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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?

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    $\begingroup$ "As a geometer": you probably mean "as an analyst". Or are you suggesting that algebraic geometry is not a form of geometry? $\endgroup$ – Mere Scribe Jan 7 at 14:02
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    $\begingroup$ 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. $\endgroup$ – Panagiotis Konstantis Jan 7 at 14:20

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