Weil-Petersson Volumes of the Moduli Spaces of CY Manifolds

In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [10]. He proved that the Chern forms of good metrics define classes of cohomology with integer coefficients on the compactified quasi-projective varieties by adding a divisor with normal crossings. Viehweg proved that the moduli space of CY manifolds is a quasi-projective variety. The proof that the volume of the moduli space of polarized CY manifolds are rational number is based on the facts that the L norm on the dualizing line bundle over the moduli space of polarized CY manifolds is a good metric. The Weil-Petersson metric is minus the Chern form of the L metric on the dualizing line bundle. This fact implies that the volumes of Weil-Petersson metric are rational numbers.