Flat connections, Higgs operators, and Einstein metrics on compact Hermitian manifolds

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 K\"ahler 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 K\"ahler manifolds [S1], [S2], [S3]. 1991 Mathematics Subject Classification: 53C07