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

Let X be a smooth complex quasi-projective variety and =₁ (X). Let R be an additive character. We prove that the ray does not belong to the BNS set () 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 ₁ (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 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.