TITLE: fundamental groups of smooth projective variety.
QUESTION [10 upvotes]: Is there a discrete group G which is the fundamental group of a compact Kahler
manifold but which is not the fundamental group of any smooth projective complex algebraic variety?
It is known that there are cohomology rings of compact Kahler manifolds not realisable by smooth projective complex algebraic varieties (some of the recent results are due to Voisin).

REPLY [8 votes]: As far as I know, still open. In the 2011 paper HOMOMORPHISMS BETWEEN FUNDAMENTAL GROUPS OF KA ̈HLER MANIFOLDS, one Botong Wang shows that there is a homomorphism between Kahler groups not realized by a holomorphic map between projective varieties, but this seems to be closest to a counterexample.