Hyperbolicity and fundamental groups of complex quasi‐projective varieties (III): Applications

This paper is Part III of a series of three. We begin by introducing the notion of ‐special varieties, which can be seen as varieties “chain‐connected by the Zariski closures of entire curves.” We prove that if is either a special smooth complex quasi‐projective variety in the sense of Campana or an ‐special variety, then for any linear representation , the image is virtually nilpotent. We also provide examples showing that this result is sharp, leading to a revised form of Campana's abelianity conjecture for smooth quasi‐projective varieties. In addition, we prove a structure theorem for quasi‐projective varieties with big and semisimple representations of the fundamental groups, thereby addressing a conjecture by Kollár in 1995. We also construct several examples of quasi‐projective varieties that are special and ‐special, highlighting certain atypical properties of the noncompact case in contrast with the projective setting.