# K\"ahler forms for families of Calabi-Yau manifolds

## TL;DR

This paper explores the relationship between Kähler-Einstein metrics and Weil-Petersson forms in families of Calabi-Yau manifolds, constructing a Kähler form on the total space with Ricci-flat fibers.

## Contribution

It establishes a connection between the curvature of Kähler-Einstein metrics and Weil-Petersson forms, enabling the construction of a global Kähler form on the family.

## Key findings

- Curvature form equals the pull-back of Weil-Petersson form up to a constant
- Constructed a Kähler form on the total space with Ricci-flat fibers
- Provided a geometric framework linking metrics and moduli of Calabi-Yau families

## Abstract

K\"ahler-Einstein metrics for polarized families of Calabi-Yau manifolds define a natural hermitian metric on the relative canonical bundle. The fact that the curvature form is equal to the pull-back of the Weil-Petersson form up to a numerical constant is being used for the construction of a K\"ahler form on the total space of a given family, whose restriction to the fibers is Ricci flat.