A Global Torelli Theorem for Calabi-Yau Manifolds

We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the Teichm\"uller space and Hodge metric completion of the Torelli space. A canonical global holomorphic section of the holomorphic $(n, 0)$ class on the Teichm\"uller space is constructed.