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HERMITIAN–EINSTEIN CONNECTIONS ON PRINCIPAL BUNDLES OVER FLAT AFFINE MANIFOLDS

INDRANIL BISWAS

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

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and

JOHN LOFTIN

Department of Mathematics and Computer Science, Rutgers University at Newark, Newark, NJ 07102, USA

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https://doi.org/10.1142/S0129167X12500395Cited by:2 (Source: Crossref)

Abstract

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle E_G over M admits a Hermitian–Einstein structure if and only if E_G is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

Keywords:

AMSC: 53C07

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