I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic topology, because they do formally seem quite similar. I'm guessing that the answers to my questions are "well-known", but I am struggling to find anything decent in the literature. Any help/references will be most useful.

The set up is this: Let $E,B$ be smooth projective complex algebraic varieties, and let $\pi:E \to B$ be a surjective flat map such that the fibres $E_b:=\pi^{-1}(b), b \in B$ are smooth projective complex algebraic varieties.

I am aware that each fibre has the same Hilbert polynomial, so cohomologically they are quite similar. But each fibre can certainly be non-isomorphic as algebraic varieties (e.g. moduli spaces). However:

Using GAGA type methods, we can consider $E$ and $B$ as complex manifolds. Is it true that $(\pi,E,B)$ is a fibration? That is, satisfies the homotopy lifting property with respect to any topological space?

Again considering $E,B$ and each fibre $E_b$ as a complex manifold, is it true that each fibre $E_b$ is homotopy equivalent to each other? What about homeomorphic/diffeomorphic?

Thanks,

Dan

localfibration $E \to B$? For example if we have a covering map $$\cup_i U_i \to X$$. Is this not a flat map? It won't be a fibration, but it will be alocalfibration. I have been told that the flat topology in algebraic geometry is similar to the surjective submersion topology on manifolds. I don't know how strong this analogy is. $\endgroup$localfibration? $\endgroup$