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

Let X be a smooth complex quasi-projective variety and Γ=π₁(X). Let χ Γ → R be an additive character. We prove that the ray [χ] does not belong to the BNS set Σ(Γ) if and only if it comes as a pullback along an algebraic fibration f X → C over a quasi-projective hyperbolic orbicurve C. We also prove that if π₁(X) admits a solvable quotient which is not virtually nilpotent, there exists a finite \'etale cover X₁ → X and a fibration f X₁ → C over a quasi-projective hyperbolic orbicurve C. Both of these results were proved by Delzant in the case when X is a compact K\"ahler manifold. We deduce that Γ 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.