The Shafarevich conjecture stipulates that the universal covering of a complex projective variety is holomorphically convex. In 2012, Eyssidieux et al. proved this conjecture for smooth projective varieties with fundamental groups being complex linear groups. In this talk, I will explain some recent progress on the extension of their work to encompass singular varieties or cases where fundamental groups are linear in positive characteristic. Additionally, I will explore the applications, notably on the Green-Griffiths-Lang conjecture and a conjecture by Claudon-Höring-Kollár regarding the structure of projective varieties with quasi-projective universal coverings. This talk is based on recent joint works with Yamanoi and, in part, with Cadorel.
