Hodge theory
The question of which groups can be fundamental groups of compact Kähler manifolds, called Kähler groups, is wide open. Hodge theory gives many restrictions on the possible Kähler groups. The simplest restriction is that the abelianization of a Kähler group must have even rank, since the Betti number "b"1 of a compact Kähler manifold is even. (For example, the integers Z cannot be the fundamental group of a compact Kähler manifold.) Extensions of the theory such as non-abelian Hodge theory give further restrictions on which groups can be Kähler groups.