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

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.