  Let $X$ be a K\"ahler manifold which is fibered over a complex manifold $Y$
such that every fiber is a Calabi-Yau manifold. Let $\omega$ be a fixed
K\"ahler form on $X$. By Yau's theorem, there exists a unique Ricci-flat
K\"ahler form $\rho\vert_{X_y}$ for each fiber, which is cohomologous to
$\omega\vert_{X_y}$. This family of Ricci-flat K\"ahler forms $\rho\vert_{X_y}$
induces a smooth $(1,1)$-form $\rho$ on $X$ with a normalization condition. In
this paper, we prove that the direct image of $\rho^{n+1}$ is positive on the
base $Y$. We also discuss several byproducts, among them the local triviality
of families of Calabi-Yau manifolds.
