This follows from elementary manipulation. First, the holomorphic coordinates $\{w_{i}\}_{i=1}^{n-1}$ can be chosen such that $g_{i\bar{j}}(0)=\delta_{ij}$ for all $1\leq i,j\leq n-1$. By the substitution of the form

(3.322)

$$\begin{cases}w_{i}=z_{i}+\frac{1}{2}\sum\limits_{j,k=2}^{n-1}C_{ijk}z_{j}z_{k}+\sum\limits_{j=2}^{n-1}D_{ij}z_{1}z_{j}+E_{i}z_{1}^{2},\ \ 2\leq i\leq n-1,\\ w_{1}=z_{1}+\sum\limits_{j=1}^{n-1}F_{j}z_{1}z_{j},\end{cases}$$

with suitable choices of coefficients, where $C_{ijk}=C_{ikj}$ for $2\leq i,j,k\leq n-1$. One can plug both the Taylor expansion of $g_{i\bar{j}}$ along $z_{k}$’s and (3.322) into $\omega_{D}$. Comparing the coefficients, then it follows that,

(3.323)

$\displaystyle\begin{cases}C_{ijk}=-\partial_{w_{k}}g_{j\bar{i}}(0),\\ D_{ij}=-\partial_{w_{j}}g_{1\bar{i}}(0),\\ E_{i}=-\frac{1}{2}\partial_{w_{1}}g_{1\bar{j}}(0),\\ F_{i}=-\partial_{w_{j}}g_{1\bar{1}}(0),\\ F_{1}=-\frac{1}{2}\partial_{w_{1}}g_{1\bar{1}}(0),\end{cases}$

where $2\leq i,j,k\leq n-1$. Then we can achieve (3.319) with $\{w_{i}\}_{i=1}^{n-1}$ replaced by $\{z_{i}\}_{i=1}^{n-1}$.

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