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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!

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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".

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Thanks, Johannes. I think the first option you presented is the closest one to what I had in mind. Too bad that there's no easily identifiable name associated to this kind of structure. –  Igor Khavkine Jan 13 '11 at 12:44
    
Now there is a follow up question that springs to mind. A vector bundle is in some sense recoverable from the module of it sections. Is there a natural algebraic object that can be associated to such a bundle of algebraic varieties in a similar way? Perhaps I should think about it and post it as a separate question. –  Igor Khavkine Jan 13 '11 at 12:55
    
A vector bundle on M is not recoverable from its module of global sections unless M is affine. You can recover a vector bundle from its sheaf of local sections instead. Likewise, you associate to any fibre bundle the sheaf of local sections (a sheaf of sets) and I would be most surprised if that sheaf does not contain all the information on the bundle. However, I do not know how to recognize the sheaf of local sections of a fibre bundle. –  Johannes Ebert Jan 13 '11 at 14:24
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First remark. I don't know a name for such a structure (though I know a context in which it may appear, see remark 2), but I can try to spell out the definition (that is maybe understood in your question).

Let $\pi: E \to M$ be a smooth fiber bundle with even-dimensional fibers $E_x=\pi^{-1}(x)$, $x \in M$, and let

$\mathcal{V}=\mathcal{V}_\pi=\mathrm{Ker}(\mathrm{d}\pi)\subseteq TE$

be the vertical distribution of the bundle, so that $\mathcal{V}|_{E_x}=TE_x$.

Suppose that there is a smooth section

$J\in \Gamma (E,\mathtt{End}(\mathcal{V}))$, where $\mathtt{End}(\mathcal{V})$ is the endomorphism bundle, such that $J^2=-\mathrm{id}_{\mathcal{V}}$. Now your $(E,J)$ is a bundle of almost-complex manifolds.

Define the Nijenhuis tensor $N_J\in \Gamma (E,\mathcal{V}^{\otimes 2}\otimes \mathcal{V})$ of $J$ in the usual way

$N_J(X,Y) = [X,Y]+J([JX,Y]+J[X,JY]) -[JX,JY]$

(I hope it's correct) for vertical vector fields $X,Y$ in terms of $J$ and Lie brackets. If $N_J$ happens to be zero, then you have a bundle $(E,J)$ of honest complex manifolds $E_x$, $x\in M$.

Now, to get a "$\mathcal{C}^{\infty}$ -bundle of algebraic varieties" you can put the condition that each fiber $E_x$ actually be an algebraic variety. Another way to do it, in case fibers are compact, could be to first endow $(\mathcal{V},J)$ with a Hermitian metric, and then impose the condition that its "associated 2-form"

$\omega\in\Gamma(E, \bigwedge^2\mathcal{V}^*)$

restricts to closed 2-forms $\omega|_x$ on the fibers, so you get a bundle of (compact) Kahler manifolds; then you require that each de Rham cohomology class $[\omega|_x]$ should come from integer cohomology, so you get a bundle of compact Hodge manifolds, hence, from Kodaira embedding theorem, of (smooth) projective varieties.

Second remark. The concept of a $\mathcal{C}^\infty$-varying family of complex manifolds has actually been studied, but in a more general context than bundles, namely that of foliations. The keyword that you should look for is trasversely holomorphic foliation, which means a differentiably-varying family of manifolds -the "leaves"- where each leaf is equipped with a complex structure. There are e.g. theorems about the topology and holonomy of the leaves when they are Kahler. A case of interest is that of one-dimensional compact leaves, cause in that case they're automatically Riemann surfaces (algebraic), and it has been studied even in the more general case in which the total space is just a topological space (the key word is holomorphic laminations, a structure that appears in holomorphic dynamics).

Edit: I should say that the case of a bundle is a special case of foliation, and the structure you address in your question (the one spelled out in my first remark) should be a particular case of transversely holomorphic foliation, i.e.:

  1. it happens to be a bundle

  2. and has algebraic leaves.

For some reasons, from the point of view of smooth foliations the particular case of a bundle is not considered very interesting: a bundle is somehow a "trivial" foliation. But I don't know in the transversally holomorphic case. As far as I remember, in the holomorphic case (i.e. when the total space of the foliation is a complex manifold, together with the leaves, and the latter vary holomorphically) in some cases a bundle structure is the only foliation that can be found on a given total space.

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