A remark on "Connections and Higgs fields on a principal bundle"
  We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection. 
Indranil Biswas and Carlos Florentino
math.DG