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

Global Torelli Theorem for Projective Manifolds of Calabi-Yau Type