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 $\rho {|}_{X_y}$ for each fiber, which is cohomologous to $\omega {|}_{X_y}$. This family of Ricci-flat Kähler forms $\rho {|}_{X_y}$ 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.

arXiv comment25 pages. Minor changes