Global Torelli Theorem for marked and polarized CY manifolds and Flat Holomorphic Structures

We will review briefly basic facts about Kodaira-Spencer-Kuranishi Theory. A description of the construction of a the moduli space of marked and polarized CY manifold will be given. Then we will define The Weil-Petersson holomorphic flat metric, give a description of the flat complex geodesics in Hodge Theoretical terms. We will show the analogue of Cartan-Hadamard Theorem, i.e. the complex holomorphic exponential map is a covering map. We will construct a canonical family of holomorphic forms and will outline the proof that the Holomorphic analogue of Cartan Hadamard Theorem implies that the periods of the family of holomorphic forms defined an embedding of the Teichmuller space into the moduli space of Variations of Polarized Hodge Structures on the fixed CY manifold. We will translate all these ideas to the language of Frobenius manifolds.
The constructions are based on the existence and uniqueness of Calabi-Yau metric. We will briefly discuss the basic properties of Ricci flat metrics.
We will define the analogue of Torelli group of CY manifold and will explain the failure of Global Torelli for the coarse moduli space of CY manifolds.
All the ideas will be briefly described in the case of marked and polarized Abelian Varieties.