Simple maps: Flat versus locally trivial

Question

In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term simple differs by literature. There are two notions of simple: as a locally trivial map and as a flat map. Are these definitions equivalent?

I should mention too that some references define an analytic family using submersions instead of simpleness for the smooth analytic case. But this definition coincides with the first one by Ehresmann's fibration theorem.

Maybe the question is trivial, but since I have never seen it written anywhere, I think I should be more careful.

Thanks in advance.

EDIT: The locally triviality is in the category of real differentiable manifolds or the category of topological manifolds. This explains why I have cited Ehresmann fibration theorem as an alternative for the statement.

Answer 1

The answer is no.

In fact, flat maps are not required to be smooth maps in general. For instance, a flat family of smooth curves degenerating to a nodal curve is clearly not locally trivial.

However, the answer is still no even if we only consider smooth maps. In fact, there is the following theorem due to Grauert and Fischer, see [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I]:

Theorem. A smooth family of compact complex manifold is locally trivial if and only if all its fibers are analytically isomorphic.

Now it is possible to give examples of smooth and flat but not locally trivial maps: the so-called Kodaira fibrations, see the same book, Chapter V.

These are smooth maps $f \colon X \to C$, where $X$ is a complex surface ad $C$ a smooth curve, such that the fibres are not analytically isomorphic. By Grauert-Fischer theorem, it follows that Kodaira fibrations are not locally trivial maps (i.e., although all their fibres are smooth, they are not complex analytic fibre bundles).

Since every smooth fibration of a surface over a curve is automatically flat, this provides the examples we were looking for.

Added. If $f \colon X \to Y$ is a flat map in the complex category, it is not true in general that it provides a submersion in the real category. Again, this is because flat map does not imply smooth map. For instance, one can consider a flat map $f \colon X \to \Delta$, where $\Delta \subset \mathbb{C}$ is the unit disk, such that the general fibre is a smooth complex conic and the central fibre is the union of two complex lines intersecting at one point. This map cannot be a submersion because of the Ehresmann theorem: in fact, the topological type of the central fibre is different from the topological type of the general fibre.