  We establish an unexpected relation among the Weil-Petersson metric, the
generalized Hodge metrics and the BCOV torsion. Using this relation, we prove
that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not
admit complete subvarieties. That is, there is no complete family for certain
class of polarized Calabi-Yau manifolds. We also give an estimate of the
complex Hessian of the BCOV torsion using the relation. After establishing a
degenerate version of the Schwarz Lemma of Yau, we prove that the complex
Hessian of the BCOV torsion is bounded by the Poincar\'e metric.
