POSITIVITY OF DIRECT IMAGES OF FIBERWISE RICCI-FLAT METRICS ON CALABI-YAU FIBRATIONS

Let X be a Kahler manifold which is fibered over a complex manifold Y such that every fiber is a Calabi-Yau manifold. Let omega be a fixed Kahler form on X. By Yau's theorem, there exists a unique Ricci-flat Kahler form omega(KE, y) on each fiber X-y for y is an element of Y which is cohomologous to omega vertical bar X-y. This family of Ricci-flat Kahler form omega(KE, y) induces a smooth (1, 1)-form rho on X under 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 including the local triviality of families of Calabi-Yau manifolds.