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 Kahler 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 Kahler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of G. 1. Introduction. Let (X, ) be a polarized simply connected Calabi-Yau manifold. That is, X is a simply connected compact Kahler manifold of dimension n with zero first Chern class and is aKform of X such that ( ) H 2 (X, Z). In this paper, we study the local properties of the moduli space of the polarized Calabi-Yau manifold (X, ). By definition is the parameter space of the complex structures over X for the fixed polarization ( ); is a quasi-projective variety by a theorem of Viehweg (17). Suppose X is a Calabi-Yau manifold. Let N =d im H 1 (X, TX) =0