ScalingStacks

Proof. Since $K$ is a compact subset in the open set $\mathcal{R}$, we can find an open set $\mathcal{U}\subset K$ properly contained in $\mathcal{R}$. This ensures that the smooth convergence in Thm. 5.6 happens uniformly on a slightly larger set $U_{s,K^{\prime}}$ than $U_{s,K}$. We assume $s\gg 1$ as ususal.

Consider a coordinate region $(s^{-1}\text{Log})^{-1}(B(x,r(x))$ contained in this larger set, which is topologically $T^{n}\times B(x,r(x))$. Here the $T^{n}$ is well defined as a homology cycle independent of the coordinates. We define the phase angles $\theta_{s}$ by requiring $\int_{T^{n}}e^{\sqrt{-1}\theta_{s}}\Omega>0.$ We consider the rescaled CY metrics $(s^{2}g_{CY,s},s^{2}\omega_{CY,s})$, so the diameter of $T^{n}$ fibres are now of order $O(1)$ by (32)(33). Within any log scale, these rescaled CY structures are $C^{\infty}$-close to the standard flat structures in section 2.7 up to constant factors. By construction the Kähler forms are exact in these coordinate charts. Thus by Zhang’s result surveyed in section 2.7, within any log scale, we can construct a SLag $T^{n}$-fibration with phase $\theta_{s}$, whose fibres are very small $C^{\infty}$-perturbations of the fibres of the map $(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$,

$$\text{Log}:(z^{m_{1}},\ldots,z^{m_{n}})\to(\log|z^{m_{1}}|,\ldots\log|z^{m_{n}}|).$$

Observe that on overlapping charts, the Log-fibres with respect to one chart are very small $C^{\infty}$-perturbations of the Log-fibres of the other chart. Then the uniqueness part of Zhang’s argument shows that on overlapping charts the SLag $T^{n}$-fibrations are in fact defined independent of charts. (It is the local universal family of SLags within the perturbative regime.) Thus the local constructions glue to a SLag fibration on a subset of $X_{s}$ containing $U_{s,K}$ as required. ∎