\cal O}_{Y})$ is the sheaf of groups of automorphisms of ${\cal O}_{Y}$. Thus, in the mirror symmetry for Calabi-Yau manifolds which are not abelian varieties, we expect a new ingredient, the cohomology class $[{\cal O}^{\prime}_{Y}]$.