| In this paper, we study the geometry of the moduli space of polarized Calabi-Yau manifolds, especially the moduli space of polarized Calabi-Yau threefolds.; Our method is differential geometric: we map the moduli space into its classifying space so that the moduli space becomes a submanifold. The Griffths transversity tells us in fact such a complex submanifold is a horizontal slice. By studying the differential geometry properties of the classifying space, we know a lot of information on the moduli space. In particular, the restriction of the natural invariant Hermitian metric on the moduli space is Kahler and its Ricci curvature is negative away from zero. We call such a metric the VHS metric.; We study the moduli space of Calabi-Yau threefold in greater detail. We established a relation between the Weil-Petersson metric to the VHS metric mentioned above. From this relation, we write out explicitly the curvature of the VHS metric. Thus we give a criterion for the moduli space, or more generally the horizontal space, to be compactified. On the other hand, we also study the Weil-Petersson metric by the relation between the Weil-Petersson metric and the VHS metric.; We also study some global properties of the moduli space: the definition of the Weil-Petersson metric and VHS metric on moduli space and horizontal slice and the local rigidity of the monodromy group representations. |
