Local isomorphisms for families of projective non-unruled manifolds

Abstract

Let $\pi\cln \cX\to S$ and $\pi\cln \cY\to S$ be two smooth families of projective non-uniruled manifolds over a Riemann surface $S$ (probably non-compact). Suppose these two families are pointwise isomorphic. We prove that there exists an open dense subset $U\subset S$ such that the two restricted families are locally isomorphic over $U$. This partially answers Wehler's question on locally isomorphic of families of compact complex manifolds.