On the Geometry of Classifying Spaces and Horizontal Slices

2005/05/26 by Zhiqin Lu
Mathematics · Physics and Astronomy · #math.DG #math-ph #math.MP #msc:32G13 #msc:58D27

paper · pdf

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

arxiv created 2005/05/26 · arxiv updated 2009/12/01

Abstract

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 D₁ be the symmetric space associated with G. Then we proved that the map from V to D₁ 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 Kähler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of G.

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