I am reading a paper. Assume on a general (compact) complex manifold, notice that I don't assume the manifold is Kahler or projective or $\partial\bar\partial $ manifold. Suppose that a holomorphic line bundle $L$ on $X$ is pseudoeffctive, it means that one can find a singular metric $h$ on $L$ such that $\sqrt{-1}\Theta_h (L)\ge 0$ in the sense of current. Suppose now the line bundle is topologically trivial, i.e., the first Chern class $c_1(L)=0$, then the author claims that $\sqrt{-1}\Theta_h (L)= 0$ (I think it also in the sense of current).
I can't see why this is right. Notice that in $\partial\bar\partial $ manifold setting, one can always find a connection such that the associated curvature form vanish. But now the manifold is arbitrary, I don't know where to go...
Any help would be appreciated. Thanks a lot!