It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is stated in the language of complex/algebraic geometry.

As a differential geometer I am struggling to understand this proof. Does someone know about a construction of $\bar\partial$-operators on complex rank $2$ vector bundles over $\mathbb C\mathbb P^2$ inducing holomorphic structures?