The BNS set of a finitely generated group is a certain canonical subset of the space of real additive characters on . It is a subtle invariant of the group that naturally comes up in different questions of geometric and homological group theory. In the case when is the fundamental group of a compact Kähler manifold , Thomas Delzant found a beautiful description of its BNS set in terms of holomorphic fibrations of over hyperbolic orbifold curves. Using it, he showed that if the fundamental group of a compact Kähler manifold is virtually solvable, it is in fact virtually nilpotent. I will explain the main ideas behind Delzant’s proof and how to generalise his theorems to the case when is a smooth complex quasi-projective variety. Time permitting, I will also discuss some applications and the case of quasi-Kähler manifolds.
