Aspherical manifolds, Mellin transformation and a question of Bobadilla-Koll\'{a}r

TL;DR

This paper addresses topological conditions for proper maps of complex algebraic varieties to be fibrations, providing positive results for specific cases and proposing conjectures related to the Singer-Hopf conjecture.

Contribution

It proves the integral and rational homology versions of Bobadilla-Kollár's question for abelian varieties and compact ball quotients, respectively, and introduces new conjectures.

Findings

Positive answer for integral homology case in abelian varieties

Positive answer for rational homology case in compact ball quotients

Proposes conjectures related to the Singer-Hopf conjecture

Abstract

In their 2012 paper, Bobadilla and Koll\'ar studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer-Hopf conjecture in the complex projective setting.