5 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…

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…

Let $G$ be an infinite Kähler group, meaning the fundamental group of a smooth complex Kähler manifold. Carlson–Toledo conjecture. There exists a finite-index subgroup $G^\circ$ of…

Let $X$ be a compact Kähler manifold. For each $g\geq 2$, let $\mathcal{E}(X)$ be the set of relevant fibrations $f\colon X\to B_f$ and define … Here $\Pi_g$ denotes the fundamenta…

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…