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

TL;DR

This paper proves the positivity of the direct image of fiberwise Ricci-flat metrics on Calabi-Yau fibrations, leading to insights about the structure and triviality of such families.

Contribution

It establishes the positivity of the direct image of Ricci-flat forms in Calabi-Yau fibrations, a novel result with implications for the geometry of these families.

Findings

Positivity of the direct image of $ ho^{n+1}$ on the base.

Implications for local triviality of Calabi-Yau families.

Connections to fiberwise Ricci-flat metrics and their geometric properties.

Abstract

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.