We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
103/2003
Kähler manifolds and fundamental groups of negatively $\delta$-pinched manifolds
Jürgen Jost and Yi-Hu Yang
Abstract
In this note, we will show that the fundamental group of any negatively $\delta$-pinched ($\delta > {\frac 1 4}$) manifold can't be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in $F_{4(-20)}$ cannot be the fundamental group of a quasi-compact K\"ahler manifold. The corresponding result for uniform lattices was proved by Carlson and Hern\'andez. Finally, we follow Gromov and Thurston to give some examples of negatively $\delta$-pinched manifolds ($\delta > {\frac 1 4}$) of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure. This shows that our result for $\delta$-pinched manifolds is a nontrivial generalization of the fact that no nonuniform lattice in $SO(n, 1) (n \ge 3)$ is the fundamental group of a quasi-compact K\"ahler manifold.
