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

Yongqiang Liu, Laurenţiu Maxim+1 more|arXiv (Cornell University)|Jun 16, 2020

Homotopy and Cohomology in Algebraic Topology19 references4 citations

TL;DR

This paper provides a positive answer to the integral and rational homology versions of the Bobadilla-Kollár question for aspherical projective manifolds: if the universal cover of a projective manifold is contractible and the pullback of the universal cover is homotopy equivalent to a finite CW complex, then the map is a Z- or Q-homology fiber bundle. The proof uses Mellin transformations, perverse sheaves, and positivity of Chern classes, with key results for abelian varieties and compact ball quotients.

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.

Motivation & Objective

• . The paper investigates whether finiteness conditions on relative covering spaces imply that a proper holomorphic map is a homotopy or homology fiber bundle.
• It addresses the integral and rational homology versions of the Bobadilla-Kollár question in the context of aspherical manifolds.
• The objective includes proving that for abelian varieties and compact ball quotients, the homological finiteness condition implies the map is a Z- or Q-homology fiber bundle.
• The work aims to connect topological finiteness conditions to fibration structures, with implications for the Singer-Hopf conjecture.
• It proposes generalizations of the Singer-Hopf conjecture to perverse sheaves on aspherical projective manifolds.

Proposed method

• . The authors use a nonabelian Mellin transformation to relate the homotopy type of the universal cover to the cohomology of the base space.
• They apply the decomposition theorem and positivity results for Chern classes of ample vector bundles to analyze the cohomological behavior of the map.
• The theory of perverse sheaves and derived categories is used to characterize when the direct image complex is locally constant.
• For abelian varieties, a key non-vanishing property of the Mellin transformation is proven using characteristic cycles and the geometry of abelian varieties.
• The proof relies on the fact that the universal cover of an aspherical projective manifold with ample cotangent bundle is Stein, under the Shafarevich conjecture.
• The argument combines complex geometry, algebraic topology, and representation theory, particularly in positive characteristic.

Experimental results

Research questions

• RQ1 . Does the finiteness of the universal cover of the total space imply that a proper holomorphic map is a Z-homology fiber bundle when the base is an abelian variety?
• RQ2 . Does the same finiteness condition imply that the map is a Q-homology fiber bundle when the base is a compact ball quotient or more generally an aspherical projective manifold with ample cotangent bundle?
• RQ3 . Can the Singer-Hopf conjecture be generalized to perverse sheaves on aspherical projective manifolds?
• RQ4 . Does the Shafarevich conjecture imply that the universal cover of an aspherical projective manifold is Stein?
• RQ5 . Does every aspherical projective manifold have a fundamental group admitting a finite-dimensional faithful linear representation?

Key findings

• . The integral Bobadilla-Kollár question has a positive answer for abelian varieties: if the universal cover of the total space is homotopy equivalent to a finite CW complex, then the Albanese map is a Z-homology fiber bundle.
• . The rational Bobadilla-Kollár question has a positive answer for aspherical projective manifolds with ample cotangent bundles, including compact ball quotients.
• . The universal cover of an aspherical projective manifold contains no positive-dimensional compact analytic subvarieties, which supports the Shafarevich conjecture.
• . The Shafarevich conjecture implies that the universal cover of an aspherical projective manifold is Stein, which is a key step toward generalizing the Singer-Hopf conjecture.
• . The paper proves that the Mellin transformation is non-vanishing on abelian varieties, a crucial technical result for the integral case.
• . The authors show that the Euler characteristic of any perverse sheaf on an aspherical projective manifold is non-negative, generalizing the Singer-Hopf conjecture.