Connections and Higgs fields on a principal bundle

Biswas, Indranil ; Gomez, Tomas L. (2008) Connections and Higgs fields on a principal bundle Annals of Global Analysis and Geometry, 33 (1). pp. 19-46. ISSN 0232-704X

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Official URL: http://www.springerlink.com/content/mj82722751xg12...

Related URL: http://dx.doi.org/10.1007/s10455-007-9072-x

Abstract

Let M be a compact connected Kahler manifold and G a connected linear algebraic group defined over . A Higgs field on a holomorphic principal G-bundle εG over M is a holomorphic section θ of ad(εG)⊗Ω1M such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (εG (L(G)), θl ) the Higgs L(G)-bundle associated with (εG , θ). The Higgs bundle (εG , θ) will be called semistable (respectively, stable) if (εG (L(G)), θ l ) is semistable (respectively, stable). A semistable Higgs G-bundle (εG , θ) will be called pseudostable if the adjoint vector bundle ad(εG (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 Kahler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.

Item Type:Article
Source:Copyright of this article belongs to Springer-Verlag.
Keywords:Pseudostable Higgs Bundle; Flat Connection; Kahler Manifold
ID Code:3440
Deposited On:11 Oct 2010 09:47
Last Modified:11 Oct 2010 09:47

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