On the geometry of classifying spaces and horizontal slices

We study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let U be a neighborhood of the moduli space. Then we know the universal covering space V of U is a smooth manifold. Suppose D is the classifying space of a polarized Calabi-Yau manifold with the automorphism group G. Then we prove that the map from V to D induced by the period map is a pluriharmonic map. We also give a Kähler metric on V, which is called the Hodge metric. We prove that the Ricci curvature of the Hodge metric is negative away from zero. We also proved the nonexistence of the Kähler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of G.