MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 11 down vote accepted

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.

share|cite|improve this answer
Dear Donu, thank you very much for this quick answer! – aglearner Jun 13 '13 at 15:28

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 (

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 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.