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

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 $\mathcal K \to S$\mathrm{.}Weshowthatthereexistsanaturalhermitianmetric$\omega$ on the fiber product $\mathcal{X}{\times }_S\mathcal{K}$. We then discuss the example of elliptic curves in some detail.