[Paper Review] Semi-positivity of fiberwise Ricci-flat metrics on Calabi-Yau fibrations

Young-Jun Choi|arXiv (Cornell University)|Aug 3, 2015

Geometry and complex manifolds19 references3 citations

TL;DR

This paper proves that the family of Ricci-flat Kähler metrics on fibers of a Calabi-Yau fibration over a complex manifold induces a smooth (1,1)-form on the total space that is semi-positive. Using Yau's theorem on Ricci-flat metrics and Kähler geometry, the authors establish the semi-positivity of the fiberwise metric family, with implications for the local triviality of such fibrations.

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 the theorem due to Yau, there exists a unique Ricci-flat K\ahler form $ ho\vert_{X_y}$ for each fiber, which is cohomologous to $\omega\vert_{X_y}$. This family of Ricci-flat K\ahler forms $ ho\vert_{X_y}$ induce a smooth $(1,1)$-form $ ho$ on $X$. In this paper, we prove that $ ho$ is semi-positive on the total space $X$. We also discuss several byproducts, among them the local triviality of families of Calabi-Yau manifolds.

Motivation & Objective

• To investigate the geometric properties of fiberwise Ricci-flat Kähler metrics on Calabi-Yau fibrations.
• To determine whether the induced family of Ricci-flat metrics on the total space is semi-positive.
• To explore the consequences of semi-positivity for the structure of families of Calabi-Yau manifolds.
• To establish conditions under which such fibrations are locally trivial.

Proposed method

• Utilizes Yau's theorem to construct a unique Ricci-flat Kähler metric on each fiber, cohomologous to the restriction of a fixed Kähler form on the total space.
• Constructs a smooth (1,1)-form on the total space X by assembling the fiberwise Ricci-flat metrics.
• Applies techniques from Kähler geometry and curvature analysis to study the curvature properties of the induced metric family.
• Employs differential geometric methods to analyze the semi-positivity of the (1,1)-form on the total space.
• Uses the curvature properties of the fibration to derive implications for the moduli space and local structure of the family.
• Analyzes the variation of the Ricci-flat metrics across fibers to deduce global geometric constraints.

Experimental results

Research questions

• RQ1 Is the family of Ricci-flat Kähler metrics on fibers of a Calabi-Yau fibration semi-positive when viewed as a form on the total space?
• RQ2 What geometric constraints does the semi-positivity of the induced metric impose on the fibration?
• RQ3 Under what conditions is a family of Calabi-Yau manifolds locally trivial?
• RQ4 How does the curvature of the total space relate to the fiberwise Ricci-flat metrics?
• RQ5 Can the semi-positivity of the metric family be used to classify or constrain the moduli of Calabi-Yau fibrations?

Key findings

• The family of Ricci-flat Kähler metrics on the fibers induces a smooth (1,1)-form on the total space X that is semi-positive.
• The semi-positivity of the induced metric is established through curvature analysis and the properties of Ricci-flat metrics on Calabi-Yau manifolds.
• The result implies that the fibration admits a non-negative curvature form on the total space, reflecting geometric control over the family.
• The semi-positivity of the metric family provides a key ingredient for studying the local structure of the fibration.
• The analysis leads to a proof of local triviality for families of Calabi-Yau manifolds under the given conditions.
• The result strengthens the understanding of moduli spaces and deformation theory for Calabi-Yau fibrations.