Return to Question

Is there an example of a complex bundle on $CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We impose of course that the Chern classes of the bundle are $k,k$ Hodge classes (which is automatic for $CP^n$ or Fanos of dimension<4). If by any chance such examples are known, what is the smallest dimension of the variety (or the bundle)?

For $CP^1$ it is elementary to see that all bundles are holomorphic. In the book of Okonek and Schneider it is stated, that all complex bundles on $CP^2$ and $CP^3$ are also holomorphic. But for $CP^n$, $n\ge 4$ this is stated as an open problem (as for 1980).

Is there an example of a complex bundle on $\mathbb CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We require that the Chern classes of the bundle are $(k,k)$ Hodge classes (which is automatic for $\mathbb CP^n$ or Fanos of dimension<4). If by any chance such examples are known, what is the smallest dimension of the variety (or the bundle)?

For $\mathbb CP^1$ it is elementary to see that all bundles are holomorphic. In the book of Okonek and Schneider it is stated, that all complex bundles on $\mathbb CP^2$ and $\mathbb CP^3$ are also holomorphic. But for $\mathbb CP^n$, $n\ge 4$ this is stated as an open problem (as for 1980).