On the Geometry of Classifying Spaces and Horizontal Slices

Details On the Geometry of Classifying Spaces and Horizontal Slices On the Geometry of Classifying Spaces and Horizontal Slices

2005-05-26

arxiv.org/abs/math/0505579v1

Amer. J. Math, vol 121, 1999, pp 177-198

Mathematics

Differential Geometry

Scientific paper

In this paper, 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 $G$. Let $D_1$ be the symmetric space associated with $G$. Then we proved that the map from $V$ to $D_1$ induced by the period map is a pluriharmonic map. We also give a Kahler metric on $V$, which is called the Hodge metric. We proved that the Ricci curvature of the Hodge metric is negative away from zero. We also proved the non-existence of the K\"ahler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of $G$.

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