As Donu mentioned, it is an open problem whether every Kähler group is projective, i.e. the fundamental group of a complex projective manifold (equivalently, a complex projective surface). This question is closely related to the Kodaira problem, which asks whether every compact Kähler manifold can be deformed to a projective variety (at least after passing to a minimal model). Using this relationship, the problem on Kähler groups has recently been answered positively in dimension three, in a series of papers.
So let $X$ be a compact Kähler threefold. We make a case distinction according to the Kodaira dimension $\kappa(X)$.
- $\kappa = -\infty$: $X$ is uniruled, and the MRC quotient $\varphi \colon X \dashrightarrow Z$ induces an isomorphism $\pi_1(X) \cong \pi_1(Z)$. Since $\dim Z \le 2$, $\pi_1(Z)$ is projective.
- $\kappa = 0$: A minimal model $X'$ of $X$ admits a locally trivial deformation to a projective variety, so $\pi_1(X) \cong \pi_1(X')$ is projective (Graf).
- $\kappa = 1$: Same line of argument as above (Lin).
- $\kappa = 2$: Claudon and, Höring and Lin proved that $\pi_1(X)$ is projective using the theory of elliptic fibrations.
- $\kappa = 3$: A Kähler manifold which is Moishezon is already projective, so there is nothing to show.
