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?
4$\begingroup$ I would say that the progress has somewhat incremental since then, but there have been some nice results proven in the intervening years. Let me see if I can put together an answer later (at least it would be nice distraction from reading the news...) $\endgroup$– Donu ArapuraMay 17, 2017 at 14:09
$\begingroup$ I like a lot this one: arxiv.org/abs/1609.08474 $\endgroup$– aglearnerMay 17, 2017 at 15:40
This is a bit off the top of my head. So if someone feels I omitted some important development, feel free to mention it to me in a comment, by email, or in your own answer. One can also look at Burger's Seminaire Bourbaki report on Kähler groups from 2010 for a more recent survey.
- A natural question, open in the 90's, is what one-relator Kähler groups look like. This was settled by Biswas-(Mahan)Mj and Kotschick. They are fundamental groups of (orbifold) curves.
- Delzant proved that solvable non-residually nilpotent groups aren't Kähler. (In the 90's Nori and I proved this under certain finiteness conditions, which were unnecessary in hindsight.)
- It's still an open problem to determine whether the class of Kähler groups is the same as the class of fundamental groups of smooth complex projective varieties. Campana-Claudon-Eyssidieux showed that for linear groups these classes are "virtually" the same.
- It's still open to decide whether mapping class groups are Kähler in general. For genus 2 and 3, they aren't (Veliche, Hain resp.)
- Infinite three manifold groups aren't Kähler (Dimca-Suciu).
$\begingroup$ Concerning point 3) in dimension three, see my answer below. $\endgroup$– pgrafMay 22, 2017 at 9:19
1$\begingroup$ Actually in point 3. above, one can remove "virtually". Claudon has written a note to improve his paper with Campana and Eyssidieux ( iecl.univ-lorraine.fr/~Benoit.Claudon/tori_equivariant.pdf ) $\endgroup$– PierreAug 24, 2017 at 3:38
It is still open whether or not all Kahler groups occur as the fundamental groups of smooth complex projective varieties. However, there has been some interesting work on which groups can occur as the fundamental groups of complex projective varieties with only mild singularities.
Kapovich-Kollar proved in
M. Kapovich and J. Kollár, Fundamental groups of links of isolated singularities. J. Amer. Math. Soc. 27 (2014), no. 4, 929–952.
that every finitely presented group is the fundamental group of a complex projective surface with simple normal crossing singularities. The varieties they construct are not irreducible; however, Kapovich proved in
Dirichlet fundamental domains and topology of projective varieties.
Invent. Math. 194 (2013), no. 3, 631–672.
that every finitely presented group is the fundamental group of a complex irreducible projective surface whose singularities are simple normal crossings and Whitney umbrellas.
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.
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