ag.algebraic geometry – Fundamental groups of primitive non-algebraic compact Kähler manifolds


Call a compact topological manifold $M$ primitive if there is no Serre fibration $Mto B$ where $B$ is a CW complex of dimension $0<d<mathrm{dim}(M)$.

Given a Kähler group $G$ does there exist a primitive non-algebraic compact Kähler manifold with $pi_1=G$?