Question

Hi, are limitations on the fundamental group for compact complex manifolds known?

Can an arbitrary (finite represantable) group be the fundamental group of a compact complex manifold?

Thanks

Answer 1

Every finitely presented group is the fundamental group of a compact complex manifold of dimension $3$.

This is proven in the book by Amoros, Burger, Corlette, Kotschick and Toledo Fundamental groups of compact Kahler manifolds, Corollary 1.66 p. 19.

The rough idea of proof is the following. Let $\Gamma$ be a finitely presented group, and let $Y$ be a smooth closed oriented $4$-manifold with $\pi_1(Y) \cong \Gamma$. Then by a result of Taubes one can find a complex $3$-fold with the same fundamental group by taking the twistor space $Z$ of $X=Y \sharp n \overline{\mathbb{C} \mathbb{P}^2}$ for $n$ sufficiently large.

Answer 2

Just to give one more refference, there is now a new proof of this theorem that does not use the deep result of Taubes, the proof is elementary and 8 pages long:

http://arxiv.org/abs/1104.4814