  In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli
spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense
of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense
of Satake). As an application, we prove the following hyperbolic version of
Campana's isotriviality conjecture: for the smooth family of canonically
polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance
of the base vanishes identically, the family must be isotrivial, that is, any
two fibers are isomorphic. We also prove that for the smooth projective family
of polarized Calabi-Yau manifolds, its variation of the family is less than or
equal to the essential dimension of the base.
