Numerically flat holomorphic bundles over non-Kähler manifolds

Abstract In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold X which admits an Astheno–Kähler metric. We prove that flatness is equivalent to approximate Hermitian and the existence of filtration by sub-bundles whose quotients are flat. This gives affirmative answer question proposed Demailly, Peternell Schneider.