Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case (Q1304193)

Summary: A flat complex vector bundle \((E,D)\) on a compact Riemannian manifold \((X,g)\) is stable (resp. polystable) in the sense of \textit{K. Corlette} [J. Differ. Geom. 28, 361-382 (1988; Zbl 0676.58007)] if it has no \(D\)-invariant subbundle (resp. if it is the \(D\)-invariant direct sum of stable subbundles). It has been shown in [loc. cit.] 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 (C. T. Simpson (1988), (1992)).