I'm looking at the cotangent bundle of $CP^{N}$ at the moment in the context of equivariance. For many reasons, it seems to me that this bundle is not $U(1)$-equivariant, or, in other words, cannot be constructed as from a representation of $U(1)$ in the standard manner (see here for example) where we consider $CP^{N}$ as the base space of the principle bundle $S^{2N}$. However, I cannot find a neat convincing argument for why this should be so.

**MY ATTEMPT**: I would guess that $U(1)$-equivariance would mean that the bundle could be expressed as a direct sum of line bundles, but I cannot see how to show that this is not the case (even though it seems most probable). Again I would guess that some sort of Chern argument comes in, but I don't know how to caclulate Chern classes without a connection. The only connection here I know is the Grassmannian, and at this point the whole thing just becomes a mess ....