On holomorphic submersions with biholomorphic fibers

Abstract

We give explicit examples of holomorphic submersions $p\colon Y\to X$ such that $Y$ and $X$ are complex manifolds, all the fibers of $p$ are biholomorphic to the unit disc in the complex plane, but the mapping $p$ is not holomorphically locally trivial. In our examples the complex manifolds $Y$ are neighborhoods of the zero section of a line bundle $L$ on $X$; the line bundle $L$ is not, in general, supposed to be positive or negative.