Given the above Lemma we first obtain that

(3.306)

$$d\bar{w}_{1}=a_{1}d\bar{y}+\bar{y}(da_{1}+2\bar{a}_{2}d\bar{y})+\widetilde{O}(|y|^{2}),$$

then

(3.307)

$$w_{1}d\bar{w}_{1}=a_{1}^{2}yd\bar{y}+|y|^{2}a_{1}da_{1}+a_{1}y(2\bar{a}_{2}\bar{y}+a_{2}y)d\bar{y}+\widetilde{O}(|y|^{3}).$$

Hence

(3.308)

$$d_{D}^{c}|w_{1}|^{2}=\sqrt{-1}a_{1}^{2}(yd\bar{y}-\bar{y}dy)+\sqrt{-1}a_{1}\bar{a}_{2}\bar{y}(2yd\bar{y}-\bar{y}dy)-\sqrt{-1}a_{1}a_{2}y(2\bar{y}dy-yd\bar{y})+\widetilde{O}(|y|^{3}).$$

On the other hand, we have

(3.309)

$$|w_{1}|^{2}=a_{1}^{2}|y|^{2}+a_{1}(a_{2}y+\bar{a}_{2}\bar{y})|y|^{2}+\widetilde{O}(|y|^{4}).$$

So

(3.310)

$$d_{D}^{c}|w_{1}|^{2}=a_{1}^{2}d_{D}^{c}|y|^{2}+|y|^{2}d_{D}^{c}a_{1}^{2}+d_{D}^{c}(a_{1}(a_{2}y+\bar{a}_{2}\bar{y})|y|^{2})+\widetilde{O}(|y|^{3}).$$

Now by Lemma 3.27,

(3.311)

$$d_{D}^{c}(a_{1}y)=d_{D}^{c}w_{1}+\widetilde{O}(|y|)=-\sqrt{-1}dw_{1}+\widetilde{O}(|y|),$$

so

(3.312)

$$d_{D}^{c}y=-\sqrt{-1}dy+\widetilde{O}(|y|).$$

Similarly, $d_{D}^{c}\bar{y}=\sqrt{-1}d\bar{y}+\widetilde{O}(|y|)$. Plugging these into (3.310), and compare with (3.308) we obtain

(3.313)

$\displaystyle d_{D}^{c}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)-2|y|^{2}d_{D}^{c}\log a_{1}+\widetilde{O}(|y|^{3}).$

Thanks to Lemma 3.27, $a_{1}=|\sigma|^{-1}$ which is a smooth function on $H$, so

(3.314)

$\displaystyle d_{D}^{c}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)+2|y|^{2}d_{H}^{c}\log|\sigma|+\widetilde{O}(|y|^{3}).$

By Lemma 3.25, $\Gamma=\frac{1}{2}d_{H}^{c}\log|\sigma|$, so we conclude

(3.315)

$$d_{D}^{c}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)+4|y|^{2}\Gamma+\widetilde{O}(|y|^{3}).$$

∎