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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.

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