Hodge metric completion of the moduli space of Calabi-Yau manifolds

In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, i.e. the Torelli space, is a complex affine manifold. As applications we prove that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the completion space is a bounded domain of holomorphy with a complete K\"ahler-Einstein metric. As a corollary we show that the period map from the moduli space of polarized Calabi-Yau manifolds with level $m$ structure is also injective.