# complex-analytic structure on the generic fibre

Here's something that's probably well known to many here, and I'd like to have a suitable reference. I've got a map $f\colon X\to Y$ of complex spaces whose restriction to dense open subspaces $f\colon U\to V$ is the projection of a fibre bundle. Suppose also that $X$ and $Y$ are affine varieties, so that I can consider the fibres of the map $f$ in two different categories. That is, if $p\in V$ is a closed point, I could refine the Zariski topology on the fibre $f^{-1}(p)$ to the complex topology, and the result (I expect) must be homeomorphic to the same fibre in the category of complex spaces. (If true, I could probably learn this much from, say, Shafarevich's Basic Algebraic Geometry.)

This leads to my question. Instead of looking at fibres over closed points, I'd probably prefer handling the fibre of $f$ over the generic point of $Y$, which I'll call $f^{-1}(\eta)$. (Does the property that $f$ restricts to a fibre bundle translate to something nice about the variety $f^{-1}(\eta)$?) But I would want to compare the generic fibre, as a topological space, with the typical fibre of $f$ (as a complex space): I'd like to say that, suitably (re)topologized, the two notions of fibre are homeomorphic. However, I don't know how to refine the Zariski topology on the generic fibre, since my base field is now a function field (instead of $\mathbb C$). Does anyone know of some good reading that might sort me out?

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You could use the Lefschetz principle to write the generic fiber as the base change of a variety defined over a finitely generated field extension $K$ of $\mathbb{Q}$, and then embed $K$ into $\mathbb{C}$ to "think of" the generic variety as a complex variety (which then has a corresponding complex analytic space). However, Serre's examples suggest the corresponding topological space may depend on the choice of embedding of $K$ into $\mathbb{C}$. –  Jason Starr Jul 13 '12 at 20:16
Thanks, that's interesting. I should probably include a reference to mathoverflow.net/questions/90551/… -- it would be great if you would like to expand. Is that to say that obtaining a complex-analytic space from the generic fibre may be possible, but not functorially? –  Graham Denham Jul 14 '12 at 2:00
Yes, I am saying that for any finite type variety over a characteristic $0$ field, there is non-canonically associated a complex analytic space. Non-surprisingly, these different complex analytic spaces will typically be non-biholomorphic. What was surprising when Serre first wrote down examples, is that these complex analytic spaces can also be non-homeomorphic. –  Jason Starr Jul 14 '12 at 11:39
Sounds good -- if you wouldn't mind giving the reference to Serre as an answer, I'll accept it, although I'm not sure yet if my fibre bundle condition makes things any more special. –  Graham Denham Jul 16 '12 at 20:27