The Bieri-Neumann-Strebel sets of quasi-projective groups

TL;DR

This paper characterizes the Bieri-Neumann-Strebel sets of quasi-projective groups via algebraic fibrations and explores their implications for the structure of fundamental groups, extending previous results from compact Kähler manifolds to quasi-projective varieties.

Contribution

It establishes a criterion for membership in the BNS set for quasi-projective groups and proves that such groups are virtually solvable if and only if they are virtually nilpotent, generalizing prior theorems.

Findings

Characterization of BNS sets via algebraic fibrations over hyperbolic orbicurves.

Virtually solvable groups are exactly those virtually nilpotent in this context.

Strengthening of results on the structure of fundamental groups of quasi-projective varieties.

Abstract

Let $X$ be a smooth complex quasi-projective variety and $\Gamma=\pi_1(X)$. Let $\chi \colon \Gamma \to \mathbb{R}$ be an additive character. We prove that the ray $[\chi]$ does not belong to the BNS set $\Sigma(\Gamma)$ if and only if it comes as a pullback along an algebraic fibration $f \colon X \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. We also prove that if $\pi_1(X)$ admits a solvable quotient which is not virtually nilpotent, there exists a finite \'etale cover $X_1 \to X$ and a fibration $f \colon X_1 \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. Both of these results were proved by Delzant in the case when $X$ is a compact K\"ahler manifold. We deduce that $\Gamma$ is virtually solvable if and only if it is virtually nilpotent, generalising the theorems of Delzant and Arapura-Nori. As a byproduct, we prove a version of Simpson's Lefschetz Theorem for the integral leaves of logarithmic $1$-forms that do not extend to any partial compactification. We give two applications of our results. First, we strengthen the recent theorem of Cadorel-Deng-Yamanoi on virtual nilpotency of fundamental groups of quasi-projective $h$-special and weakly special manifolds. Second, we prove the sharpness of Suciu's tropical bound for the fundamental groups of smooth quasi-projective varieties and answer a question of Suciu on the topology of hyperplane arrangements.