What is the geography of Kähler manifolds with negative sectional curvature? More precisely, can any hyperbolic group be realized as the fundamental group of a Kähler manifold with negative sectional curvature?

## 1 Answer

$\DeclareMathOperator\PO{PO}\DeclareMathOperator\PU{PU}$This is more a comment than an answer, but if one does not impose compactness of the manifold (but still asks for the Kähler manifold to be complete, with pinched negative curvature or even more to be complete locally symmetric of rank 1) then one sees that certain hyperbolic groups occurs in this way whereas they cannot occur as fundamental groups of any closed Kähler manifold.

More precisely take a cocompact torsion free lattice $\Gamma$ in the group $\PO(n,1)$ with $n\ge 3$. Then $\Gamma$ is not isomorphic to the fundamental group of a closed Kähler manifold by an old theorem of Carlson and Toledo (see their paper Harmonic mappings of Kähler manifolds to locally symmetric spaces). Yet by embedding $\PO(n,1)$ into $\PU(n,1)$ and letting $\Gamma$ act on complex hyperbolic space of (complex) dimension $n$, one sees that $\Gamma$ is the fundamental group of a negatively curved Kähler manifold.

So without any further assumption on the negatively curved Kähler manifold one really gets more examples of hyperbolic groups than by simply looking at fundamental groups of closed Kähler manifolds and asking them to be Gromov hyperbolic.

By the way, even studying convex cocompact subgroups of $\PU(n,1)$ is a difficult topic and there are few examples (see Granier - Groupes discrets en géométrie hyperbolique — Aspects effectifs for an interesting one).

compactKähler manifolds. $\endgroup$everycountable group can be realized as the fundamental group of a Kahler manifold of negative sectional curvature. $\endgroup$