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

Abstract

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