Irrational pencils and Betti numbers

TL;DR

This paper investigates irrational pencils on complex manifolds, showing that nonempty critical points imply non-finitely generated homology of certain kernels, and constructs new examples of varieties with complex fundamental groups.

Contribution

It generalizes previous results by linking critical points of irrational pencils to homology properties and provides new examples using the Cartwright-Steger surface.

Findings

Nonempty critical points imply non-finitely generated homology.

Construction of new smooth projective varieties with complex fundamental groups.

Extension of previous results by Dimca, Papadima, and Suciu.

Abstract

We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright-Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology.