# Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

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?

• "As a geometer": you probably mean "as an analyst". Or are you suggesting that algebraic geometry is not a form of geometry? – Mere Scribe Jan 7 at 14:02
• Thank you for this remark! Of course algebraic geometry is a (beautiful) form of geometry. I was thinking more as a differential geometer. Changed that. – Panagiotis Konstantis Jan 7 at 14:20