There is a natural family of complex structures $J_{s}$ on the model torus fibration $h\colon W\to\Sigma\backslash D$. Namely, for a given $s$ we can take $dy_{i}+\frac{\sqrt{-1}}{\log|s|}d\theta_{i}$ to be the holomorphic 1-forms on $W$, where $\{y_{i}\}$ are the affine coordinates on $\Sigma\backslash D$ and $\{\theta_{i}\}$ are the corresponding coordinates on the torus fibers. Also, given a Riemannian metric $\sum g_{ij}dy_{i}\otimes dy_{j}$ on $\Sigma\backslash D$, one can define the Kähler metric on $W$ by $\omega_{ij}=\sum g_{ij}(dy_{i}+\frac{\sqrt{-1}}{\log|s|}d\theta_{i})\otimes(dy_{j}-\frac{\sqrt{-1}}{\log|s|}d\theta_{j})$. If, in addition, $g_{ij}$ satisfy the real Monge-Ampère equation: $\det g_{ij}\equiv 1$, then the induced metric on $W$ is Ricci-flat.

In the second part of the paper we will show that $H_{s}^{\mathrm{sm}}$ embeds into $(W,J_{s})$ “almost” holomorphically. Moreover, we will construct a family of $\mathbb{T}$-invariant Kähler forms $\omega_{s}$ on $X_{\Delta_{\nu}}$ in the class $\frac{[\nu]}{\log|s|}$, such that the pairs $(H_{s},H_{s}^{\mathrm{sing}})$ with the induced metrics converge in the Gromov-Hausdorff sense to the pair $(\Sigma,D)$. Here $\Sigma$ will carry a compact metric space structure which will restrict to a Riemannian metric on $\Sigma\backslash D$.