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?