Holomorphic fiber bundle with Stein base and Stein fibers

TL;DR

This paper proves that a complex manifold with a Stein base and Stein fibers, under certain local conditions, must itself be Stein, extending the understanding of Stein manifolds in complex geometry.

Contribution

It establishes that a surjective holomorphic map with Stein base and fibers, satisfying local Stein conditions, implies the total space is Stein, providing a new criterion for Steinness.

Findings

If the base space is Stein and fibers are Stein, then the total space is Stein under local conditions.

The result applies to complex manifolds of dimension at least 3.

The theorem extends previous criteria for Stein manifolds in complex geometry.

Abstract

In this article, we prove that if $\Pi: X\to \Omega$ is a surjective holomorphic map, with $\Omega$ a Stein space and $X$ a complex manifold of dimension $n\geq 3,$ and if, for every $x\in \Omega$ there exists an open neighborhood $U$ such that $\Pi^{-1}(U)$ is Stein, then $X$ is Stein