On the Weil-Petersson Geometry of Calabi-Yau Moduli
  In this paper we proved that the Weil-Petersson volume of the Chern class of any order over the moduli space of Calabi-Yau manifolds is a rational number. We also found the necessary and sufficient condition of the incompleteness of Weil-Petersson metric in several variables case. 
Zhiqin Lu and Eisuke Natsukawa