Computing fundamental groups and singular cohomology of projective varieties

This is an interesting question. To repeat some of the earlier answers, one should be able to get one's hands on a triangulation algorithmically using real algebro-geometric methods, and thereby compute singular cohomology and (a presentation for) the fundamental group. But this should probably be a last resort in practice. For smooth projective varieties, as people have noted, one can compute the Hodge numbers by writing down a presentation for the sheaf p-forms and then apply standard Groebner basis techniques to compute sheaf cohomology. This does work pretty well on a computer. For specific classes, there are better methods. For smooth complete intersections, there is a generating function for Hodge numbers due to Hirzebruch (SGA 7, exp XI), which is extremely efficient to use.

As for the fundamental group, if I had to compute it for a general smooth projective variety, I would probably use a Lefschetz pencil to write down a presentation.

For singular varieties, one can still define Hodge numbers using the mixed Hodge structure on cohomology. The sum of these numbers are still the Betti numbers. I expect these Hodge numbers are still computable, but it would somewhat unpleasant to write down a general algorithm. The first step is to build a simplicial resolution using resolution of singularities. My colleagues who know about resolutions assure me that this can be done algorithmically now days.

(This is my first reply in this forum. Hopefully it'll go through.)

I just want to assure you that everything in this situation is computable. For any real semi-algebraic set, there is an algorithm called cylindrical decomposition which breaks it into contractible pieces, glued along contractible pieces. See Algorithms in Real Algebraic Geometry, by Basu, Pollack and Roy. The $\mathbb{C}$-points of a $\mathbb{C}$-variety are, in particular, a semi-algebraic set, by restriction of scalars.

So you can compute cohomology, and you can compute a presentation of $\pi_1$. Of course, as always when dealing with groups in terms of generators and relations, it will probably not be computable to determine whether that group is trivial, or is isomorphic to some other group given by generators and relations.

I am pretty sure that this is not how anyone actually computes these things though. I hope someone will give an answer that reflects the actual state of the art.

Regarding your third paragraph, let p : X ---> B be a smooth, proper map of varieties over C, and for the heck of it say B is smooth. Here I'm thinking B is your space of possible coefficients in the equation, and the fibers of p are the varieties you're talking about. Then on complex points, p is proper submersion between manifolds, and hence a fibration of topological spaces; thus the fibers of p will be homotopy equivalent provided B is connected, canonically homotopy equivalent (up to 2nd order homotopy) if B is simply-connected, and really canonically homotopy equivalent if B is contractible, e.g. an affine space.