A natural Hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class (Q664927)

Let \(\pi:X \to S\) be a locally universal family of compact Kähler manifolds with zero first Chern class over a smooth base. The complexified Kähler cones of the manifolds in this family give rise to a holomorphic fiber bundle \(K\to S\). A natural Hermitian metric is shown to exist on the fiber product \(X\times_S K\) and the case of families of elliptic curves is discussed in detail.