A gentle introduction to the non-abelian Hodge correspondence

Abstract

We aim at giving a pedagogical introduction to the non-abelian Hodge correspondence, a bridge between algebra, geometric structures, and complex geometry. The correspondence links representations of a fundamental group, the character variety, to the theory of holomorphic bundles. We focus on motivations, key ideas, links between the concepts and applications. Among others, we discuss the Riemann–Hilbert correspondence, Goldman’s symplectic structure via the Atiyah–Bott reduction, the Narasimhan–Seshadri theorem, Higgs bundles, harmonic bundles, and hyperkähler manifolds.