Local and Global Theory of the Moduli of Polarized Calabi-Yau Manifolds

In this paper we review the moduli theory of polarized CY manifolds. We briefly sketched Kodaira-Spencer-Kuranishi local deformation theory developed by the author and G. Tian. We also construct the Teichmüller space of polarized CY manifolds following the ideas of I. R. Shafarevich and I. I. Piatetski-Shapiro. We review the fundamental result of E. Viehweg about the existence of the course moduli space of polarized CY manifolds as a quasi-projective variety. Recently S. Donaldson computed the moment map for the action of the group of symplectic diffeomorphisms on the space of Kähler metrics with fixed class of cohomology. Combining this results with the solution of Calabi conjecture by Yau one obtain a very conceptual proof of the existence of the coarse moduli space for a large class of varieties. We follow the approach developed in [24] to study the global properties of the moduli of polarized CY manifolds. We discuss the latest results connecting the discriminant locus in the moduli space of polarized odd dimensional CY manifolds with the Bismut-GilletSoule-Quillen-Donaldson Theory of Determinant line bundles.