On the Curvature Tensor of the Hodge Metric of Moduli Space of Polarized Calabi-yau Threefolds

This paper is the continuation of the paper [6] of our study of the Moduli space of polarized Calabi-Yau threefold. A polarized Calabi-Yau manifold is a pair (X,ω) of a compact algebraic manifold X with zero first Chern class and a Kähler form ω ∈ H(X,Z). The form ω is called a polarization. Let U be the universal deformation space of (X,ω). U is smooth by a theorem of Tian [12]. By [15], we may assume that each X ′ ∈ U is a Kähler-Einstein manifold. i.e. the associated Kähler metric (g αβ ) is Ricci flat. The tangent space TX′U of U at X ′ can be identified with H(X , TX′)ω where H(X , TX′)ω = {φ ∈ H(X , TX′)|φyω = 0} The Weil-Petersson metric GPW on U is defined by