Fundamental groups of normal complex quasi-projective varieties

Question

I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-projective variety?

Answer 1

Yes, sure. Take any polycyclic group which is not virtually nilpotent (e.g. upper triangular matrices with entries in $GL_n(\mathbb{Z})$, $n\ge 2$). This cannot be the fundamental group of a normal quasiprojective variety by a theorem of Nori and myself [Solvable fundamental groups of algebraic varieties..., Composito 1999]. There are many other sorts of examples.

Answer 2

More obstructions for this realizability problem can be found in the paper On the fundamental groups of normal varieties by Donu Arapura, Alexandru Dimca, and Richard Hain (http://arxiv.org/abs/1412.1483).

For an explicit example, take the right-angled Artin group corresponding to a path on 4 vertices, $$ G=\langle a,b,c,d \mid [a,b]=[b,c]=[c,d]=1 \rangle. $$ It is known from http://arxiv.org/abs/0902.1250 that this group (or, for that matter, any RAAG whose associated graph is not a complete multipartite graph) is not the fundamental group of a smooth complex quasi-projective variety. In the paper quoted above, the authors show that such a RAAG cannot be realized as the fundamental group of a normal complex quasi-projective variety.