Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case
A flat complex vector bundle on a compact Riemannian manifold is stable (resp. polystable) in the sense of Corlette [C] if it has no -invariant subbundle (resp. if it is the -invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of in this sense is equivalent to the existence of a so-called harmonic metric in . In this paper we consider flat complex vector bundles on compact Hermitian manifolds . We propose new notions of -(poly-)stability of such bundles, and of -Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if is a Kähler metric, but are different in general. Our main result is that the -polystability in our sense is equivalent to the existence of a -Hermitian-Einstein metric. Our notion of a -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. 487–512DOI 10.4171/DM/66