A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a neighbourhood $ U$ such that $\pi ^{-1} (U) $ is $S$-diffeomorphic (=fiber preserving diffeomorphism) to $U\times F$ for some manifold $F$, called the typical fiber. Of course the holomorphic analogon is completely false: deformation theory might be described as the study of this failure!

To give a concrete example, consider the family $X \subset S\times \mathbb P^2 (\mathbb C) $ of elliptic curves $y^2z=x(x-z)(x-\lambda z)$ with $\lambda \in S=\mathbb C \setminus \{0,1\} $ and the corresponding proper holomorphic submersion $\pi: X \to S: (\lambda , [x:y:z]) \mapsto \lambda $. This $\pi$ is certainly not locally trivial downstairs because its fibers are not mutually isomorphic.

However, in the proper case, non-isomorphy of fibers is the only obstruction to being locally trivial. Indeed, Fischer and Grauert proved that a proper holomorphic submersion having all its fibers isomorphic is locally trivial downstairs. I wonder what can be salvaged of their theorem in the non-proper case:

**Question:** Is there a class $\mathcal C$ of non-compact complex manifolds such that the following holds. If a holomorphic submersion $\pi: X\to S$ between complex manifolds has all its fibers $\pi^{-1}(s)\;(s\in S)$ isomorphic to the same $F \in \mathcal C$, then $\pi$ is locally trivial downstairs with typical fiber $F$.

**Bibliography** Fischer and Grauert unfortunately published their result in a rather confidential journal: W. Fischer, H. Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten,
Nachr. Akad. Wiss. G¨ottingen Math.-Phys. Kl. II (1965), 89–94.

(Please note that the Fischer here is Grauert's doctoral student *Wolfgang* Fischer, not the complex geometer Gerd Fischer. )

Compact Complex Surfacesin Chapter I,Theorem 10.1 on page 29. In a way, I'm happy about your doubts: it shows how unbelievable Fischer-Grauert's theorem is! $\endgroup$ – Georges Elencwajg Feb 26 '11 at 16:23