Let $\pi : X \to B$ be a proper, surjective holomorphic submersion, where both $X$ and $B$ are compact Kahler manifolds. Assume that all the fibers $X\_b = \pi^{-1}(b)$ are smooth. Is the family $\pi : X \to B$ then locally trivial?


The answer is "no" when formulated like this: Let $\pi : X \to B$ be the family where $X\_b$ is the blowup of $B$ at the point $b$. If the automorphism group of $B$ does not act transitively on $B$ then the resuling family cannot be locally trivial but all of its fibers are smooth.


This blowup example (and its trivial variants: blowup more points, things of positive dimension if you can get the smoothness hypothesis to work, ...) is however the only counterexample I've been able to come up with, so I wonder if the statement could be true if we make some positivity hypotheses on $K\_X$? For example, suppose in addition that $K\_X$ is nef. Is the family then locally trivial?



Is a holomorphic family whose fibers are all smooth locally trivial?
By Grauert-Fischer Theorem, a smooth family of compact complex manifold is locally trivial if and only if all the fibers are analytically isomorphic.


However, there exist smooth families $\pi \colon X \to B$ such that the fibres are *not* isomorphic, and so they are *not* locally trivial.


The easiest examples occur already for $\dim X =2$ and $\dim B =1$: they are the so-called *Kodaira fibrations*. In such fibrations, all the fibres are smooth curves but their complex structure varies. 


If $\pi \colon X \to B$ is a Kodaira fibration, then $B$ is a curve of genus at least $2$. Furthermore, $X$ is a minimal surface of general type, hence it is algebraic and $K\_X$ is ample. 


So, even imposing a positivity condition on $K\_X$, the answer to your question is *no*.


A reference is [Barth-Hulek-Peters-Van de Ven, *Compact Complex Surfaces*], Chapter V.


For the Grauert-Fischer theorem see the same book, Chapter I.


