Now we are ready to explain the reason for the choice of the function $q(z)$ and the rescaling factor $T^{\frac{n-2}{n}}$ in the above definition of $\omega$. These are chosen to make the Kähler metric $(\omega,\Omega)$ approximately Calabi-Yau in the following sense:

1. (1)

   Applying (3.349) and (4.16), we have $\mathrm{Err}_{CY}=T^{-2}\epsilon(z(\bm{x}))$ for $\bm{x}\in\mathcal{M}^{*}$ satisfying $|z(\bm{x})|\geq C$.

2. (2)

   Applying (3.316) and (4.17), we have $\mathrm{Err}_{CY}=O(T^{-2})$ for $\bm{x}\in\mathcal{M}^{*}$ satisfying $|z(\bm{x})|\leq C$ and $d_{Q}(\bm{x},P)\geq d_{0}>0$, where $d_{0}>0$ is some definite constant.

We will need a more precise weighted estimate on $\mathrm{Err}_{CY}$. See Proposition 4.23.