Connections and Higgs fields on a principal bundle

Let M be a compact connected Kähler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle ε _G over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (ε _G (L(G)), θ _l ) the Higgs L(G)-bundle associated with (ε _G , θ). The Higgs bundle (ε _G , θ) will be called semistable (respectively, stable) if (ε _G (L(G)), θ _l ) is semistable (respectively, stable). A semistable Higgs G-bundle (ε _G , θ) will be called pseudostable if the adjoint vector bundle ad(ε _G (L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.