Quasi-K\"ahler Bestvina-Brady groups

A finite simple graph \G determines a right-angled Artin group G_\G, with one generator for each vertex v, and with one commutator relation vw=wv for each pair of vertices joined by an edge. The Bestvina-Brady group N_\G is the kernel of the projection G_\G \to \Z, which sends each generator v to 1. We establish precisely which graphs \G give rise to quasi-K\"ahler (respectively, K\"ahler) groups N_\G. This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasi-projective variety.