Let $({\cal X}_{mer},vol)=(X_{q},vol_{q})$ be a 1-parameter family of maximally degenerate Calabi-Yau manifolds, and $X_{q}^{new}$ be the family with rescaled metrics, as before. There exist a constant $C>0$ and a function $t(q)$ such that Kähler manifolds $X_{q}^{new}$ and $X^{\varepsilon(q),t(q)}$ with $\varepsilon(q)=C(log|q|)^{-1}$ are close to each other (as $q\to 0$) in the following sense:

for any $\delta>0$ there exist a decomposition $X_{q}=X_{q}^{sm}\sqcup X_{q}^{sing}$ and an embedding of smooth manifolds $j_{q}:X_{q}\to p_{\varepsilon(q),t(q)}^{-1}(Y\setminus(Y^{sing})^{\delta})$, where $(Y^{sing})^{\delta}$ is a $\delta$-neighborhood of $Y^{sing}$, such that:

a) $(X_{q},X_{q}^{sing})$ converges in the Gromov-Hausdorff metric to the pair $(\overline{Y},Y^{sing})$.

b) $j_{q}$ identifies up to $o(1)$ terms, uniformly in $x\in X_{q}^{sm}$, the scalar products and complex structures on the tangent spaces $T_{x}X_{q}$ and $T_{j_{q}(x)}X^{\varepsilon(q),t(q)}$.