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](http://en.wikipedia.org/wiki/Word_problem_for_groups) 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](https://mathoverflow.net/questions/15087/computing-fundamental-groups-and-singular-cohomology-of-projective-varieties) show that presentations for $\pi\_1(X)$ are computable, so there are no issues there.



The word problem for fundamental groups of smooth projective varieties
I was going through my old questions and realized that this one did not have a good answer (as far as I know, the reference that Ben Wieland gave in his answer does not work). I've since learned that it is a well-known open question. However, I thought I'd point out the recent paper


Kapovich, Michael,
Dirichlet fundamental domains and topology of projective varieties. 
Invent. Math. 194 (2013), no. 3, 631–672. 


which proves that every finitely presentable group is the fundamental group of an irreducible projective variety with very mild singularities (only normal crossings and Whitney umbrellas), and thus the word problem is unsolvable for the fundamental groups of such varieties. If you don't require irreducibility, the paper


Kapovich, Michael and Kollár, János, 
Fundamental groups of links of isolated singularities. 
J. Amer. Math. Soc. 27 (2014), no. 4, 929–952. 


manages to do this with only normal crossing singularities.


