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On the Hopf Problem and a Conjecture of Liu-Maxim-Wang

Luca F. Di Cerbo, Rita Pardini

TL;DR

This work investigates the Hopf problem for aspherical, smooth projective varieties within the Liu–Maxim–Wang framework that connects the conjecture to the Shafarevich conjecture and nefness of the cotangent bundle. It provides multiple proofs that certain abelian branched covers of abelian varieties are not aspherical, including topological/Hodge arguments and a purely analytic degeneration approach in dimension two, and extends these insights to a surface-case argument via moduli and Severi-type considerations. A key implication is that, under the Liu–Maxim–Wang conjecture, aspherical surfaces of general type should satisfy $K_S^2\ge 6\chi(\mathcal{O}_S)$, offering a concrete geography constraint and a testbed for the conjecture. Overall, the results reinforce the viability of Shafarevich-driven strategies in the Hopf problem and delineate clear directions for future search and obstruction of aspherical examples in the general type category.

Abstract

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu, Maxim, and Wang in [LMW21]. In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.

On the Hopf Problem and a Conjecture of Liu-Maxim-Wang

TL;DR

This work investigates the Hopf problem for aspherical, smooth projective varieties within the Liu–Maxim–Wang framework that connects the conjecture to the Shafarevich conjecture and nefness of the cotangent bundle. It provides multiple proofs that certain abelian branched covers of abelian varieties are not aspherical, including topological/Hodge arguments and a purely analytic degeneration approach in dimension two, and extends these insights to a surface-case argument via moduli and Severi-type considerations. A key implication is that, under the Liu–Maxim–Wang conjecture, aspherical surfaces of general type should satisfy , offering a concrete geography constraint and a testbed for the conjecture. Overall, the results reinforce the viability of Shafarevich-driven strategies in the Hopf problem and delineate clear directions for future search and obstruction of aspherical examples in the general type category.

Abstract

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu, Maxim, and Wang in [LMW21]. In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.
Paper Structure (3 sections, 5 theorems, 11 equations)

This paper contains 3 sections, 5 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Main Result -- Several Proofs
  3. The Case of Surfaces

Key Result

Theorem 2.2

If $n\ge 2$, the $n$-dimensional smooth projective variety $X$ is not aspherical.

Theorems & Definitions (15)

  • Conjecture 1.1: Hopf Conjecture
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 5 more