A natural hermitian metric associated with local universal families of compact Ricci-flat Kähler manifolds

Abstract

Let $\pi :\mathcal{X}\to S$ be a local universal family of compact Ricci-flat Kähler manifolds over a smooth base $S$. The complexified Kähler 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.

arXiv comment5 pages. This note is to appear in Comptes Rendus - Mathématique