I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles.

Let me outline a situation that is a bit more concrete, to help focus this query. Suppose that $A$ is a holomorphic line bundle over $\mathbf{P}^m$ and that $B$ is a holomorphic vector bundle of rank $k$ over $\mathbf{P}^{m-k+1}$, where $m,k$ are positive integers. The total space of $A$ and the total space of $B$ are both $m+1$ dimensional. Can there exist a (holomorphic) injection of $A$ into $B$? (Of course, I am not asking about maps of bundles. I am only asking about maps between the total spaces as complex varieties.)

Of course, the answer can be "yes" when $k=1$, but what about for $k>1$?

I have purposefully not specified anything in regards to degrees, Chern classes of the bundles. If these have a bearing on the existence of maps, please feel free to comment on this.

Any information will be greatly appreciated.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.