We choose coordinates $x_{1},\cdots,x_{n}$ on $\bar{S}$ and $y_{1},\cdots,y_{n}$ on $\bar{S}^{\perp}$ such that

If $\mathcal{G}$ is a subgroup of the fundamental group $\pi_{1}(X)=\pi_{1}(S)$, then $\mathcal{G}$ acts on $\mathbb{C}^{n}$ preserving $g_{E}$, $\omega_{E}$ and $\Omega_{E}$, and $\bar{S}$ is a invariant subspace. For any $\gamma\in\mathcal{G}$, we have $\gamma\cdot(v+w)=G_{\gamma}(v+w)+b_{\gamma}$, where $G_{\gamma}\in U(\mathbb{C}^{n})$, $b_{\gamma}\in\bar{S}$, $v\in\bar{S}$ and $w\in\bar{S}^{\perp}$. Since $\bar{S}$ is invariant, we obtain then $G_{\gamma}(v+w)=A_{\gamma}v+B_{\gamma}w+C_{\gamma}w$ where $A_{\gamma}\in SO(\bar{S})$, $B_{\gamma}\in SO(\bar{S}^{\perp})$, and $C_{\gamma}\in Hom(\bar{S}^{\perp},\bar{S})$. Moreover, $G_{\gamma}\in SO(\mathbb{R}^{2n})$ implies $C_{\gamma}=0$. Since $\omega_{E}(G_{\gamma}(v+w),G_{\gamma}(v+w))=\omega_{E}(v+w,v+w)$, we have $B_{\gamma}=A_{\gamma}^{-1,T}=A_{\gamma}$, and $\gamma\cdot(v+w)=A_{\gamma}(v+w)+b_{\gamma}$. Thus $\pi_{1}(\tilde{X})\cong\Lambda$ acts on $\mathbb{C}^{n}$ given by $\gamma\cdot(v+w)=v+w+b_{\gamma}$, $b_{\gamma}\in\Lambda$, for any $v\in\bar{S}$ and $w\in\bar{S}^{\perp}$. This implies that $\tilde{X}=\mathbb{C}^{n}/\pi_{1}(\tilde{X})\cong\bar{S}/\Lambda\times\bar{S}^{\perp}=\tilde{S}\times\bar{S}^{\perp}$, and $\pi^{*}g_{0}=h+h_{E}$ where $h_{E}=g_{E}|_{\bar{S}^{\perp}}$, and $h$ is the standard flat metric on $\tilde{S}$ induced by $g_{E}|_{\bar{S}}$.

The $\pi_{1}(X)$-action on $\mathbb{C}^{n}$ descents to a $\Gamma$-action on $\tilde{X}$, which is a product action since the $\pi_{1}(X)$-action is so. Moreover, $\tilde{S}\times\{0\}$ is a invariant set as $\bar{S}\times\{0\}$ is invariant under the $\pi_{1}(X)$-action. If we denote the quotient map $\mathfrak{q}_{1}:\mathbb{C}^{n}\longrightarrow\mathbb{C}^{n}/\Lambda=\tilde{X}$, then $\bar{\pi}=\pi\circ\mathfrak{q}_{1}$, $g_{E}=\mathfrak{q}_{1}^{*}\pi^{*}g_{0}$, $\omega_{E}=\mathfrak{q}_{1}^{*}\pi^{*}\omega_{0}$, and $\Omega_{E}=\mathfrak{q}_{1}^{*}\pi^{*}\Omega_{0}$. Since $\omega_{E}|_{\bar{S}\times\{y\}}=0$ and $e^{\sqrt{-1}\theta_{0}}\Omega_{E}|_{\bar{S}\times\{y\}}=0$ for $y\in\bar{S}^{\perp}$, we obtain that

for a constant $\theta_{0}\in\mathbb{R}$. ∎