Abstract
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.
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Results of this paper were announced in the conference on “Algebraic Groups and Homogeneous Spaces” held at T.I.F.R. in 2004.
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Biswas, I., Gómez, T.L. Connections and Higgs fields on a principal bundle. Ann Glob Anal Geom 33, 19–46 (2008). https://doi.org/10.1007/s10455-007-9072-x
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DOI: https://doi.org/10.1007/s10455-007-9072-x