Yang–Mills bar connection and holomorphic structure

In this note, we study the Yang–Mills bar connection A , i.e., the curvature of A obeys \bar\partial_A^\astF_A^0,2 = 0 , on a principal G -bundle P over a compact complex manifold X . According to the Koszul–Malgrange criterion, any holomorphic structure on P can be seen as a solution to this equation. Suppose that G = SU(2) or SO(3) and X is a complex surface with H^1(X,\mathbbZ_2) = 0 . We then prove that the (0,2) -part curvature of an irreducible Yang–Mills bar connection vanishes, i.e., (P,\bar\partial_A) is holomorphic.