Weil-petersson Geometry on Moduli Space of Polarized Calabi-yau Manifolds

Moduli spaces of general polarized algebraic varieties are studied extensively by algebraic geometers. However, there are two classes of moduli spaces where the methods of differential geometry are equally powerful. These are the moduli spaces of curves and the moduli spaces of polarized Calabi-Yau manifolds. Both spaces are complex orbifolds. The Weil-Petersson metric is the main tool for investigating the geometry of such moduli spaces. Under the Weil-Petersson metrics, these moduli spaces are Kähler orbifolds. The GIT construction of the (coarse) moduli space (see [27]) of Mumford is as follows: let X be a Calabi-Yau manifold and let L be an ample line bundle over X. The pair (X,L) is called a polarized Calabi-Yau manifold. Choose a large m such that Lm is very ample. In this way X is embedded into a complex projecive space CPN . Let Hilb(X) be the Hilbert scheme of X. It is a compact complex variety. The group G = PSL(N + 1, C) acts on Hilb(X) and the moduli space M is the quotient of the stable points of Hilb(X) by the group G. For the purpose of this paper, we assume that M is connected. The curvature of these moduli spaces with respect to the Weil-Petersson metric has been studied by many people. For the moduli space of curves, Wolpert [30] gave an explicit formula for the curvature and proved that the (Riemannian) sectional curvature of the Weil-Petersson is negative. Siu [23] generalized the result to the moduli spaces of Kähler-Einstein manifolds with c1 < 0. Schumacher [21], using Siu’s methods, computed the curvature tensor of the moduli spaces of Kähler-Einstein manifolds in the case of c1 > 0 and c1 < 0 respectively. 1 Furthermore,