A natural hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class

Abstract

Let π:X→S be a local universal family of compact Kähler manifolds with zero first Chern class over a smooth base S. The complexified Kähler cones of each manifold of the family form a holomorphic fiber bundle K→S. We show that there exists a natural hermitian metric ω on the fiber product X×SK. We then discuss the example of elliptic curves in some detail.