Consider a smooth manifold $M$ and a bundle $\pi\colon E\to M$ over it, where each fibre of $E$ is an algebraic variety. Is there a special name for this kind of bundle? The idea I have is that questions about the topology of the fibres can then be turned into algebraic ones. One example that I have in mind is of an $SL(n)$-bundle, which could be seen as a sub-bundle of an $M\_n$-bundle (with $M\_n$ denoting the space $n$-dimensional square matrices). The latter is both a vector bundle and a smooth bundle, while the former is a smooth but obviously not a vector bundle. However, both $M\_n$ and $SL(n)$, being algebraic varieties, have more structure than just smooth manifolds. It seems like an awful waste to throw all that structure away just to consider them as bundles of the same class.


It strikes me that these kinds of bundles must have already been studied in the context of algebraic geometry, where $M$ would likely also be taken as an algebraic variety. Unfortunately, my knowledge of algebraic geometry not good enough to penetrate most of the literature and figure out how to translate what's known to the case when $M$ is just smooth (being the case that I understand).


Thanks for any help!



Name for bundle of algebraic varieties over a smooth manifold
I do not know names for these kind of structures, but here are some hints. There are two cases you might want to consider:


1.) all the fibres are isomorphic as algebraic varieties. Let $Aut(V)\subset Diff(V)$ be the group of all automorphisms of a variety $V$, with the $C^{\infty}$-topology and $G \subset Aut(V)$ be a subgroup. Then you can consider bundles with structural group $G$ and fibre $V$. Examples: $V=\mathbb{CP}^n$, $G=PGL\_{n+1}(\mathbb{C})$. Or you can study a complex linear algebraic group $G$ and study $G$-principal bundles. Or $V=M\_n$, $G=PGL\_n (\mathbb{C})$, acting by conjugation. 


2.) the fibres are not isomorphic as varieties. If you have two smooth algebraic varieties $E, M$ over $\mathbb{C}$, you can talk about a proper smooth morphism $\pi:E \to M$. When you forget the complex structures, then $\pi:E \to M$ is a proper submersion and a (fairly elementary) result from differetial topology (Ehresmann's fibration lemma) says that $\pi:E \to M$ is a differentiable fibre bundle. The fibres are complete varieties, and they are all diffeomorphic, but not isomorphic as algebraic varieties. Examples of this type show up in moduli problems, for example Teichmueller theory, where one has a tautological bundle of Riemann surfaces over the Teichmueller space. For more details, you can consult Voisin: "Hodge theory and algebraic geometry".


