Question

While attending a very nice talk on the geometric group theory of fundamental groups of Kahler manifolds by Pierre Py last weekend, I realized that I don't know the answer to the following question. Let $X$ be a smooth projective variety over $\mathbb{C}$. Is the word problem for $\pi_1(X)$ solvable?

Here are a couple of relevant facts. Taubes proved that every finitely presentable group is the fundamental group of a compact complex manifold of complex dimension 3. Earlier, Gompf proved that every finitely presentable group is the fundamental group of a compact symplectic manifold of real dimension 4. Thus the word problem is not solvable for fundamental groups of compact complex manifolds. Also, Toledo has an example of a smooth compact projective variety whose fundamental group is not residually finite. This rules out using maps to finite groups to solve the word problem, and also shows that $\pi_1(X)$ need not be linear.

EDIT : Another relevant remark is that the answers to the question here show that presentations for $\pi_1(X)$ are computable, so there are no issues there.

Answer 1

Try Bogomolov and Katzarkov (google books or an earlier paper). I don't understand the statements, but I think that for every finitely presented group, they find a extensions of surface groups by the given group that are "approximated" by projective groups. The quality of the approximations is not clear, but I suspect that they preserve uncomputability.