Communications in Analysis and Geometry
Volume 15 (2007)
Weil-Petersson volumes of the moduli spaces of CY manifolds
Pages: 381 – 405
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 Hirzebruch’s proportionality principle in the non-compact case (D. Mumford, Inv. Math. 42 (1977), 239-272). 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 $L\sp 2$ 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 $L\sp 2$ metric on the dualizing line bundle. This fact implies that the volumes of Weil-Petersson metric are rational numbers. Also we get that the Weil-Petersson metric is a good metric. Therefore, all the Chern forms define integer classes of cohomologies.