TITLE: Fundamental groups of compact Kähler manifolds
QUESTION [10 upvotes]: This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics books on any subject) is fantastic, but is over twenty years old now. Can someone summarize how the state of the art in that field has advanced in the intervening period?

REPLY [5 votes]: 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 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.