By definition

(3.290)

$$\sigma=f\partial_{y}=\partial_{w_{1}}-\sum_{j\geq 2}\mu_{j}\partial_{w_{j}},$$

where $f=|\sigma|>0$ is local real valued function on $H$, and $\mu_{2},\cdots,\mu_{n-1}$ are local complex valued function on $H$. The key property we will use is that along $H$, $\nabla_{\partial_{\bar{w}_{k}}}\sigma$ is tangential to $H$ for $k\geq 2$. In fact, the Kähler condition implies $\nabla_{\bar{\partial}_{w_{k}}}\partial_{w_{j}}=0$ for all $j$, and hence

(3.291)

$$\nabla_{\partial_{\bar{w}_{k}}}\sigma=\nabla_{\partial_{\bar{w}_{k}}}\Big(\partial_{w_{1}}-\sum_{j=2}^{n-1}\mu_{j}\partial_{w_{j}}\Big)=-\sum_{j=2}^{n-1}\partial_{\bar{w}_{k}}(\mu_{j})\partial_{w_{j}}.$$

Therefore,

(3.292)

$$\partial_{w_{k}}f=\partial_{w_{k}}\langle\partial_{y},\sigma\rangle=\langle\nabla_{\partial_{w_{k}}}\partial_{y},f\partial_{y}\rangle+\langle\partial_{y},\nabla_{\partial_{\bar{w}_{k}}}\sigma\rangle=f\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle,$$

and hence

(3.293)

$$\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle=f^{-1}\partial_{w_{k}}f=\partial_{w_{k}}(\log f).$$

Differentiating $|\partial_{y}|^{2}=1$, we get

(3.294)

$$\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle+\langle\partial_{y},\nabla_{\partial_{\bar{w}_{k}}}\partial_{y}\rangle=0,$$

which implies

(3.295)

$$\langle\nabla_{\partial_{\bar{w}_{k}}}\partial_{y},\partial_{y}\rangle=-\partial_{\bar{w}_{k}}\log f.$$

Therefore,

	$\displaystyle\Gamma$	$\displaystyle=-\frac{\sqrt{-1}}{2}(\sum_{k\geq 2}\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle dw_{k}+\sum_{k\geq 2}\langle\nabla_{\partial_{\bar{w}_{k}}}\partial_{y},\partial_{y}\rangle d\bar{w}_{k})$
		$\displaystyle=-\frac{\sqrt{-1}}{2}(\sum_{k\geq 2}\partial_{w_{k}}(\log f)dw_{k}-\sum_{k\geq 2}\partial_{\bar{w}_{k}}(\log f)d\bar{w}_{k})$
		$\displaystyle=-\frac{\sqrt{-1}}{2}(\partial_{H}\log f-\bar{\partial}_{H}\log f)$
(3.296)			$\displaystyle=\frac{1}{2}d^{c}_{H}\log f.$

∎