Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case

  • M. Lübke

Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case cover
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Abstract

A flat complex vector bundle (E,D)(E,D) on a compact Riemannian manifold (X,g)(X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no DD-invariant subbundle (resp. if it is the DD-invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D)(E,D) in this sense is equivalent to the existence of a so-called harmonic metric in EE. In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g)(X,g). We propose new notions of gg-(poly-)stability of such bundles, and of gg-Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if gg is a Kähler metric, but are different in general. Our main result is that the gg-polystability in our sense is equivalent to the existence of a gg-Hermitian-Einstein metric. Our notion of a gg-Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

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M. Lübke, Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case. Doc. Math. 4 (1999), pp. 487–512

DOI 10.4171/DM/66