Non-abelian Hodge Theory

We study the non-abelian Hodge decomposition of Higgs bundles (Chapter 3), which generalises the Hodge decomposition of complex manifolds (Chapter 1). In order to construct this decomposition, we cover the Narasimhan-Seshadri correspondence between representations of fundamental groups and stable holomorphic vector bundles, and the Kobayashi–Hitchin correspondence between stable holomorphic vector bundles and vector bundles with Einstein-Hermitian metrics (Chapter 2). Lastly, we present the isosingularity theorem of Simpson (Chapter 4), which is slightly stronger than the results usually understood as non-abelian Hodge theory. This result was used by Deligne to construct the twistor space of the hyperkähler manifold of Higgs bundles through λ-connections, which smoothly deform Higgs bundles and flat connections. The correspondence has also seen some recent applications, e.g. in Andrea Tirelli’s paper on symplectic singularities of the moduli space of Higgs bundles.