On the Geometry of Classifying Spaces and Horizontal Slices

Let (X,ω) be a polarized simply connected Calabi-Yau manifold. That is, X is a simply connected compact Kähler manifold of dimension n with zero first Chern class and ω is a Kähler form of X such that [ω] ∈ H(X,Z). In this paper, we study the local properties of the moduli space M of the polarized Calabi-Yau manifold (X,ω). By definition M is the parameter space of the complex structures over X for the fixed polarization [ω]. M is a quasi-projective variety by a theorem of Viehweg [17] Suppose X ∈ M is the Calabi-Yau manifold. Let N = dim{η ∈ H(X, TX)|ηyω = 0} where TX is the holomorphic tangent bundle of X. By a theorem of Tian [16], we know that the polarized universal deformation space of the Calabi-Yau manifold is smooth and has dimension N . Since for each X ′ ∈ M, there are no nonzero holomorphic tangent vectors on X , we concluded that in each neighborhood Ũof X , there is an open neighborhood U in C such that U is a finite covering of Ũ . Thus the moduli space is a complex orbifold. A good reference of the theorem of Tian can be found in [5].