On the Weil-petersson Volume and the First Chern Class of the Moduli Space of Calabi-yau Manifolds

In this paper, we continue our study of the Weil-Petersson geometry as in the previous paper [10], in which we have proved the boundedness of the Weil-Petersson volume, among the other results. The main results of this paper are that the volume and the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space are rational numbers. In particular, the Ricci curvature defines the first Chern class of the moduli space in the sense of Mumford [11]. It was a classical result of Mumford [11] that for a noncompact Kähler manifold M with M being a smooth compactification of M and M\M being a divisor D of normal crossings, and for any Hermitian bundle (E,h) overM , one can define the Chern classes ck(E) provided the metric h is “good” defined by Mumford [11, Section 1]. Roughly speaking, a metric is “good” if the metric matrix have log bound, and the local connection form and the curvature have Poincaré type growth. It was verified that the natural bundles over locally Hermitian symmetric spaces are “good” (cf. [11]). For the moduli space of curves of genus greater than or equal to 2, the metric induced by the Weil-Petersson metric on the determinant bundle of the log extension of the cotangent bundle is good [19]. However, for the moduli space of polarized Calabi-Yau manifolds, it is not clear that the Weil-Petersson metric or the volume form of the Weil-Petersson metric is “good”. By [1], the Weil-Petersson potential is related to the analytic torsion of the moduli space. While the Hessian of the torsion is known to be related to the Weil-Petersson metric and the generalized Hodge metric([3]), it is not easy to find the asymptotic behavior of the BOCV torsion itself. Thus we can not use the theorem of Mumford directly to prove that the integrations are rational numbers. In this paper, we avoided using the BCOV torsion by the careful analysis of the asymptotic behavior of the Hodge bundles at infinity. Using the Nilpotent Orbit theorem of Schmid [12], we can give another explicit (local) representation of the Weil-Petersson potential. The potential