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



Fundamental Groups of compact Complex manifolds?
*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](http://rads.stackoverflow.com/amzn/click/0821804987), 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. 


