let $G_{P}$ be a Green’s current for $P\equiv H\times\{0\}$ in $Q$, then locally

(3.262)

$$G_{P}=\psi\wedge dz+\mathcal{R},$$

where $\mathcal{R}\in C^{\infty}$ and $\psi$ is family of real-valued $(1,1)$-forms on $D$ parametrized by $z$, satisfying

(3.263)

$$\psi(-z)-\psi(z)\in C^{\infty}.$$

Moreover, in terms of the above local coordinates we can write

(3.264)

$$\psi=\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y}+\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+r\cdot d\Gamma+r^{-3}\Pi_{2}^{(4)}+O^{\prime}(r^{2}),$$

where $\Gamma$ is a smooth real-valued $1$-form locally defined on $H$ given by

(3.265)

$$\Gamma(v)\equiv-\frac{\sqrt{-1}}{2}\langle\nabla_{v}\partial_{y},\partial_{y}\rangle\Big|_{H},\ v\in T_{p}H,$$

and $\Pi_{2}^{(4)}$ is the $2$-form given by Notation 3.9 such that it contains at least one of the $dy$ or $d\bar{y}$.