ScalingStacks

The geometric meaning is that inside the generic region, the CY metric $\omega_{CY,t}$ is $C^{\infty}$-close to a semiflat metric:

$$\omega_{CY,t}\sim dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}=\frac{1}{4\pi|\log|t||^{2}}\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j}.$$

In terms of the Riemannian metric tensors,

$$g_{CY,t}\sim\frac{1}{2\pi|\log|t||^{2}}\text{Re}\{\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}d\log z_{i}\otimes d\log\bar{z}_{j}\}.$$

(11)

The name ‘semiflat’ means the metric restricted to the $T^{n}$-fibres are flat Euclidean. The $T^{n}$-fibres are precisely special Lagrangian in the model case

$$\begin{cases}\omega_{semiflat}=\frac{1}{4\pi|\log|t||^{2}}\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j},\\ \Omega_{semiflat}=\text{const}\cdot\prod_{1}^{n}d\log z_{i},\end{cases}$$

or equivalently

$$\text{Log}_{\mathcal{X}}:(z_{1},\ldots z_{n})\mapsto\frac{1}{\log|t|}(\log|z_{1}|,\ldots\log|z_{n}|)$$

is a special Lagrangian fibration in the model case for some choice of the phase angle. Since $(\omega_{CY,t},\Omega_{t})$ is $C^{\infty}$-close to the model case, standard perturbation theory allows one to perturb the $T^{n}$-fibres into special Lagrangians with respect to $(\omega_{CY,t},\Omega_{t})$ in the generic region, to obtain a new special Lagrangian fibration. The details are carried out in [49], and more expositions can be found in [32].