 define the complete complex manifold $\mathcal{T}^H=\mathcal{T}^H_{_m}$, the holomorphic map $i_{\mathcal{T}}: \,\mathcal{T}\to \mathcal{T}^H$ by $i_{\mathcal{T}}=i_m$, and the extended period map $\Phi^H:\, \mathcal{T}^H\rightarrow D$ by $\Phi^H=\Phi^H_{_m}$ for any $m\geq 0$. 