How do fibers of the functor Algebraic Varieties $\to$ Complex Analytic Spaces look like?

There's already a question (which got several interesting answers) asking about examples of the phenomenon of non (essential) injectivity of the functor $U:Alg\to AnEsp$, mapping each algebraic variety to its associated (reduced) complex analytic space.

I would like to complement that question by asking:

As far as it is currently known, do the fibers have any particular structure? Are there invariants classifying different algebraic structures on the same analytic space (like e.g. some cohomology space which bears some correspondence with algebraic structures, and is trivial iff the structure is unique)?

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Could you link to the question you mention about non-injectivity please? Thanks. –  Ruadhaí Dervan Dec 3 '12 at 0:09
Sure - I forgot to put the link. –  Qfwfq Dec 3 '12 at 0:11
Another related question: mathoverflow.net/questions/86000/… –  jvp Dec 3 '12 at 10:58