Non-abelian Hodge theory for non-proper varieties and the linear Shafarevich conjecture

Abstract

We survey recent advances in non-abelian Hodge theory in the "mixed" setting of non-proper algebraic varieties. We then describe how these tools are used to construct algebraic Shafarevich morphisms and prove a version of the linear Shafarevich conjecture for any algebraic variety.

24 pages, to appear in the Proceedings of the ICM 2026. Comments welcome!