Non-abelian Hodge theory translates between three different structures living on a complex algebraic variety: flat connections (differential equations), Higgs bundles (algebraic objects with a twisted endomorphism), and local systems (representations of the fundamental group). For compact varieties, the correspondence is classical. For non-compact varieties — where the boundary introduces singularities — the theory requires controlling the behavior at infinity.
The paper on non-abelian Hodge theory for non-proper varieties (arXiv: 2603.23467) surveys how extending this correspondence to open varieties enables the construction of algebraic Shafarevich morphisms: maps that detect the fundamental group's linear representations geometrically.
The Shafarevich conjecture, in its linear form, predicts that every variety admits a map to a lower-dimensional target such that the fibers have finite linear image of the fundamental group. The non-abelian Hodge machinery provides the tool: the moduli space of local systems has an algebraic structure (via Higgs bundles) that can be analyzed with algebro-geometric techniques. The Shafarevich morphism factors out the “linear” part of the fundamental group.
The through-claim: transcendental data becomes algebraically tractable through translation. The fundamental group is a topological invariant — hard to work with algebraically. But non-abelian Hodge theory translates its representations into algebraic objects (Higgs bundles), where the construction of the Shafarevich morphism becomes an algebro-geometric problem. The conjecture that topology can be detected geometrically is proven by first converting the topology into algebra, then doing geometry on the algebraic side.
2603.23467. Algebraic geometry / non-abelian Hodge theory / Shafarevich conjecture / fundamental groups / Higgs bundles.