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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.

Received:
Dec 8, 2003
Published:
Dec 8, 2003

Related publications

inJournal
2004 Repository Open Access
Jürgen Jost and Yi-Hu Yang

Kähler manifolds and fundamental groups of negatively delta-pinched manifolds

In: International journal of mathematics, 15 (2004) 2, pp. 151-167