Positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations

Abstract:

Let $X$ be a Kähler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $\omega$ be a fixed Kähler form on $X$. By Yau’s theorem, there exists a unique Ricci-flat Kähler form $\omega _{KE,y}$ on each fiber $X_y$ for $y\in Y$ which is cohomologous to $\omega \vert _{X_y}$. This family of Ricci-flat Kähler forms $\omega _{KE,y}$ 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.