The punctured cotangent bundle (excluding the zero section) of the complex projective space of dimension $n$ can be identified with the space:


$(P,Q) | P^2=P, tr(P)=1, PQ+QP = Q, tr(Q) = 0, Q\ne 0$,


Where $P$ and $Q$ are $(n+1)$ dimensional Hermitian matrices.
Please see Furutani and  Tanaka's [article][1](Proposition 2.3.).

Sketch of the proof: The space of the matrices $P$ is isomorphic as a
homogeneous space to $CP^n$, since the action of $U(n+1)$ on it is
transitive and the isotropy group of any point is $U(n)$. The conditions
on the matrix $Q$ are obtained by taking the derivatives of the conditions
on $P$, and the identification of $T^*CP^n$ and $TCP^n$ vuia the
Fubini-Study metric.


  [1]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kjm/1250518882