Global Torelli Theorem for Projective Manifolds of Calabi-Yau Type

We prove that the period maps from the Torelli space and the moduli space with level $m$ structure of Calabi-Yau type manifolds to the corresponding period domain of polarized Hodge structures are injective. The proof is based on the construction of holomorphic affine structure on the Teichm\"uller space and the construction of the completion space of the Torelli space with respect to the Hodge metric.