First, we prove Item (1). Since $Q$ is a Riemannian product, we have for $z\neq 0$,

(3.382)

$$\frac{d^{2}}{dz^{2}}\psi(z)=\Delta_{D}\psi(z)=dd^{*}\psi(z)$$

is exact, which implies that the cohomology class $[\psi(z)]\in H^{2}(D;\mathbb{R})$ is locally constant for $z\in\mathbb{R}\setminus\{0\}$. On the other hand, by the exponential decay property in (3.349) we see that

(3.383)

$$\lim_{z\rightarrow\pm\infty}[\psi(z)]=\lim_{z\rightarrow\infty}k_{\pm}[\omega_{D}].$$

For Item (2), denote

(3.384)

$$\tilde{\psi}(z)\equiv\psi(-z)+(k_{-}+k_{+})z\omega_{D}.$$

Then $\tilde{\psi}\wedge dz$ is also a Green current for $P$ and it is also asymptotic to $k_{\pm}z\cdot\omega_{D}$ as $z\rightarrow\pm\infty$. Therefore by uniqueness, $\tilde{\psi}(z)=\psi(z)$. Taking the $z$-derivative at $z=0$ we get the conclusion. ∎