I know there are indeed some examples show that two line bundles over some projective manifold is topological equivalent but not holomorphic equivalent, but I find I give a "proof" shows that two bundles over a projective manifold are topological equivalent=holomorphic equivalent, but I can't figure out what wrong with my proof.
Suppose our projective manifold is $M$, First for any pair of topological equivalent line bundle $L_1$ and $L_2$, suppose their curvature are $\Omega_1$ and $\Omega_2$, and we know that $\Omega_1=\Omega_2+d\alpha$ for some $alpha\in \Omega^1(M)$.
Secondly we know the Kahler identity: $d\alpha=\partial\bar{\partial}f$ for some $f\in C^{\infty}(M)$.
Now consider complex gauge transformation $g=\exp{\frac{f}{2i}}$, in curvature level we have $g^*\Omega_1=\Omega_1+\partial\bar{\partial}f$, so we find a complex gauge transformation(holomorphic endomorphism) from $L_1$ to $L_2$.
Do I misunderstand something? Thanks you for your answer.