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Einstein-Hermitian connections on principal bundles and stability.

There is the well known (algebraic) notion of stability for holomorphic vector bundles on a compact Kahler manifold, due to D. Mumford. For a compact Riemann surface, Narasimhan and Seshadri proved that a holomorphic vector bündle is stable if and only if it is associated to an irreducible projective unitary representation of the fundamental group (cf. [10]). As a higher dimensional formulation of this correspondence, Kobayashi proved that a holomorphic vector bündle is quasi-stable (i.e. is a direct sum of stable vector bundles of the same slope) if it admits a hermitian metric such that the curvature of the corresponding hermitian connection is Einstein (i.e. when contracted with the Kahler form it is a constant scalar) and conjectured that every stable vector bündle carries an Einstein-Hermitian metric (cf. [7]). This conjecture has been proved by Donaldson (cf. [3]) in the algebraic case.