As a by-product we can also recover the formula of the Calabi model metric in terms of Kähler potentials as mentioned in Section 2.2. In this case as in (2.30) we take $\tilde{\omega}=z\omega_{D}$ and $h=z^{n-1}$. Then we can write

(4.152)

$$\tilde{\omega}=dd^{c}\phi$$

with

(4.153)

$$\phi=\int_{0}^{z}u^{n}du=\frac{1}{n+1}z^{n+1}$$

To match with the formula for Calabi ansatz in (2.32), we notice that $z^{n+1}=(-\log|\xi|)^{2}$, and there is a factor of $\frac{n}{2}$ due to the normalization of the Calabi-Yau equation and that $dd^{c}=2\sqrt{-1}\partial\bar{\partial}$.