13 problems

For $g\geq 3$, let ${\rm Mod}_g$ be the mapping class group of a closed genus-$g$ surface. Kähler-group conjecture. The group ${\rm Mod}_g$ is a Kähler group, meaning that it is is…

Let $\Gamma$ be a Kähler group, let $X_\Gamma$ be its character variety of homomorphisms $\Gamma\to\mathbb{C}^*$, and let … An isolated point of $X^1_\Gamma$ is an isolated charact…

Let ${\mathcal K}$ denote the class of Kähler groups. Let ${\bf NNF}$ denote the class of groups that do not have normal finitely generated free subgroups of rank at least $2$. Käh…

Ramachandran's conjecture. One of the following holds: the universal cover $\widetilde{X}$ has the Bochner–Hartogs property, or $\pi_{1}(X)$ is commensurable with the fundamental g…

A 3-manifold group means the fundamental group of a compact 3-manifold, and a group is Kähler if it is the fundamental group of a compact Kähler manifold. Goldman–Donaldson–Rezniko…

Kähler-group conjecture. The sets of Kähler and projective groups are equal:

Reduced projective resonance and Chen ranks conjecture. The projective resonance ${\bf R}(G)$ of any Kähler group $G$ is reduced. Moreover, if $X$ is a compact Kähler manifold, the…

Let $\Gamma$ be a non-abelian K34hler group, that is, the fundamental group of a compact K34hler manifold. A group is flawless when no strong deformation retraction of…

Let ${\mathcal M}_g$ be the moduli space of smooth curves of genus $g$, and let $X$ be a smooth complete subvariety of ${\mathcal M}_g$. Let $\pi_1(X)$ map to the mapping class gro…

A finitely generated group is called Sasaki if it is the fundamental group of a compact Sasakian manifold, and a Kähler group is the fundamental group of a compact Kähler manifold.…

Let a Kähler group mean the fundamental group of a compact Kähler manifold, and let a compact Hodge manifold be a compact manifold whose fundamental group is under consideration as…

Donaldson–Goldman conjecture. $Q$ cannot be a Kähler group.

Let $\Gamma$ be an infinite Kähler group. Carlson–Toledo conjecture. There exists a compact Kähler manifold $M$ with $\pi_1(M)$ a finite-index subgroup of $\Gamma$ such that the Hu…