Let $\bar{S}^{\perp}$ be the orthogonal complement of $\bar{S}$ in $\mathbb{C}^{n}$, i.e. $\mathbb{C}^{n}=\bar{S}\oplus\bar{S}^{\perp}$, and $g_{E}(v,w)=0$, for any $v\in\bar{S}$ and $w\in\bar{S}^{\perp}$. Then

• i)

  $(\tilde{X},\pi^{*}g_{0})$ is isometric to $(T^{n}\times\bar{S}^{\perp},h+h_{E})$, where $T^{n}=\bar{S}/\Lambda=\tilde{S}$, $h_{E}=g_{E}|_{\bar{S}^{\perp}}$, and $h$ is the standard flat metric on $T^{n}$ induced by $g_{E}|_{\bar{S}}$.

• ii)

  The action of $\Gamma$ on $\tilde{X}$ is a product action, i.e. there are $\Gamma$-actions on $T^{n}$ and $\bar{S}^{\perp}$ such that $\gamma\cdot(x,y)=(\gamma\cdot x,\gamma\cdot y)$ for any $\gamma\in\Gamma$, $x\in T^{n}$ and $y\in\bar{S}^{\perp}$. Furthermore, $T^{n}\times\{0\}$ is $\Gamma$-invariant, and $S=\pi(T^{n}\times\{0\})=(T^{n}\times\{0\})/\Gamma$.

• iii)

  $$\pi^{*}\omega_{0}|_{T^{n}\times\{y\}}\equiv 0,\ \ {\rm and}\ \ \pi^{*}{\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{0}|_{T^{n}\times\{y\}}\equiv 0,$$

  for any $y\in\bar{S}^{\perp}$, and a constant $\theta_{0}\in\mathbb{R}$.