In the case $b_{1}(D)\neq 0$, by Remark 4.8.2 to ensure $\mathcal{M}_{T}$ is holomorphically embedded in $\mathcal{N}^{0}$, we need an appropriate choice of the connection 1-form in the construction of the Kähler metrics $\omega_{T}$. It is not difficult to see this is always achievable.