A natural hermitian metric associated with local universal families of compact Ricci-flat K\"ahler manifolds

TL;DR

This paper constructs a natural hermitian metric on the fiber product of a universal family of Ricci-flat K"ahler manifolds and their complexified K"ahler cones, with detailed examination of elliptic curves.

Contribution

It introduces a new hermitian metric on the fiber product of universal Ricci-flat K"ahler families and their K"ahler cones, enriching the geometric understanding of such families.

Findings

Existence of a natural hermitian metric on the fiber product.

Application to the case of elliptic curves.

Insights into the geometry of Ricci-flat K"ahler families.

Abstract

Let $\pi : \mathcal X \to S$ be a local universal family of compact Ricci-flat K\"ahler manifolds over a smooth base $S$. The complexified K\"ahler cones of each fiber of the family form a holomorphic fiber bundle $\mathcal K \to S$. We show that there exists a natural hermitian metric $\omega$ on the fiber product $\mathcal X \times_S \mathcal K$. We then discuss the example of elliptic curves in some detail.