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

TL;DR

This paper reviews recent progress in non-abelian Hodge theory for non-proper varieties and demonstrates how these developments help establish the linear Shafarevich conjecture for all algebraic varieties.

Contribution

It introduces new methods connecting non-abelian Hodge theory with the linear Shafarevich conjecture, extending results to non-proper algebraic varieties.

Findings

Construction of algebraic Shafarevich morphisms

Proof of the linear Shafarevich conjecture for any algebraic variety

Application of non-abelian Hodge theory to non-proper varieties

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.