ScalingStacks

In the face type region case, up to $C^{\infty}$-small error in the $s^{-1}\log z^{m_{i}}$ coordinates,

$$\begin{split}&\omega_{CY,s}=\sqrt{-1}\partial\bar{\partial}\varphi_{CY,s,0}\\ \approx&\sqrt{-1}\partial\bar{\partial}u_{\infty}\circ s^{-1}\text{Log}=\frac{1}{4}\frac{\partial^{2}u_{\infty}}{\partial x^{m_{i}}\partial x^{m_{j}}}\sqrt{-1}s^{-1}d\log z^{m_{i}}\wedge s^{-1}d\overline{\log z^{m_{j}}},\end{split}$$

hence the CY metrics $g_{CY,s}$ is up to $C^{\infty}$-small error

$$g_{CY,s}\approx\text{Re}\{\frac{1}{2}\frac{\partial^{2}u_{\infty}}{\partial x^{m_{i}}\partial x^{m_{j}}}s^{-1}d\log z^{m_{i}}\otimes s^{-1}d\overline{\log z^{m_{j}}}\}.$$

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