Can one compute the fundamental group of a complex variety? Other topological invariants? Given a system of polynomial equations with rational coefficients, is there an algorithm to compute the geometric fundamental group of the variety defined by these equations? I'm interested in both the affine case and the projective case. By "compute" I mean "express in terms of generators and relations."
Can one at least compute the unipotent or profinite completion in some manner?
Some possible approaches:
If the equations define a smooth, projective variety, then one can compute the Hodge cohomology, which tells you the rational de Rham cohomology. By formality, one can use the rational de Rham cohomology to compute the unipotent completion of the fundamental group.
Another approach to computing the cohomology is by looking at numbers of points over various finite fields and using the Weil conjectures.
Is there, for example, a way to get a CW complex homotopy equivalent to the complex points of the variety?
 A: Suppose that $X\subseteq\mathbb{C}P^n$ is a smooth variety.  There are various ways to define families of smooth maps $\mathbb{C}P^n\to\mathbb{R}$; for example, we can choose a line $L_0<\mathbb{C}^{n+1}$ and define $f(L)$ to be the norm of the orthogonal projection $L\to L_0$.  A generic choice of $f$ should restrict to give a Morse-Smale function on $X$.  If we choose a particular $f$, we can use Grobner basis methods to check whether it is Morse, and to find the critical points and their indices.  It is not so obvious how to check the Smale condition; I am nt sure whether that would cause trouble. There is a CW structure with a cell of dimension $d$ for each critical point of index $d$.  To calculate the fundamental group, you just need to understand the cells of dimension at most two, and their attaching maps.  You can search for "Morse homology" to find more about this sort of thing.
A: There are algorithms to compute triangulations of real algebraic varieties, and thus of complex algebraic varieties. Sadly, the algorithms tend to be doubly exponential in the size of the input data. For more, see:
Basu, Saugata, Algorithms in real algebraic geometry: a survey,  ZBL06843372.
In fact, I don't think this is particularly new, so you can also look at the very nice book:
Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise, Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics 10. Berlin: Springer (ISBN 3-540-33098-4/hbk). x, 662 p. (2006). ZBL1102.14041.
